Automated asteroseismic peak detections

Automated asteroseismic peak detections Abstract Space observatories such as Kepler have provided data that can potentially revolutionize our understanding of stars. Through detailed asteroseismic analyses we are capable of determining fundamental stellar parameters and reveal the stellar internal structure with unprecedented accuracy. However, such detailed analyses, known as peak bagging, have so far been obtained for only a small percentage of the observed stars while most of the scientific potential of the available data remains unexplored. One of the major challenges in peak bagging is identifying how many solar-like oscillation modes are visible in a power density spectrum. Identification of oscillation modes is usually done by visual inspection that is time-consuming and has a degree of subjectivity. Here, we present a peak-detection algorithm especially suited for the detection of solar-like oscillations. It reliably characterizes the solar-like oscillations in a power density spectrum and estimates their parameters without human intervention. Furthermore, we provide a metric to characterize the false positive and false negative rates to provide further information about the reliability of a detected oscillation mode or the significance of a lack of detected oscillation modes. The algorithm presented here opens the possibility for detailed and automated peak bagging of the thousands of solar-like oscillators observed by Kepler. asteroseismology, methods: data analysis, Sun: helioseismology, stars: oscillations 1 INTRODUCTION With the advent of space observatories CoRoT (Baglin et al. 2006) and Kepler (Borucki et al. 2010), there are high-quality, near-uninterrupted and long (>100  d) photometric time series measurements for an unprecedented number of stars available. For solar-like oscillators, the power density spectrum (PDS) of time series data contains rich information that allows for a precise determination of fundamental stellar properties provided that individual oscillation mode parameters are measured accurately (Christensen-Dalsgaard 2004). However, most of the currently existing methods to analyse in detail the PDS require substantial human intervention. For this reason, the scientific potential of the large amount of available data has not been fully exploited. Furthermore, since current methods rely on human input, they carry a considerable degree of subjectivity. Therefore, a method to reliably extract all the relevant information from the measured PDS in an automated way is greatly needed. This issue will be fundamental to take advantage of upcoming missions such as TESS (Ricker et al. 2014) and PLATO (Rauer et al. 2014). It is expected that TESS and PLATO will detect approximately 3 × 105 and 2 × 105 stars with solar-like oscillations (Huber 2018), respectively. This will greatly increase the amount of available data. The PDS of solar-like oscillators have a complex structure stemming from a combination of the granulation background and the stochastically excited global oscillation modes. This makes it challenging to model the PDS accurately. The functional form of the granulation background component has been the topic of several studies (e.g. Harvey 1985; Michel et al. 2009; Kallinger et al. 2014) and it takes the form of a superposition of super-Lorentzian profiles to take into account granulation at different time-scales. Additionally, each individual global stochastic oscillation mode in the power excess is well described by a Lorentzian profile, in case the mode width is larger than the frequency resolution (Kumar, Franklin & Goldreich 1988), or by a sinc2 function otherwise (Christensen-Dalsgaard 2004). One of the main challenges in modelling the stochastic oscillations resides in identifying how many oscillation modes are visible in a given PDS realization. Furthermore, in practice, it is desirable to have at least a crude estimate of all the parameters describing each oscillation mode. For example in a maximum likelihood estimation (MLE), having a set of adequate initial values for all parameters is necessary to avoid local minima (Toutain & Appourchaux 1994). The Bayesian estimations using the Markov chain Monte Carlo (MCMC) methods require an adequate prior probability distribution for each parameter that can be constructed from the initial estimates (Handberg & Campante 2011). These initial estimates can be obtained by visual inspection of the PDS since the oscillation modes produce peaks with recognisable patterns in the PDS. However, this method is not scalable to the large number of stars observed by space missions. Additionally, a visual inspection inevitably introduces a degree of subjectivity tied to the person doing the analysis. This subjectivity can partially be mitigated by assessing the statistical significance of each oscillation mode found in the PDS, which prevents the inclusion of non-significant peaks in the PDS model. However, there might still be significant peaks that escape the visual inspection and are never tested for inclusion. Therefore, a reliable peak-detection method that is free from significant human input would greatly benefit the analysis of large samples of stars. A number of peak-detection algorithms have been developed over the last decades. A common approach is to search for local maxima with a signal-to-noise ratio (SNR) above a certain threshold with the SNR depending only on the peak height (e.g. Appourchaux et al. 2012). A major problem in this approach is that noise can have larger heights than some peaks caused by oscillations; this could produce a large number of either false positives or false negatives depending on the chosen SNR threshold. Furthermore, tests on the peak height would miss significant peaks that are wide but have a small height. Since most peaks in the PDS of solar-like oscillators have a width larger than the frequency resolution, this issue can be partially mitigated by smoothing before attempting a peak detection. However, this approach is very sensitive to the width of the mode (i.e. its lifetime) and the amount of smoothing applied; peaks with different widths are more prominent with different amounts of smoothing. So with this approach, it is impossible to choose a unique best strategy to correctly identify all features of interest. In the current work, we present a different approach based on an extension of the peak-detection algorithm proposed by Du, Kibbe & Lin (2006) in the context of mass spectrometry. This algorithm uses a continuous wavelet transform (CWT)-based pattern-matching algorithm where there is no need for smoothing and most features are correctly identified while keeping the false positive rate low (Cruz-Marcelo et al. 2008). The CWT serves as a pattern-matching function where the signal is compared to a wavelet function specifically chosen to have similar features as the most common peaks that contain signal. The CWT has two parameters, location and scale, that regulate the position on which the matching is being calculated and the width of the feature being matched, respectively. The CWT has a similar effect as a smoothing where the amount of smoothing is variable and controlled by the scale parameter. This approach is similar to searching for narrow features with little smoothing and wide features with a larger amount of smoothing simultaneously. Solar-like oscillators produce peaks in their PDS that are similar to the peaks studied by Du et al. (2006), which makes the CWT-based pattern-matching algorithm adequate for this context. We find that the original formulation of the algorithm is reliable when the peaks are well separated; however, it fails to correctly identify them when the modes have significant overlap, which is a common scenario in the PDS of solar-like oscillators. Furthermore, the original algorithm by Du et al. (2006) only estimates the location of the peak and not its height and width. We propose an extension that is more tolerant to peak overlap and also estimates the height and width of the peaks. To achieve these improvements, we incorporate the additional assumption that the individual peaks are described by Lorentzian functions, as is the case for solar-like oscillations (Anderson, Duvall & Jefferies 1990). Additionally, we require that the PDS contains only information from the stochastically excited oscillation modes, i.e. that the PDS has been normalized to remove the granulation contributions. In Section 2, we describe the algorithm by Du et al. (2006) and the adaptations we made for the case of analysing the PDS of solar-like oscillators. We then characterize the performance of the proposed algorithm by looking at the number false positive (Section 3) and false negative (Section 4) peak detections. In Section 5, we compare our estimation with previous studies of helioseismic data from the Birmingham Solar Oscillations Network (BiSON; Davies et al. 2014; Hale et al. 2016), and in Section 6 we compare our results as obtained from photometric measurements made by the Kepler space observatory with the results from Corsaro, De Ridder & García (2015) for 19 red giants. In each case, we find similar values for the number of peaks present in the background-normalized PDS as well as their location, height, and width albeit with some minor discrepancies. In contrast to previous approaches, the procedure presented here requires no human intervention, which makes it more objective and suitable for the analysis of present and upcoming large data sets.1 To study the internal structure of solar-like oscillators, the frequencies of the oscillations are not sufficient. We also need to characterize each mode by its spherical degree and azimuthal order. An automated procedure to obtain this mode identification will be presented in a forthcoming paper. 2 PEAK-DETECTION METHOD The peak-detection method proposed by Du et al. (2006) is based on using the CWT of a signal as a pattern-matching function. The CWT of a function s(x), denoted as $$\mathcal {W}_\psi [s]$$, is an integral transform that depends on two parameters, (a, b), usually referred to as scale and location, respectively. It is defined as   \begin{equation} \mathcal {W}_\psi [s]\ (a,b) = \frac{1}{\sqrt{a}} \int _{-\infty }^\infty s(x^{\prime })\ \overline{\psi } \left( \frac{x^{\prime }-b}{a} \right) \text{d}x^{\prime }\ , \end{equation} (1)where ψ(x) is a continuous function called the mother wavelet and the overline denotes complex conjugation. Intuitively, the CWT value reflects the pattern matching between the signal s(x΄) and $$\overline{\psi } (\frac{x^{\prime }-b}{a})$$, with larger values representing a better match. The mother wavelet is intentionally chosen to have properties useful for a specific analysis. An appropriate mother wavelet for peak detection is the Mexican-hat wavelet, also known as Ricker wavelet (Ricker 1944), given by   \begin{equation} \psi (x)=\frac{2}{\sqrt{3\sigma }\pi ^{1/4}}\left(1 - \frac{x^2}{\sigma ^2}\right)\text{e}^{-x^2/2\sigma ^2}. \end{equation} (2)It is the second derivative of a Gaussian function with variance σ2 and a normalization factor such that $$\int _\infty ^\infty |\psi (x^{\prime })|^2 \text{d}x^{\prime }=1$$ (see Fig. 1 ). It was shown by Du et al. (2006) that this wavelet is useful for peak finding in spectra since it has the basic features of the most common peaks in a spectrum: approximate symmetry, a major positive peak, and finite width. Figure 1. View largeDownload slide Mexican-hat mother wavelet as defined by equation (2) with σ = 1. Figure 1. View largeDownload slide Mexican-hat mother wavelet as defined by equation (2) with σ = 1. 2.1 Single peak We describe the peak-detection method by Du et al. (2006) by illustrating it on a PDS realization originating from a single stochastically excited global oscillation mode. We assume that there is no contribution to the PDS other than the oscillation mode, i.e. we have a background-normalized PDS. The limit PDS in this case can be described by a Lorentzian profile. As an example, we consider a Lorentzian profile with a central frequency νk, a height of Ik = 10, and a half-width at half-maximum of γk = 0.5, all parameters are given in arbitrary units. The frequency resolution is taken as δν = 0.01 also in arbitrary units. We construct a possible PDS realization by multiplying each frequency bin by a random number drawn from an exponentially decaying probability distribution (see the top panel of Fig. 2). This corresponds to a scaled χ2 distribution with 2 degrees of freedom, which is the probability density function (PDF) of noise in a PDS. This example PDS is similar to the PDS of a single stochastically excited and damped oscillation mode. We then compute the CWT of the PDS using a Mexican-hat wavelet as the mother wavelet with scale values ranging from δν to the frequency range of the PDS (see the middle panel of Fig. 2). Finally, we look at all the local maxima in the CWT values as a function of location (b) for each scale (a). At small scales, the CWT is sensitive to narrow features, which can be seen by the large number of local maxima at small values of a. Towards larger scales, the CWT map becomes smoother and more sensitive to wider features. Figure 2. View largeDownload slide Top panel: simulated PDS (black) realization for a limit PDS (red) described by a Lorentzian profile with a half-width at half-maximum γk = 0.5 and height Ik = 10 (arbitrary units). Middle panel: Mexican-hat CWT of the simulated PDS realization in the top panel as a colour map with values indicated in the side bar. The small dots are at the local maxima across each scale (a) with nearby points of the same colour being identified as belonging to the same ridge. The larger orange dot indicated the maximum CWT value for the longest ridge. Bottom panel: CWT as a function of scale (a) for points in the longest ridge from the middle panel. The horizontal dashed line is placed at an SNR of 1 as defined by Du et al. (2006). The orange dot is the maximum of the ridge that has an SNR close to 14. Figure 2. View largeDownload slide Top panel: simulated PDS (black) realization for a limit PDS (red) described by a Lorentzian profile with a half-width at half-maximum γk = 0.5 and height Ik = 10 (arbitrary units). Middle panel: Mexican-hat CWT of the simulated PDS realization in the top panel as a colour map with values indicated in the side bar. The small dots are at the local maxima across each scale (a) with nearby points of the same colour being identified as belonging to the same ridge. The larger orange dot indicated the maximum CWT value for the longest ridge. Bottom panel: CWT as a function of scale (a) for points in the longest ridge from the middle panel. The horizontal dashed line is placed at an SNR of 1 as defined by Du et al. (2006). The orange dot is the maximum of the ridge that has an SNR close to 14. Du et al. (2006) noted that the local maxima can be connected in ridges. The ridges produced by noise are short and have low absolute CWT values while peaks with a resolved width produce longer ridges with larger CWT values. Thus, peaks are identified by looking at ridges longer than a certain threshold, which have a maximum CWT value larger than a chosen SNR. To take into account the possibility of a different baseline in each peak, Du et al. (2006) proposed an SNR definition that is local for each ridge. It is defined as the maximum CWT value on the ridge divided by the 95 percentile of the absolute value of all the CWT values at the smallest scale in a frequency range close to the ridge location. The optimal frequency range for this local noise definition depends on the problem. Here we used the implementation by Constantine & Percival (2016) in which the range for the local noise is 10 times the frequency resolution (see the bottom panel of Fig. 2). As already mentioned, the algorithm by Du et al. (2006) was proposed in the context of mass spectrometry with no assumption of a particular peak shape other than being positive, approximately symmetric, and with a finite width. For the particular case of finding peaks in the background-normalized PDS of solar-like oscillators, we can assume that resolved oscillation modes can be modelled by Lorentzian profiles. With this assumption, we can find an approximate relationship between the three parameters of the limit PDS Lorentzian (νk, Ik, γk) and the value of the CWT maximum $$\mathcal {W}_{\max }$$, its scale amax , and location bmax . By making 5000 simulations of this single-peak scenario with representative values, in arbitrary units, of Ik ∈ (10, 200) and γk ∈ (5, 30), we derived the following empirical relationships:   \begin{eqnarray} \nu _k \simeq b_{\max } \end{eqnarray} (3)  \begin{eqnarray} \gamma _k \simeq 1.26 + 0.32\ a_{\max } \end{eqnarray} (4)  \begin{eqnarray} I_k \simeq 2.30 \left(\mathcal {W}_{\max }\sqrt{\delta \nu / a_{\max }}\right)^{0.93}\ , \end{eqnarray} (5)where δν is the PDS frequency resolution simulated to be in the range δν ∈ (0.01, 1). Equations (3)–(5) are not relevant for the algorithm as formulated by Du et al. (2006); we will however use them later as a first characterization of the oscillation modes in solar-like oscillators. 2.2 Multiple peaks Unlike the example in Fig. 2 with one mode, solar-like oscillators show several oscillation modes in their PDS. The limit PDS is thus a superposition of several Lorentzian profiles. The peak-detection method by Du et al. (2006) is adequate when the overlap between these Lorentzian profiles is small; however, it is less reliable when the overlap is significant. When the separation between two peaks is large, there are two ridges in the CWT, each one with a large SNR (see the left-hand panel of Fig. 3). As the central frequencies get closer, the SNR of the ridge corresponding to the peak with smaller amplitude decreases considerably until it has an SNR similar to the noise even though it is visible by eye in the PDS realization (see the right-hand panel of Fig. 3). Reducing the SNR threshold for such cases usually produces several false positive identifications. Figure 3. View largeDownload slide Left-hand panels: the upper part shows a limit PDS represented by two Lorentzian profiles with heights I1 = 120 and I2 = 150, half-widths at half-maximum γ1 = 0.5, γ2 = 0.75, and central frequencies ν2 − ν1 = 5 (red) and the simulated PDS realization (black). The bottom part shows a colour map of the CWT of the simulated PDS realizations with the identified ridges as small dots. The bigger orange dots are at the maximum CWT values on each ridge that have an SNR greater than 3 as defined by Du et al. (2006). Right-hand panels: same as before with a reduced distance between the central frequencies to ν2 − ν1 = 3. The black dots are at the maximum CWT value for ridges with SNR greater than or equal to 1.84 and smaller than 3. Figure 3. View largeDownload slide Left-hand panels: the upper part shows a limit PDS represented by two Lorentzian profiles with heights I1 = 120 and I2 = 150, half-widths at half-maximum γ1 = 0.5, γ2 = 0.75, and central frequencies ν2 − ν1 = 5 (red) and the simulated PDS realization (black). The bottom part shows a colour map of the CWT of the simulated PDS realizations with the identified ridges as small dots. The bigger orange dots are at the maximum CWT values on each ridge that have an SNR greater than 3 as defined by Du et al. (2006). Right-hand panels: same as before with a reduced distance between the central frequencies to ν2 − ν1 = 3. The black dots are at the maximum CWT value for ridges with SNR greater than or equal to 1.84 and smaller than 3. To overcome this limitation, we now propose an adjustment to the peak-detection algorithm by Du et al. (2006) for the context of solar-like oscillators. We use the same CWT with the Mexican-hat mother wavelet and use the same approach to find the ridges of local CWT maxima. We modify the SNR definition using our knowledge about the statistical distribution of noise in a PDS realization. In this modified SNR definition, we consider a global instead of a local noise level since a background-normalized PDS has a constant baseline. We simulated a PDS realization of pure white noise, which has a constant limit PDS with value 1, and defined as noise the 95 percentile of the absolute values of all the CWT coefficients at the lowest scale, which we choose as the frequency resolution δν. With this SNR definition, the noise has a value of approximately 2 and thus the SNR is defined as half the maximum CWT value in a ridge. This definition is adopted throughout this work. Furthermore, in contrast to the approach taken by Du et al. (2006) that considers only the global maximum on each ridge, we consider all local maxima since their occurrence frequently indicates a significant peak overlap. By using equations (3)–(5), each one of these local maxima represents a possible Lorentzian profile in the limit PDS (see Fig. 4). Thus, each combination of local maxima in the CWT ridges is considered as a possible PDS model. However, when there are multiple peaks partially overlapping, the CWT values close to the location of a peak are reduced by a negative contribution from neighbouring peaks. This usually results in a displacement of some local maxima in a ridge to lower scales and, subsequently, an underestimation of γk for some peaks when using equations (3)–(5). We correct for this underestimation by an MLE parameter optimization using the values obtained from equations (3)–(5) as initial estimates (see Fig. 5). Finally, to select the most appropriate model for the PDS from all the possible combinations, we use the Akaike information criterion (AIC) that penalizes the likelihood of the model with its complexity to avoid overfitting (Akaike 1998). Since a lower AIC value is indicative of a better model, we select the most appropriate model as the one with the lowest AIC. Figure 4. View largeDownload slide Top panel: CWT of the simulated PDS realization from the right-hand panel of Fig. 3 as a colour map with the identified ridges as small dots. The larger orange dots are at the local CWT maxima of the two longest ridges. The horizontal dashed lines span 4γk for each point as inferred from equation (4). Middle and bottom panels: CWT as a function of scale (a) for the longest and second longest ridges in the CWT map from the top panel, respectively. The orange points denote the local maxima with the labels being the same as in the top panel. Figure 4. View largeDownload slide Top panel: CWT of the simulated PDS realization from the right-hand panel of Fig. 3 as a colour map with the identified ridges as small dots. The larger orange dots are at the local CWT maxima of the two longest ridges. The horizontal dashed lines span 4γk for each point as inferred from equation (4). Middle and bottom panels: CWT as a function of scale (a) for the longest and second longest ridges in the CWT map from the top panel, respectively. The orange points denote the local maxima with the labels being the same as in the top panel. Figure 5. View largeDownload slide Same simulated PDS realization (black) and limit PDS (red) as the right-hand panel of Fig. 3. The vertical dashed lines are at νk ± 2γk as obtained from equations (3)–(5) for point 4 from Fig. 4. In orange a PDS model is shown based on equations (3)–(5) using point 4 (top panel) and points 3 and 6 (bottom panel). In blue is the model obtained after an MLE parameter optimization using the previous values as initial guesses in the range νk ± 2γk. The AIC values of the MLE are calculated in the PDS region delimited by the vertical dashed lines. A lower AIC indicates a better model. Figure 5. View largeDownload slide Same simulated PDS realization (black) and limit PDS (red) as the right-hand panel of Fig. 3. The vertical dashed lines are at νk ± 2γk as obtained from equations (3)–(5) for point 4 from Fig. 4. In orange a PDS model is shown based on equations (3)–(5) using point 4 (top panel) and points 3 and 6 (bottom panel). In blue is the model obtained after an MLE parameter optimization using the previous values as initial guesses in the range νk ± 2γk. The AIC values of the MLE are calculated in the PDS region delimited by the vertical dashed lines. A lower AIC indicates a better model. Since the total number of local maxima in all the ridges is usually of the order of 102–103, depending on the chosen SNR threshold, comparing all combinations of these points as possible PDS models is computationally expensive. We reduce the computational time by considering different segments of the PDS separately. In the first step, we select the local CWT maxima that has the highest scale. According to equations (3)–(5), this corresponds to a peak located at a certain frequency νk and a linewidth γk. We define a PDS region local to this peak as the PDS in the frequency range νk ± 2γk. Finally, we consider all the local maxima in this region and select the best model as described before. The algorithm presented here works optimally for oscillations that have a width larger than the frequency resolution of the PDS. This might not be the case for some of the oscillations. In such circumstances, the peak is best modelled by a sinc2 profile. We find these oscillations by performing the peak detection as described above and looking at the residuals of this fit with a false-alarm probability test for a single frequency bin (Appourchaux et al. 2012). 3 FALSE POSITIVE PEAK DETECTIONS Since noise in the PDS can generate false positive detections, it is useful to assess the probability that a given peak can be generated only by random chance. A significance test can be made by the ratio between the likelihood of the observed PDS assuming a model with the peak included and its likelihood if the peak is omitted from the model. Alternatively, the AIC difference between both models provides a similar and more robust test that penalizes for the number of degrees of freedom in the model. Complementary to these tests, the wavelet-based SNR described previously can also be used as a significance test that is sensitive to different features of the PDS. Since our algorithm uses a Mexican-hat mother wavelet, which has the general shape of the most common peaks, the SNR is particularly suited to assess the significance of solar-like oscillations. The wavelet-based SNR is, on average, a monotonically increasing function of the AIC difference between a PDS model with the peak and without it. However, their relationship is non-linear and has considerable spread so they provide to some extent complementary information. In this section, we describe the statistical properties of the SNR for peaks generated purely by random noise in the PDS and quantify the number of such peaks that we can expect to detect in any given PDS realization. To estimate the chance of false positive peak detections, we generated 105 PDS, each one containing 104 frequency bins with only noise having the same distribution as the noise in a background-normalized PDS, i.e. a scaled χ2 distribution with 2 degrees of freedom. We applied the wavelet-based peak-detection method described here to all the generated PDS and aggregated the results. We found that there is a 10−3 chance per frequency bin in a PDS of having a false positive detection with an SNR greater than 1.1 and an AIC difference greater than 0. For a typical Kepler PDS, this amounts to approximately 30 false positives in the whole PDS. However, the number is reduced when considering only the power excess region (see below). The false-positive detections have different amplitudes and their number depends on the SNR threshold. To quantitatively characterize this dependence, we define Nfp(A, s) to be the number of false-positive peak detections that have an amplitude greater than or equal to A and an SNR greater than or equal to s. Since the number Nfp(A, s) also depends on the number of frequency bins, n, in the PDS, it is more convenient to normalize it by n (see Fig. 6, Nfp/n). Additionally, to take into account the PDS frequency resolution, δν, we calculate Nfp as a function of $$A^{\prime } = A / \sqrt{\delta \nu }$$. By using symbolic regression (Searson 2015), we discovered that Nfp can be described by a function that is symmetric in A΄ and s and has the form   \begin{eqnarray} \log \left(N_\mathrm{fp} \right) &=& c_0 + c_1\,s + c_2 A^{\prime } \nonumber \\ &+& c_3 s^2 + c_4 A^{\prime 2} + c_5\,s A^{\prime } \nonumber\\ &+& c_6 (sA^{\prime })^2 + c_7 (A^{\prime }/s)^2 + c_8 (s/A^{\prime })^2 + \log (n) . \end{eqnarray} (6)We further refined the symbolic regression estimate of the coefficients ci by using a multiple linear regression with the same model. That is, we considered {s, A΄, s2, A΄2, sA΄, (sA΄)2, (A΄/s)2, (s/A΄)2} as linear predictors for log (Nfp/n) and performed a least-squares regression to estimate ci. The obtained coefficient values, their standard error, and the null-hypothesis probability for each term (p-value) are reported in Table 1. Figure 6. View largeDownload slide Expected number of false-positive detections Nfp per number of frequency bins n of peaks having an SNR greater than or equal to s and an amplitude greater than or equal to A. The left-hand panel shows A as the independent variable with s as a colour code while in the right-hand panel the roles are inverted to show the symmetry between A and s. Figure 6. View largeDownload slide Expected number of false-positive detections Nfp per number of frequency bins n of peaks having an SNR greater than or equal to s and an amplitude greater than or equal to A. The left-hand panel shows A as the independent variable with s as a colour code while in the right-hand panel the roles are inverted to show the symmetry between A and s. Table 1. Least-squares linear regression estimate for the parameter values, standard errors, and null-hypothesis probability (p-value) for each parameter in equation (6).   Estimate  Standard error  p-value  c0  −9.63  1.18  2.67× 10−14  c1  3.99  0.75  2.13 × 10−7  c2  2.24  0.36  1.88 × 10−9  c3  −0.21  0.06  1.94 × 10−4  c4  −0.07  0.01  1.49 × 10−6  c5  −1.93  0.19  7.99 × 10−20  c6  0.027  0.003  1.73 × 10−15  c7  −0.14  0.01  4.30 × 10−32  c8  −2.78  0.24  1.71 × 10−25    Estimate  Standard error  p-value  c0  −9.63  1.18  2.67× 10−14  c1  3.99  0.75  2.13 × 10−7  c2  2.24  0.36  1.88 × 10−9  c3  −0.21  0.06  1.94 × 10−4  c4  −0.07  0.01  1.49 × 10−6  c5  −1.93  0.19  7.99 × 10−20  c6  0.027  0.003  1.73 × 10−15  c7  −0.14  0.01  4.30 × 10−32  c8  −2.78  0.24  1.71 × 10−25  View Large To calculate Nfp, we must also provide the number of frequency bins n under consideration. The value of Nfp will change considerably whether we take n as the number of frequency bins in the whole PDS or only in the power excess region. Since we only expect stochastically excited global oscillations in the power excess region, we adopt n as the number of frequency bins in the power excess region. To define this region precisely, we adopt the global description of the power excess as a Gaussian function centred at νmax with a width of the Gaussian σenv and take n as the number of frequency bins in νmax ± 4σenv. This definition is adopted throughout this work. In summary, given a peak amplitude A and SNR s, we can estimate Nfp using equation (6) using $$A^{\prime } = A/\sqrt{\delta \nu }$$. The number Nfp is the expected number of false positives in the power excess region that have an amplitude equal to or greater than A and an SNR equal to or greater than s. A value of Nfp greater than 1 indicates that the peak is more likely to have been generated by noise than from a real signal. Conversely, if Nfp is smaller than 1, it is more probable that the peak originates from a process different than noise. Smaller values of Nfp denote that a peak is less likely to be a false positive detection. 4 DETECTION PROBABILITY Complementary to assessing the significance of a peak detection, we now address the significance of the lack of detections, that is, for a given peak amplitude, we estimate the probability that the algorithm presented here can recognize it among the noise. To estimate this detection probability, we simulated 105 different PDS each one with a Lorentzian peak multiplied by noise as described in Section 2.1. The Lorentzian peaks were generated with parameters in the ranges γk/δν ∈ (2, 10) and I ∈ (1, 10) in arbitrary units. We then applied the algorithm presented here to the PDS using an SNR of 1.1 and attempted to recover the input peak. From these simulations, we estimated the proportion of times P that a peak with linewidth γk/δν was not successfully recovered (see Fig. 7). It can be seen from Fig. 7 that P can be described as a function of γk/δν alone. In this case, it is not necessary to use a model discovery technique, like symbolic regression. Instead, we propose a polynomial model in the form   \begin{equation} P = \sum _{i=0}^N d_i \left( \frac{\gamma _k}{\delta \nu } \right)^i \end{equation} (7)for some coefficients di, with N being the degree of the polynomial. We estimated the coefficients di by a multiple linear regression, similar to Section 3, for increasing values of N and found that the model with the lowest AIC is a third-degree polynomial (N = 3). We adopt this model throughout this work and give in Table 2 the estimated coefficients di, their standard error, and null-hypothesis probability for each term (p-value). Figure 7. View largeDownload slide Probability (P) of not finding a peak of a given linewidth γk where δν is the PDS frequency resolution. The blue line is the fit using equation (7). Figure 7. View largeDownload slide Probability (P) of not finding a peak of a given linewidth γk where δν is the PDS frequency resolution. The blue line is the fit using equation (7). Table 2. Same as Table 1 for equation (7).   Estimate  Standard error  p-value  d0  0.09  <10−4  3.11 × 10−130  d1  −0.49  <10−4  1.34 × 10−107  d2  0.18  <10−4  5.94 × 10−67  d3  −0.06  <10−4  6.71 × 10−26    Estimate  Standard error  p-value  d0  0.09  <10−4  3.11 × 10−130  d1  −0.49  <10−4  1.34 × 10−107  d2  0.18  <10−4  5.94 × 10−67  d3  −0.06  <10−4  6.71 × 10−26  View Large In summary, for a predicted peak with linewidth γk and a PDS frequency resolution δν, we can calculate the probability that the algorithm will not find the peak, P, using equation (7). 5 SOLAR FREQUENCIES OBSERVED BY BISON As a proof of concept for the peak-detection algorithm presented here, we analyse the solar stochastically excited global oscillation modes obtained from radial velocity variations measured by the BiSON. We used the time series from the 1991 January 1 to the 2015 December 31 optimized for fill (Davies et al. 2014; Hale et al. 2016). We calculated the PDS from the discrete Fourier transform of the time series normalized using the spectral window function (Kallinger et al. 2014, , see Fig. 8). Figure 8. View largeDownload slide PDS (grey) of the radial velocity measurements by BiSON with a 0.1 μHz binned version (black). The background function is shown in red with the individual granulation components in blue and the white noise in green. Figure 8. View largeDownload slide PDS (grey) of the radial velocity measurements by BiSON with a 0.1 μHz binned version (black). The background function is shown in red with the individual granulation components in blue and the white noise in green. For the CWT-based peak-detection algorithm, we need to normalize the PDS by an estimation of the granulation background. As an estimation of this background, we used a superposition of three granulation components in the form of super-Lorentzians Ai/[1 + (ν/bi)4] and a white noise $$P_n^{\prime }$$. The oscillation power excess region in this data set is difficult to model accurately so we excluded it from our fit. Specifically, during the parameter estimations, we computed the likelihood of the PDS model only for the region outside the frequency interval ranging from 1200 to 5000 μHz. Thus, we fitted a PDS model containing only the granulation background in the form   \begin{equation} P(\nu ) = P_n^{\prime } + \eta (\nu )^2\left[\sum _{i=1}^3 \frac{A_i}{1+(\nu /b_i)^4} \right], \end{equation} (8)where $$\eta (\nu ) = \mathrm{sinc}(\pi \nu /2\nu _\mathrm{nq})$$ is a frequency-dependent damping, consequence of the measurement discretization process (Chaplin et al. 2011), and νnq is the Nyquist frequency. To estimate the model parameters, we used the affine-invariant MCMC algorithm by Goodman & Weare (2010), as implemented by Foreman-Mackey et al. (2013). We estimated the posterior PDF for each parameter using the chains after convergence was reached. The expectation value for each parameter was estimated as the median of its PDF. Additionally, we estimated the uncertainties by the 16–84 percentiles of each PDF. The obtained parameter estimations with their uncertainties are   \begin{eqnarray*} P_n &=& 2.06_{-0.02}^{+0.02}\times 10^2 (m/s)^2/\mu \mathrm{Hz}, \\ A_1 &=& 6.09_{-0.60}^{+0.77}\times 10^4 (m/s)^2/\mu \mathrm{Hz}, \\ b_1 &=& 1.45_{-0.08}^{+0.08}\times 10^2\,\mu \mathrm{Hz}, \\ A_2 &=& 5.04_{-0.59}^{+0.61}\times 10^3 (m/s)^2/\mu \mathrm{Hz}, \\ b_2 &=& 4.95_{-0.22}^{+0.26}\times 10^2\,\mu \mathrm{Hz}, \\ A_3 &=& 6.06_{-0.38}^{+0.35}\times 10^2 (m/s)^2/\mu \mathrm{Hz}, \\ b_3 &=& 4.60_{-0.10}^{+0.12}\times 10^3\,\mu \mathrm{Hz}. \end{eqnarray*} Even though this granulation background description is too simplistic to accurately describe all phenomena contributing to the BiSON PDS, it is sufficiently accurate for our purpose of finding the most relevant signatures of the stochastically excited global oscillations (see Fig. 8). Having obtained a background-normalized PDS, we proceeded to apply our peak-detection algorithm to it. However, due to the large size of the BiSON data set, it was not computationally feasible to use the peak-detection algorithm on the full PDS. Instead, we analysed segments of the PDS spanning 500 μHz each in steps of 150 μHz so that there is a 350 μHz overlap between each consecutive segment, to avoid possible edge effects in the CWT, and combined the resulting peaks. Fig. 9 shows the frequencies of the identified peaks, νk, as a function of γk (top) and Ik (bottom). The points follow the same frequency-dependent γk as found by Chaplin et al. (1997). The red solid line in the top panel of Fig. 9 is a smoothing cubic spline that shows the overall trend; the dashed red lines are the spline fit multiplied by e±1.5 and delimit a region where most points lie. The outliers from this trend have been coloured in orange in order to distinguish them from the rest. Most outliers from this trend have a relatively small height (see the bottom panel of Fig. 9). We interpret the wide outliers as not being stochastically excited global oscillations but originating as a compensation for an incomplete description of the granulation background. Figure 9. View largeDownload slide Half-width at half-maxima (γk, top) and heights (Ik, bottom) as functions of frequency for the peaks identified in the PDS of the solar radial velocity measurements by BiSON. The solid red line in the top panel is a smoothing cubic spline to the data; the dashed red lines are the spline multiplied by e±1.5. The points outside the region delimited by the dashed lines are coloured orange. The square symbols for the outliers denote peaks with a larger width while the triangular symbols denote a smaller width. Figure 9. View largeDownload slide Half-width at half-maxima (γk, top) and heights (Ik, bottom) as functions of frequency for the peaks identified in the PDS of the solar radial velocity measurements by BiSON. The solid red line in the top panel is a smoothing cubic spline to the data; the dashed red lines are the spline multiplied by e±1.5. The points outside the region delimited by the dashed lines are coloured orange. The square symbols for the outliers denote peaks with a larger width while the triangular symbols denote a smaller width. In Fig. 10, we use an échelle diagram to compare the frequency of the solar-like oscillation modes found with the peak-detection method presented here with the results obtained by Chaplin et al. (1996), Broomhall et al. (2009), and Davies et al. (2014). All oscillations with a frequency higher than ∼ 1400 μHz are correctly identified. The lowest frequency modes are not found with the CWT-based method because their width is small and the CWT-based pattern matching is optimal for peaks with a width that can be resolved. It is also visible that, due to rotational splitting, most oscillation modes with l > 0 are detected as having more than one peak. Fig. 11 shows an example of a rotational split l = 2 mode compared with the reported values by Davies et al. (2014). It should be noted that Davies et al. (2014) fit for each spherical degree, l, a model described by the central frequency νn, l and the rotational splitting δνn, l whereas the method presented here finds each oscillation frequency individually. Figure 10. View largeDownload slide Échelle diagram of the peaks shown in Fig. 9 with the same colouring and symbol scheme. In red are a combination of the oscillations found by Chaplin et al. (1996), Broomhall et al. (2009), and Davies et al. (2014); the symbol identifies the spherical harmonic degree l according to the legend. Figure 10. View largeDownload slide Échelle diagram of the peaks shown in Fig. 9 with the same colouring and symbol scheme. In red are a combination of the oscillations found by Chaplin et al. (1996), Broomhall et al. (2009), and Davies et al. (2014); the symbol identifies the spherical harmonic degree l according to the legend. Figure 11. View largeDownload slide PDS (black) of the radial velocity measurements by BiSON. The dashed red lines are located at the frequencies δn, l + mδνn, l for n = 10 and l = 2 as reported by Davies et al. (2014). The blue dots are located at the frequencies of the peaks identified in this work. Figure 11. View largeDownload slide PDS (black) of the radial velocity measurements by BiSON. The dashed red lines are located at the frequencies δn, l + mδνn, l for n = 10 and l = 2 as reported by Davies et al. (2014). The blue dots are located at the frequencies of the peaks identified in this work. There are also several ridges detected that are not reported in the literature as stochastically excited oscillation modes. The origin of these peaks can be traced back to the window function of BiSON observations, which produces prominent aliases of the oscillation modes. We find more oscillation modes at higher frequencies than reported in the literature. 6 RED GIANTS OBSERVED BY KEPLER The CWT-based peak detection presented here is capable of identifying the solar-like oscillations in the red giant stars observed by Kepler. We use the set of 19 red giant stars studied in detail by Corsaro et al. (2015) to compare our results. The Kepler light curves used in this work have been extracted using the pixel data following the methods described in Mathur et al. (in preparation) and corrected following Garcia et al. (2011).2 We calculated the PDS from the corrected time series using a Lomb–Scargle periodogram (Scargle 1982) normalized using the spectral window function (Kallinger et al. 2014). To obtain a background-normalized PDS, we fit the following model:   \begin{equation} P(\nu ) = P_n^{\prime } + \eta (\nu )^2\left[\sum _{i=1}^3 \frac{A_i}{1+(\nu /b_i)^4} + P_g\exp \left(\frac{\nu -\nu _\mathrm{max}}{\sigma _\mathrm{env}}\right)^2 \right] \!\!\!\!\!\!\! \end{equation} (9)to the observed PDS. To estimate the model parameters, we adopt a Bayesian framework and use the MCMC algorithm of Goodman & Weare (2010) as implemented by Foreman-Mackey et al. (2013) to estimate the posterior density function of each parameter. The expectation value for each parameter is approximated as the median of its posterior density function and the standard errors are approximated as the 16–84 percentiles. An example of the resulting background fit is shown in Fig. 12 for KIC 12008916. The estimated background parameter values and their uncertainties are given in Appendix A. Figure 12. View largeDownload slide Background fit of KIC 12008916. The PDS is shown in grey with a binned version in black. The solid red line is the background without the power excess while the dotted red line is the background fit plus the power excess. The individual granulation components are shown in blue, the power excess in orange, and the white noise in green dashed lines. Figure 12. View largeDownload slide Background fit of KIC 12008916. The PDS is shown in grey with a binned version in black. The solid red line is the background without the power excess while the dotted red line is the background fit plus the power excess. The individual granulation components are shown in blue, the power excess in orange, and the white noise in green dashed lines. We subsequently identified the resolved oscillation modes in the background-normalized PDS using the peak-detection algorithm presented here with an SNR threshold of 1.1 discarding any detections with an AIC smaller than 0 and an Nfp greater than 1. As discussed in Section 2, the CWT is suitable to detect peaks with a width larger than the frequency resolution. However, in some red giant stars, there are gravity-dominated mixed modes visible that can have peaks narrower than the frequency resolution, i.e. they are unresolved. To highlight the unresolved oscillation modes, we make a PDS model with the peaks detected with the CWT and divide the background-normalized PDS by this model. The resulting PDS (see the middle panel of Fig. 13) contains only the unresolved oscillations on top of the noise. The peaks that have a false-alarm probability lower than a certain threshold, which we choose to be 10−4, are deemed significant and fitted with a sinc2 function. Figure 13. View largeDownload slide Top panel: background-normalized PDS for KIC 12008916. The blue circles indicate the location of the oscillation modes identified with the peak-detection algorithm presented here. Middle panel: PDS from the top panel divided by a PDS model containing the peaks identified previously. The blue circles indicate the location of points having a false-alarm probability lower than 10−4. Bottom panel: residual power obtained by dividing the PDS from the top panel by a model containing all the resolved and unresolved oscillations identified with their parameters optimized with an MLE on the whole background-normalized PDS. The dashed horizontal line is placed at a level corresponding to a false-alarm probability of 10−4. Figure 13. View largeDownload slide Top panel: background-normalized PDS for KIC 12008916. The blue circles indicate the location of the oscillation modes identified with the peak-detection algorithm presented here. Middle panel: PDS from the top panel divided by a PDS model containing the peaks identified previously. The blue circles indicate the location of points having a false-alarm probability lower than 10−4. Bottom panel: residual power obtained by dividing the PDS from the top panel by a model containing all the resolved and unresolved oscillations identified with their parameters optimized with an MLE on the whole background-normalized PDS. The dashed horizontal line is placed at a level corresponding to a false-alarm probability of 10−4. To mitigate possible power leakage between the peaks in the PDS, we apply a final MLE to the full background-normalized PDS taking the detections of both resolved and unresolved oscillation modes into account. This provides our final identification of the solar-like oscillations visible in the PDS. We compared our peak identifications with the values reported by Corsaro et al. (2015). Both methods agree on most of the detections, although there are some discrepancies especially when the peaks have low amplitude or significant overlap (see Figs 14 and 15 ). The resolved and unresolved oscillation modes detected for KIC 12008916 are shown in Tables B37 and B38, respectively. We provide the parameters of the detected oscillations for the remaining stars in the sample analysed by Corsaro et al. (2015) in Appendix B. Figure 14. View largeDownload slide Échelle diagram with the frequencies of the solar-like oscillation modes of KIC 12008916 using Δν = 12.9 μHz. Red symbols by Corsaro et al. (2015) with their shape identifying the spherical degree. The blue circles are the frequencies detected by the method presented here. Figure 14. View largeDownload slide Échelle diagram with the frequencies of the solar-like oscillation modes of KIC 12008916 using Δν = 12.9 μHz. Red symbols by Corsaro et al. (2015) with their shape identifying the spherical degree. The blue circles are the frequencies detected by the method presented here. Figure 15. View largeDownload slide Background-normalized PDS (grey) for Kepler target KIC 12008916 with a 0.05 μHz binned version (black) in the region where the solar-like oscillations are located. The vertical red lines are placed at the frequencies of the oscillation modes obtained by Corsaro et al. (2015) with a detection probability greater than 0.9. The blue line is an MLE fit using the heights, locations, and linewidths obtained with the peak-detection results as input parameters to build a PDS model. We also indicate the AIC difference with the null-hypothesis model (red), the SNR (blue), and Nfp (green) as described in Section 3 for each oscillation. The residuals are calculated by dividing the background-normalized PDS by the fitted model. Figure 15. View largeDownload slide Background-normalized PDS (grey) for Kepler target KIC 12008916 with a 0.05 μHz binned version (black) in the region where the solar-like oscillations are located. The vertical red lines are placed at the frequencies of the oscillation modes obtained by Corsaro et al. (2015) with a detection probability greater than 0.9. The blue line is an MLE fit using the heights, locations, and linewidths obtained with the peak-detection results as input parameters to build a PDS model. We also indicate the AIC difference with the null-hypothesis model (red), the SNR (blue), and Nfp (green) as described in Section 3 for each oscillation. The residuals are calculated by dividing the background-normalized PDS by the fitted model. 7 DISCUSSION AND CONCLUSIONS Here we have addressed the problem of finding the significant solar-like oscillations in a background-normalized PDS. We presented a peak finding algorithm capable of finding the oscillation modes in a PDS and provide an estimate of their model parameters provided that the PDS background description is adequate. Furthermore, for each detected mode, we assess its probability of being a false positive detection. Additionally, for a predicted peak, we can estimate the probability of the algorithm being able to detect it. In contrast to other approaches, the algorithm is fast (less than a minute for a typical Kepler long-cadence PDS) and does not require human intervention making it suitable for the analysis of large samples of stars. Since the algorithm does not rely on human intervention, it provides an objective criterion for peak selection in the fitted PDS model. We have shown that the results obtained from the peak-detection algorithm presented here are comparable to previous approaches. The peak-detection algorithm presented here together with an adequate background description, and the spherical degree, and azimuthal order determination for each mode enables the study of the internal structure of the large number of solar-like oscillators that are already observed and yet remain unexplored. ACKNOWLEDGEMENTS The research leading to the presented results has received funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 338251 (StellarAges). This work partially used data analysed under the NASA grant NNX12AE17. The authors also acknowledge the anonymous referee for the helpful and constructive comments. 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The obtained parameter values for the white noise and granulation background are reported in Table A1 and for the power excess in Table A2. The expected value for each parameter is estimated by the median of the posterior density function while the lower and upper uncertainties are estimated as the 18 and 64 percentiles of the posterior density function. Table A1. Estimated PDS fit parameters for the white noise and granulation background in the 19 red giant stars analysed by Corsaro et al. (2015) using equation (9). The estimated parameter value is the median of the posterior distribution. The lower and upper uncertainties for each parameter are the 16 and 84 percentiles of its posterior distribution. KIC  Pn (ppm2/μHz)  A1 (ppm2/μHz)  b1 (μHz)  A2 (ppm2/μHz)  b2 (μHz)  A3 (ppm2/μHz)  b3 (μHz)  3744043  $$8.23_{-0.22}^{+0.20}\times 10^{0}$$  $$1.73_{-0.05}^{+0.06}\times 10^{3}$$  $$2.41_{-0.06}^{+0.06}\times 10^{1}$$  $$3.00_{-0.74}^{+0.27}\times 10^{2}$$  $$9.44_{-0.52}^{+0.24}\times 10^{1}$$  $$3.73_{-2.58}^{+8.47}\times 10^{1}$$  $$1.10_{-0.08}^{+0.18}\times 10^{2}$$  6117517  $$1.20_{-0.03}^{+0.03}\times 10^{1}$$  $$4.03_{-0.56}^{+0.66}\times 10^{3}$$  $$1.77_{-0.22}^{+0.26}\times 10^{0}$$  $$1.67_{-0.05}^{+0.06}\times 10^{3}$$  $$2.94_{-0.07}^{+0.07}\times 10^{1}$$  $$4.01_{-0.14}^{+0.13}\times 10^{2}$$  $$1.04_{-0.01}^{+0.01}\times 10^{2}$$  6144777  $$1.05_{-0.03}^{+0.03}\times 10^{1}$$  $$4.06_{-0.56}^{+0.61}\times 10^{3}$$  $$1.32_{-0.13}^{+0.15}\times 10^{0}$$  $$1.16_{-0.03}^{+0.04}\times 10^{3}$$  $$3.31_{-0.08}^{+0.08}\times 10^{1}$$  $$2.67_{-0.09}^{+0.09}\times 10^{2}$$  $$1.20_{-0.02}^{+0.02}\times 10^{2}$$  7060732  $$3.12_{-0.07}^{+0.07}\times 10^{1}$$  $$8.03_{-0.99}^{+1.12}\times 10^{3}$$  $$1.86_{-0.19}^{+0.23}\times 10^{0}$$  $$1.07_{-0.04}^{+0.04}\times 10^{3}$$  $$3.14_{-0.08}^{+0.09}\times 10^{1}$$  $$2.60_{-0.09}^{+0.10}\times 10^{2}$$  $$1.18_{-0.03}^{+0.03}\times 10^{2}$$  7619745  $$1.78_{-0.10}^{+0.11}\times 10^{1}$$  $$3.60_{-0.44}^{+0.52}\times 10^{3}$$  $$1.23_{-0.10}^{+0.11}\times 10^{0}$$  $$5.46_{-0.13}^{+0.13}\times 10^{2}$$  $$4.31_{-0.08}^{+0.08}\times 10^{1}$$  $$1.14_{-0.04}^{+0.04}\times 10^{2}$$  $$1.70_{-0.06}^{+0.06}\times 10^{2}$$  8366239  $$1.98_{-0.18}^{+0.15}\times 10^{1}$$  $$1.68_{-0.19}^{+0.22}\times 10^{3}$$  $$2.18_{-0.22}^{+0.24}\times 10^{0}$$  $$4.07_{-0.11}^{+0.11}\times 10^{2}$$  $$4.69_{-0.11}^{+0.11}\times 10^{1}$$  $$9.64_{-0.35}^{+0.40}\times 10^{1}$$  $$1.77_{-0.12}^{+0.13}\times 10^{2}$$  8475025  $$2.49_{-0.05}^{+0.05}\times 10^{1}$$  $$1.76_{-0.18}^{+0.22}\times 10^{4}$$  $$1.34_{-0.08}^{+0.08}\times 10^{0}$$  $$1.55_{-0.05}^{+0.05}\times 10^{3}$$  $$2.81_{-0.07}^{+0.07}\times 10^{1}$$  $$3.33_{-0.13}^{+0.13}\times 10^{2}$$  $$1.03_{-0.02}^{+0.02}\times 10^{2}$$  8718745  $$2.10_{-0.06}^{+0.06}\times 10^{1}$$  $$9.19_{-0.97}^{+1.14}\times 10^{4}$$  $$1.08_{-0.06}^{+0.06}\times 10^{0}$$  $$1.57_{-0.05}^{+0.05}\times 10^{3}$$  $$3.03_{-0.07}^{+0.07}\times 10^{1}$$  $$3.05_{-0.10}^{+0.10}\times 10^{2}$$  $$1.19_{-0.02}^{+0.02}\times 10^{2}$$  9145955  $$7.96_{-0.26}^{+0.25}\times 10^{0}$$  $$4.84_{-0.45}^{+0.55}\times 10^{3}$$  $$2.16_{-0.15}^{+0.15}\times 10^{0}$$  $$8.52_{-0.27}^{+0.27}\times 10^{2}$$  $$3.33_{-0.08}^{+0.08}\times 10^{1}$$  $$2.00_{-0.07}^{+0.08}\times 10^{2}$$  $$1.15_{-0.02}^{+0.02}\times 10^{2}$$  9267654  $$2.20_{-0.04}^{+0.04}\times 10^{1}$$  $$5.12_{-0.65}^{+0.74}\times 10^{3}$$  $$1.28_{-0.10}^{+0.12}\times 10^{0}$$  $$9.87_{-0.31}^{+0.31}\times 10^{2}$$  $$2.95_{-0.08}^{+0.07}\times 10^{1}$$  $$2.47_{-0.09}^{+0.09}\times 10^{2}$$  $$1.05_{-0.02}^{+0.02}\times 10^{2}$$  9475697  $$1.50_{-0.03}^{+0.03}\times 10^{1}$$  $$2.00_{-0.07}^{+0.06}\times 10^{3}$$  $$2.60_{-0.08}^{+0.06}\times 10^{1}$$  $$3.04_{-0.72}^{+0.52}\times 10^{2}$$  $$9.61_{-0.97}^{+0.30}\times 10^{1}$$  $$1.05_{-0.65}^{+0.79}\times 10^{2}$$  $$1.08_{-0.05}^{+0.08}\times 10^{2}$$  9882316  $$1.60_{-0.19}^{+0.17}\times 10^{1}$$  $$1.88_{-0.19}^{+0.22}\times 10^{3}$$  $$1.99_{-0.16}^{+0.18}\times 10^{0}$$  $$3.72_{-0.09}^{+0.09}\times 10^{2}$$  $$4.53_{-0.09}^{+0.09}\times 10^{1}$$  $$8.20_{-0.24}^{+0.23}\times 10^{1}$$  $$2.10_{-0.11}^{+0.11}\times 10^{2}$$  10123207  $$2.16_{-0.10}^{+0.10}\times 10^{1}$$  $$9.36_{-1.07}^{+1.24}\times 10^{3}$$  $$1.06_{-0.08}^{+0.08}\times 10^{0}$$  $$8.84_{-0.24}^{+0.23}\times 10^{2}$$  $$3.83_{-0.08}^{+0.08}\times 10^{1}$$  $$1.86_{-0.06}^{+0.06}\times 10^{2}$$  $$1.55_{-0.04}^{+0.04}\times 10^{2}$$  10200377  $$2.61_{-0.07}^{+0.07}\times 10^{1}$$  $$7.52_{-0.89}^{+1.00}\times 10^{3}$$  $$1.37_{-0.12}^{+0.14}\times 10^{0}$$  $$8.77_{-0.26}^{+0.27}\times 10^{2}$$  $$3.54_{-0.09}^{+0.09}\times 10^{1}$$  $$2.25_{-0.10}^{+0.10}\times 10^{2}$$  $$1.17_{-0.03}^{+0.03}\times 10^{2}$$  10257278  $$2.92_{-0.09}^{+0.10}\times 10^{1}$$  $$5.75_{-0.73}^{+0.87}\times 10^{3}$$  $$1.05_{-0.09}^{+0.10}\times 10^{0}$$  $$9.58_{-0.25}^{+0.27}\times 10^{2}$$  $$3.61_{-0.09}^{+0.09}\times 10^{1}$$  $$2.27_{-0.08}^{+0.08}\times 10^{2}$$  $$1.36_{-0.04}^{+0.04}\times 10^{2}$$  11353313  $$2.44_{-0.05}^{+0.05}\times 10^{1}$$  $$7.63_{-0.88}^{+1.04}\times 10^{3}$$  $$1.36_{-0.11}^{+0.12}\times 10^{0}$$  $$9.74_{-0.30}^{+0.31}\times 10^{2}$$  $$3.17_{-0.07}^{+0.08}\times 10^{1}$$  $$2.24_{-0.08}^{+0.08}\times 10^{2}$$  $$1.13_{-0.02}^{+0.02}\times 10^{2}$$  11913545  $$2.39_{-0.05}^{+0.05}\times 10^{1}$$  $$2.05_{-0.06}^{+0.06}\times 10^{3}$$  $$2.63_{-0.06}^{+0.06}\times 10^{1}$$  $$3.51_{-0.86}^{+0.51}\times 10^{2}$$  $$1.04_{-0.03}^{+0.02}\times 10^{2}$$  $$8.88_{-5.05}^{+8.99}\times 10^{1}$$  $$1.14_{-0.05}^{+0.09}\times 10^{2}$$  11968334  $$2.92_{-0.08}^{+0.08}\times 10^{1}$$  $$7.60_{-0.88}^{+0.99}\times 10^{3}$$  $$1.16_{-0.08}^{+0.09}\times 10^{0}$$  $$9.92_{-0.27}^{+0.27}\times 10^{2}$$  $$3.62_{-0.07}^{+0.07}\times 10^{1}$$  $$2.17_{-0.07}^{+0.07}\times 10^{2}$$  $$1.42_{-0.03}^{+0.03}\times 10^{2}$$  12008916  $$2.04_{-0.11}^{+0.11}\times 10^{1}$$  $$1.31_{-0.16}^{+0.18}\times 10^{4}$$  $$1.06_{-0.07}^{+0.07}\times 10^{0}$$  $$7.31_{-0.18}^{+0.19}\times 10^{2}$$  $$4.11_{-0.09}^{+0.08}\times 10^{1}$$  $$1.61_{-0.05}^{+0.05}\times 10^{2}$$  $$1.62_{-0.05}^{+0.05}\times 10^{2}$$  KIC  Pn (ppm2/μHz)  A1 (ppm2/μHz)  b1 (μHz)  A2 (ppm2/μHz)  b2 (μHz)  A3 (ppm2/μHz)  b3 (μHz)  3744043  $$8.23_{-0.22}^{+0.20}\times 10^{0}$$  $$1.73_{-0.05}^{+0.06}\times 10^{3}$$  $$2.41_{-0.06}^{+0.06}\times 10^{1}$$  $$3.00_{-0.74}^{+0.27}\times 10^{2}$$  $$9.44_{-0.52}^{+0.24}\times 10^{1}$$  $$3.73_{-2.58}^{+8.47}\times 10^{1}$$  $$1.10_{-0.08}^{+0.18}\times 10^{2}$$  6117517  $$1.20_{-0.03}^{+0.03}\times 10^{1}$$  $$4.03_{-0.56}^{+0.66}\times 10^{3}$$  $$1.77_{-0.22}^{+0.26}\times 10^{0}$$  $$1.67_{-0.05}^{+0.06}\times 10^{3}$$  $$2.94_{-0.07}^{+0.07}\times 10^{1}$$  $$4.01_{-0.14}^{+0.13}\times 10^{2}$$  $$1.04_{-0.01}^{+0.01}\times 10^{2}$$  6144777  $$1.05_{-0.03}^{+0.03}\times 10^{1}$$  $$4.06_{-0.56}^{+0.61}\times 10^{3}$$  $$1.32_{-0.13}^{+0.15}\times 10^{0}$$  $$1.16_{-0.03}^{+0.04}\times 10^{3}$$  $$3.31_{-0.08}^{+0.08}\times 10^{1}$$  $$2.67_{-0.09}^{+0.09}\times 10^{2}$$  $$1.20_{-0.02}^{+0.02}\times 10^{2}$$  7060732  $$3.12_{-0.07}^{+0.07}\times 10^{1}$$  $$8.03_{-0.99}^{+1.12}\times 10^{3}$$  $$1.86_{-0.19}^{+0.23}\times 10^{0}$$  $$1.07_{-0.04}^{+0.04}\times 10^{3}$$  $$3.14_{-0.08}^{+0.09}\times 10^{1}$$  $$2.60_{-0.09}^{+0.10}\times 10^{2}$$  $$1.18_{-0.03}^{+0.03}\times 10^{2}$$  7619745  $$1.78_{-0.10}^{+0.11}\times 10^{1}$$  $$3.60_{-0.44}^{+0.52}\times 10^{3}$$  $$1.23_{-0.10}^{+0.11}\times 10^{0}$$  $$5.46_{-0.13}^{+0.13}\times 10^{2}$$  $$4.31_{-0.08}^{+0.08}\times 10^{1}$$  $$1.14_{-0.04}^{+0.04}\times 10^{2}$$  $$1.70_{-0.06}^{+0.06}\times 10^{2}$$  8366239  $$1.98_{-0.18}^{+0.15}\times 10^{1}$$  $$1.68_{-0.19}^{+0.22}\times 10^{3}$$  $$2.18_{-0.22}^{+0.24}\times 10^{0}$$  $$4.07_{-0.11}^{+0.11}\times 10^{2}$$  $$4.69_{-0.11}^{+0.11}\times 10^{1}$$  $$9.64_{-0.35}^{+0.40}\times 10^{1}$$  $$1.77_{-0.12}^{+0.13}\times 10^{2}$$  8475025  $$2.49_{-0.05}^{+0.05}\times 10^{1}$$  $$1.76_{-0.18}^{+0.22}\times 10^{4}$$  $$1.34_{-0.08}^{+0.08}\times 10^{0}$$  $$1.55_{-0.05}^{+0.05}\times 10^{3}$$  $$2.81_{-0.07}^{+0.07}\times 10^{1}$$  $$3.33_{-0.13}^{+0.13}\times 10^{2}$$  $$1.03_{-0.02}^{+0.02}\times 10^{2}$$  8718745  $$2.10_{-0.06}^{+0.06}\times 10^{1}$$  $$9.19_{-0.97}^{+1.14}\times 10^{4}$$  $$1.08_{-0.06}^{+0.06}\times 10^{0}$$  $$1.57_{-0.05}^{+0.05}\times 10^{3}$$  $$3.03_{-0.07}^{+0.07}\times 10^{1}$$  $$3.05_{-0.10}^{+0.10}\times 10^{2}$$  $$1.19_{-0.02}^{+0.02}\times 10^{2}$$  9145955  $$7.96_{-0.26}^{+0.25}\times 10^{0}$$  $$4.84_{-0.45}^{+0.55}\times 10^{3}$$  $$2.16_{-0.15}^{+0.15}\times 10^{0}$$  $$8.52_{-0.27}^{+0.27}\times 10^{2}$$  $$3.33_{-0.08}^{+0.08}\times 10^{1}$$  $$2.00_{-0.07}^{+0.08}\times 10^{2}$$  $$1.15_{-0.02}^{+0.02}\times 10^{2}$$  9267654  $$2.20_{-0.04}^{+0.04}\times 10^{1}$$  $$5.12_{-0.65}^{+0.74}\times 10^{3}$$  $$1.28_{-0.10}^{+0.12}\times 10^{0}$$  $$9.87_{-0.31}^{+0.31}\times 10^{2}$$  $$2.95_{-0.08}^{+0.07}\times 10^{1}$$  $$2.47_{-0.09}^{+0.09}\times 10^{2}$$  $$1.05_{-0.02}^{+0.02}\times 10^{2}$$  9475697  $$1.50_{-0.03}^{+0.03}\times 10^{1}$$  $$2.00_{-0.07}^{+0.06}\times 10^{3}$$  $$2.60_{-0.08}^{+0.06}\times 10^{1}$$  $$3.04_{-0.72}^{+0.52}\times 10^{2}$$  $$9.61_{-0.97}^{+0.30}\times 10^{1}$$  $$1.05_{-0.65}^{+0.79}\times 10^{2}$$  $$1.08_{-0.05}^{+0.08}\times 10^{2}$$  9882316  $$1.60_{-0.19}^{+0.17}\times 10^{1}$$  $$1.88_{-0.19}^{+0.22}\times 10^{3}$$  $$1.99_{-0.16}^{+0.18}\times 10^{0}$$  $$3.72_{-0.09}^{+0.09}\times 10^{2}$$  $$4.53_{-0.09}^{+0.09}\times 10^{1}$$  $$8.20_{-0.24}^{+0.23}\times 10^{1}$$  $$2.10_{-0.11}^{+0.11}\times 10^{2}$$  10123207  $$2.16_{-0.10}^{+0.10}\times 10^{1}$$  $$9.36_{-1.07}^{+1.24}\times 10^{3}$$  $$1.06_{-0.08}^{+0.08}\times 10^{0}$$  $$8.84_{-0.24}^{+0.23}\times 10^{2}$$  $$3.83_{-0.08}^{+0.08}\times 10^{1}$$  $$1.86_{-0.06}^{+0.06}\times 10^{2}$$  $$1.55_{-0.04}^{+0.04}\times 10^{2}$$  10200377  $$2.61_{-0.07}^{+0.07}\times 10^{1}$$  $$7.52_{-0.89}^{+1.00}\times 10^{3}$$  $$1.37_{-0.12}^{+0.14}\times 10^{0}$$  $$8.77_{-0.26}^{+0.27}\times 10^{2}$$  $$3.54_{-0.09}^{+0.09}\times 10^{1}$$  $$2.25_{-0.10}^{+0.10}\times 10^{2}$$  $$1.17_{-0.03}^{+0.03}\times 10^{2}$$  10257278  $$2.92_{-0.09}^{+0.10}\times 10^{1}$$  $$5.75_{-0.73}^{+0.87}\times 10^{3}$$  $$1.05_{-0.09}^{+0.10}\times 10^{0}$$  $$9.58_{-0.25}^{+0.27}\times 10^{2}$$  $$3.61_{-0.09}^{+0.09}\times 10^{1}$$  $$2.27_{-0.08}^{+0.08}\times 10^{2}$$  $$1.36_{-0.04}^{+0.04}\times 10^{2}$$  11353313  $$2.44_{-0.05}^{+0.05}\times 10^{1}$$  $$7.63_{-0.88}^{+1.04}\times 10^{3}$$  $$1.36_{-0.11}^{+0.12}\times 10^{0}$$  $$9.74_{-0.30}^{+0.31}\times 10^{2}$$  $$3.17_{-0.07}^{+0.08}\times 10^{1}$$  $$2.24_{-0.08}^{+0.08}\times 10^{2}$$  $$1.13_{-0.02}^{+0.02}\times 10^{2}$$  11913545  $$2.39_{-0.05}^{+0.05}\times 10^{1}$$  $$2.05_{-0.06}^{+0.06}\times 10^{3}$$  $$2.63_{-0.06}^{+0.06}\times 10^{1}$$  $$3.51_{-0.86}^{+0.51}\times 10^{2}$$  $$1.04_{-0.03}^{+0.02}\times 10^{2}$$  $$8.88_{-5.05}^{+8.99}\times 10^{1}$$  $$1.14_{-0.05}^{+0.09}\times 10^{2}$$  11968334  $$2.92_{-0.08}^{+0.08}\times 10^{1}$$  $$7.60_{-0.88}^{+0.99}\times 10^{3}$$  $$1.16_{-0.08}^{+0.09}\times 10^{0}$$  $$9.92_{-0.27}^{+0.27}\times 10^{2}$$  $$3.62_{-0.07}^{+0.07}\times 10^{1}$$  $$2.17_{-0.07}^{+0.07}\times 10^{2}$$  $$1.42_{-0.03}^{+0.03}\times 10^{2}$$  12008916  $$2.04_{-0.11}^{+0.11}\times 10^{1}$$  $$1.31_{-0.16}^{+0.18}\times 10^{4}$$  $$1.06_{-0.07}^{+0.07}\times 10^{0}$$  $$7.31_{-0.18}^{+0.19}\times 10^{2}$$  $$4.11_{-0.09}^{+0.08}\times 10^{1}$$  $$1.61_{-0.05}^{+0.05}\times 10^{2}$$  $$1.62_{-0.05}^{+0.05}\times 10^{2}$$  View Large Table A2. Same as Table A1 for the oscillation power excess parameters. KIC  Pg (ppm2/μHz)  νmax (μHz)  σenv (μHz)  3744043  $$5.28_{-0.14}^{0.15} \times 10^{2}$$  $$1.125_{-0.003}^{0.003} \times 10^{2}$$  $$1.21_{-0.03}^{0.03} \times 10^{1}$$  6117517  $$6.14_{-0.17}^{0.17} \times 10^{2}$$  $$1.203_{-0.003}^{0.003} \times 10^{2}$$  $$1.33_{-0.03}^{0.03} \times 10^{1}$$  6144777  $$5.65_{-0.15}^{0.15} \times 10^{2}$$  $$1.297_{-0.003}^{0.003} \times 10^{2}$$  $$1.27_{-0.03}^{0.03} \times 10^{1}$$  7060732  $$4.23_{-0.13}^{0.13} \times 10^{2}$$  $$1.323_{-0.003}^{0.003} \times 10^{2}$$  $$1.27_{-0.03}^{0.04} \times 10^{1}$$  7619745  $$2.17_{-0.06}^{0.06} \times 10^{2}$$  $$1.707_{-0.004}^{0.004} \times 10^{2}$$  $$1.49_{-0.04}^{0.04} \times 10^{1}$$  8366239  $$1.43_{-0.04}^{0.05} \times 10^{2}$$  $$1.857_{-0.005}^{0.005} \times 10^{2}$$  $$1.69_{-0.07}^{0.07} \times 10^{1}$$  8475025  $$5.81_{-0.17}^{0.18} \times 10^{2}$$  $$1.129_{-0.003}^{0.003} \times 10^{2}$$  $$1.09_{-0.03}^{0.03} \times 10^{1}$$  8718745  $$5.72_{-0.16}^{0.17} \times 10^{2}$$  $$1.296_{-0.003}^{0.003} \times 10^{2}$$  $$1.15_{-0.03}^{0.03} \times 10^{1}$$  9145955  $$2.84_{-0.07}^{0.07} \times 10^{2}$$  $$1.320_{-0.004}^{0.004} \times 10^{2}$$  $$1.50_{-0.03}^{0.04} \times 10^{1}$$  9267654  $$4.65_{-0.13}^{0.14} \times 10^{2}$$  $$1.184_{-0.003}^{0.003} \times 10^{2}$$  $$1.14_{-0.03}^{0.03} \times 10^{1}$$  9475697  $$5.82_{-0.16}^{0.15} \times 10^{2}$$  $$1.150_{-0.003}^{0.003} \times 10^{2}$$  $$1.29_{-0.03}^{0.04} \times 10^{1}$$  9882316  $$1.06_{-0.04}^{0.04} \times 10^{2}$$  $$1.823_{-0.005}^{0.005} \times 10^{2}$$  $$1.52_{-0.07}^{0.07} \times 10^{1}$$  10123207  $$4.46_{-0.12}^{0.12} \times 10^{2}$$  $$1.607_{-0.003}^{0.003} \times 10^{2}$$  $$1.26_{-0.03}^{0.03} \times 10^{1}$$  10200377  $$3.56_{-0.09}^{0.09} \times 10^{2}$$  $$1.433_{-0.003}^{0.003} \times 10^{2}$$  $$1.47_{-0.31}^{0.04} \times 10^{1}$$  10257278  $$4.11_{-0.12}^{0.12} \times 10^{2}$$  $$1.503_{-0.003}^{0.003} \times 10^{2}$$  $$1.24_{-0.04}^{0.04} \times 10^{1}$$  11353313  $$3.75_{-0.10}^{0.11} \times 10^{2}$$  $$1.264_{-0.003}^{0.003} \times 10^{2}$$  $$1.25_{-0.03}^{0.03} \times 10^{1}$$  11913545  $$7.85_{-0.24}^{0.26} \times 10^{2}$$  $$1.173_{-0.003}^{0.003} \times 10^{2}$$  $$1.08_{-0.02}^{0.03} \times 10^{1}$$  11968334  $$4.55_{-0.14}^{0.15} \times 10^{2}$$  $$1.413_{-0.003}^{0.003} \times 10^{2}$$  $$1.10_{-0.03}^{0.03} \times 10^{1}$$  12008916  $$3.02_{-0.08}^{0.08} \times 10^{2}$$  $$1.613_{-0.003}^{0.004} \times 10^{2}$$  $$1.49_{-0.04}^{0.04} \times 10^{1}$$  KIC  Pg (ppm2/μHz)  νmax (μHz)  σenv (μHz)  3744043  $$5.28_{-0.14}^{0.15} \times 10^{2}$$  $$1.125_{-0.003}^{0.003} \times 10^{2}$$  $$1.21_{-0.03}^{0.03} \times 10^{1}$$  6117517  $$6.14_{-0.17}^{0.17} \times 10^{2}$$  $$1.203_{-0.003}^{0.003} \times 10^{2}$$  $$1.33_{-0.03}^{0.03} \times 10^{1}$$  6144777  $$5.65_{-0.15}^{0.15} \times 10^{2}$$  $$1.297_{-0.003}^{0.003} \times 10^{2}$$  $$1.27_{-0.03}^{0.03} \times 10^{1}$$  7060732  $$4.23_{-0.13}^{0.13} \times 10^{2}$$  $$1.323_{-0.003}^{0.003} \times 10^{2}$$  $$1.27_{-0.03}^{0.04} \times 10^{1}$$  7619745  $$2.17_{-0.06}^{0.06} \times 10^{2}$$  $$1.707_{-0.004}^{0.004} \times 10^{2}$$  $$1.49_{-0.04}^{0.04} \times 10^{1}$$  8366239  $$1.43_{-0.04}^{0.05} \times 10^{2}$$  $$1.857_{-0.005}^{0.005} \times 10^{2}$$  $$1.69_{-0.07}^{0.07} \times 10^{1}$$  8475025  $$5.81_{-0.17}^{0.18} \times 10^{2}$$  $$1.129_{-0.003}^{0.003} \times 10^{2}$$  $$1.09_{-0.03}^{0.03} \times 10^{1}$$  8718745  $$5.72_{-0.16}^{0.17} \times 10^{2}$$  $$1.296_{-0.003}^{0.003} \times 10^{2}$$  $$1.15_{-0.03}^{0.03} \times 10^{1}$$  9145955  $$2.84_{-0.07}^{0.07} \times 10^{2}$$  $$1.320_{-0.004}^{0.004} \times 10^{2}$$  $$1.50_{-0.03}^{0.04} \times 10^{1}$$  9267654  $$4.65_{-0.13}^{0.14} \times 10^{2}$$  $$1.184_{-0.003}^{0.003} \times 10^{2}$$  $$1.14_{-0.03}^{0.03} \times 10^{1}$$  9475697  $$5.82_{-0.16}^{0.15} \times 10^{2}$$  $$1.150_{-0.003}^{0.003} \times 10^{2}$$  $$1.29_{-0.03}^{0.04} \times 10^{1}$$  9882316  $$1.06_{-0.04}^{0.04} \times 10^{2}$$  $$1.823_{-0.005}^{0.005} \times 10^{2}$$  $$1.52_{-0.07}^{0.07} \times 10^{1}$$  10123207  $$4.46_{-0.12}^{0.12} \times 10^{2}$$  $$1.607_{-0.003}^{0.003} \times 10^{2}$$  $$1.26_{-0.03}^{0.03} \times 10^{1}$$  10200377  $$3.56_{-0.09}^{0.09} \times 10^{2}$$  $$1.433_{-0.003}^{0.003} \times 10^{2}$$  $$1.47_{-0.31}^{0.04} \times 10^{1}$$  10257278  $$4.11_{-0.12}^{0.12} \times 10^{2}$$  $$1.503_{-0.003}^{0.003} \times 10^{2}$$  $$1.24_{-0.04}^{0.04} \times 10^{1}$$  11353313  $$3.75_{-0.10}^{0.11} \times 10^{2}$$  $$1.264_{-0.003}^{0.003} \times 10^{2}$$  $$1.25_{-0.03}^{0.03} \times 10^{1}$$  11913545  $$7.85_{-0.24}^{0.26} \times 10^{2}$$  $$1.173_{-0.003}^{0.003} \times 10^{2}$$  $$1.08_{-0.02}^{0.03} \times 10^{1}$$  11968334  $$4.55_{-0.14}^{0.15} \times 10^{2}$$  $$1.413_{-0.003}^{0.003} \times 10^{2}$$  $$1.10_{-0.03}^{0.03} \times 10^{1}$$  12008916  $$3.02_{-0.08}^{0.08} \times 10^{2}$$  $$1.613_{-0.003}^{0.004} \times 10^{2}$$  $$1.49_{-0.04}^{0.04} \times 10^{1}$$  View Large APPENDIX B: RESULTS FOR THE OSCILLATION MODES We applied the peak finding procedure to the 19 red giant stars analysed by Corsaro et al. (2015) as described in Section 6. We then used the identified peaks as initial values for a parameter optimization using MLE. When fitting a Lorentzian profile in a PDS, the height and linewidth show high correlations that could hinder the MLE optimization. For this reason, it is more stable to optimize the amplitude instead of the height. That is, during the MLE optimization, each Lorentzian profile is described by  \begin{equation} O_k^\mathrm{resolved} = \frac{A_k^2}{\pi \gamma _k\left[1 + \left(\frac{\nu -\nu _k}{\gamma _k}\right)^2\right]}, \end{equation} (B1)where $$A_k=\sqrt{\pi I_k \gamma _k}$$ is the amplitude of a Lorentzian profile centred at νk, with height Ik and half-width at half-maximum γk. We described the unresolved oscillations in a PDS having frequency resolution of δν as   \begin{equation} O_k^\mathrm{unresolved} = H_k \mathrm{sinc}^2\left(\frac{\nu - \nu _k}{\delta \nu }\right), \end{equation} (B2)where Hk is the height and νk is the location of the oscillation. The unresolved oscillations are detected by identified single points in the PDS with a false-alarm probability lower than 10−4. When a significant oscillation is detected in this way, it is fitted with both equation (B1) and (B2) and the model with the lower AIC is chosen. If model B1 is preferred, it is reported as a resolved oscillation; however, the SNR value is missing in the table. In Tables B1–B38, we report the obtained parameters for the resolved and unresolved oscillations using equations (B1) and (B2), respectively. The uncertainties are derived from the covariance matrix as estimated by the inverse of the Hessian matrix evaluated at the MLE. The reported AIC value corresponds to the difference in AIC between a PDS model having and omitting that oscillation. Figs B1–B19 show a comparison between our method and the method used by Corsaro et al. (2015) for the 19 stars in this sample. Figure B1. View largeDownload slide Same as Fig. 14 for KIC 3744043. Figure B1. View largeDownload slide Same as Fig. 14 for KIC 3744043. Figure B2. View largeDownload slide Same as Fig. 14 for KIC 6117517. Figure B2. View largeDownload slide Same as Fig. 14 for KIC 6117517. Figure B3. View largeDownload slide Same as Fig. 14 for KIC 6144777. Figure B3. View largeDownload slide Same as Fig. 14 for KIC 6144777. Figure B4. View largeDownload slide Same as Fig. 14 for KIC 7060732. Figure B4. View largeDownload slide Same as Fig. 14 for KIC 7060732. Figure B5. View largeDownload slide Same as Fig. 14 for KIC 7619745. Figure B5. View largeDownload slide Same as Fig. 14 for KIC 7619745. Figure B6. View largeDownload slide Same as Fig. 14 for KIC 8366239. Figure B6. View largeDownload slide Same as Fig. 14 for KIC 8366239. Figure B7. View largeDownload slide Same as Fig. 14 for KIC 8475025. Figure B7. View largeDownload slide Same as Fig. 14 for KIC 8475025. Figure B8. View largeDownload slide Same as Fig. 14 for KIC 8718745. Figure B8. View largeDownload slide Same as Fig. 14 for KIC 8718745. Figure B9. View largeDownload slide Same as Fig. 14 for KIC 9145955. Figure B9. View largeDownload slide Same as Fig. 14 for KIC 9145955. Figure B10. View largeDownload slide Same as Fig. 14 for KIC 9267654. Figure B10. View largeDownload slide Same as Fig. 14 for KIC 9267654. Figure B11. View largeDownload slide Same as Fig. 14 for KIC 9475697. Figure B11. View largeDownload slide Same as Fig. 14 for KIC 9475697. Figure B12. View largeDownload slide Same as Fig. 14 for KIC 9882316. Figure B12. View largeDownload slide Same as Fig. 14 for KIC 9882316. Figure B13. View largeDownload slide Same as Fig. 14 for KIC 10123207. Figure B13. View largeDownload slide Same as Fig. 14 for KIC 10123207. Figure B14. View largeDownload slide Same as Fig. 14 for KIC 10200377. Figure B14. View largeDownload slide Same as Fig. 14 for KIC 10200377. Figure B15. View largeDownload slide Same as Fig. 14 for KIC 10257278. Figure B15. View largeDownload slide Same as Fig. 14 for KIC 10257278. Figure B16. View largeDownload slide Same as Fig. 14 for KIC 11353313. Figure B16. View largeDownload slide Same as Fig. 14 for KIC 11353313. Figure B17. View largeDownload slide Same as Fig. 14 for KIC 11913545. Figure B17. View largeDownload slide Same as Fig. 14 for KIC 11913545. Figure B18. View largeDownload slide Same as Fig. 14 for KIC 11968334. Figure B18. View largeDownload slide Same as Fig. 14 for KIC 11968334. Figure B19. View largeDownload slide Same as Fig. 14 repeated here for completeness. Figure B19. View largeDownload slide Same as Fig. 14 repeated here for completeness. Table B1. Resolved oscillations for KIC 3744043. νk  Ak  γk  SNR  AIC  77.346±0.013  0.408±0.117  0.016±0.013  2.23  5.58  80.273±0.016  0.424±0.113  0.02±0.015  2.24  5.35  81.711±0.02  0.888±0.118  0.051±0.018  6.30  59.13  86.586±0.005  0.735±0.174  0.009±0.006  9.09  62.09  89.771±0.036  0.919±0.121  0.107±0.045  3.87  42.59  91.109±0.014  1.323±0.147  0.048±0.015  11.96  178.70  96.065±0.142  1.458±0.182  0.4±0.158  2.08  25.66  96.276±0.192  1.528±0.168  0.4±0.108  2.90  30.72  98.752±0.014  0.426±0.128  0.016±0.014  5.56  3.28  99.406±0.037  1.203±0.201  0.104±0.06  2.52  44.72  99.585±0.011  1.456±0.219  0.027±0.012  6.70  97.67  100.892±0.011  2.43±0.242  0.045±0.012  19.33  782.22  102.424±0.005  0.283±0.071  0.01±NaN  42.01  3.54  102.749±0.009  0.587±0.138  0.015±0.013  1.91  23.72  104.797±0.006  0.538±0.153  0.008±0.006  4.42  19.86  105.311±0.008  0.827±0.168  0.016±0.011  5.56  43.82  105.544±0.008  1.185±0.208  0.017±0.008  8.23  104.40  105.743±0.004  0.802±0.267  0.006±0.006  16.51  31.28  106.11±0.004  1.438±0.428  0.005±0.004    51.76  106.207±0.067  1.531±0.186  0.249±0.112    40.17  106.492±0.014  0.537±0.171  0.013±0.01  5.43  1.67  109.437±0.015  3.392±0.254  0.081±0.016  59.91  1315.23  110.746±0.013  3.926±0.311  0.068±0.014  88.56  1936.95  112.688±0.013  0.999±0.169  0.035±0.029  5.73  77.42  114.824±0.008  1.174±0.195  0.02±0.01  15.76  102.34  115.617±0.016  3.089±0.224  0.092±0.019  45.39  1071.80  116.228±0.058  1.017±0.237  0.094±0.082    2.37  116.237±0.003  1.956±0.608  0.005±0.004    107.30  116.448±0.01  0.55±0.171  0.011±0.01  4.96  4.80  117.112±0.005  0.757±0.208  0.007±0.005  10.38  51.62  117.376±0.007  0.489±0.152  0.006±0.005  5.08  15.39  119.324±0.018  2.807±0.195  0.105±0.021  36.71  846.37  120.609±0.011  3.423±0.309  0.053±0.012  76.99  1464.76  122.528±0.052  1.184±0.122  0.175±0.051  5.35  65.47  125.524±0.015  2.139±0.188  0.068±0.017  26.29  470.58  126.266±0.02  1.623±0.152  0.081±0.024  12.72  204.09  126.997±0.011  0.517±0.133  0.015±0.011  3.65  10.76  127.292±0.009  0.744±0.151  0.015±0.007  7.92  43.37  129.403±0.022  2.303±0.163  0.12±0.025  22.12  487.64  130.681±0.031  2.164±0.141  0.177±0.034  15.18  380.50  132.825±0.022  0.586±0.119  0.038±0.022  2.92  13.20  134.588±0.019  0.585±0.127  0.034±0.023  2.71  12.91  134.786±0.004  0.449±0.169  0.005±0.007  4.06  12.18  135.793±0.03  1.626±0.13  0.145±0.036  9.96  195.09  136.802±0.052  1.155±0.126  0.173±0.054  4.12  56.87  139.227±0.015  0.625±0.132  0.028±0.019  3.32  17.78  139.613±0.018  0.848±0.143  0.045±0.029  5.08  39.21  140.808±0.092  1.452±0.126  0.398±0.103  3.48  89.35  143.357±0.023  0.43±0.129  0.033±0.031  1.50  2.38  146.199±0.043  1.075±0.116  0.143±0.044  5.20  60.98  147.289±0.027  0.713±0.117  0.057±0.025  3.67  23.13  149.464±0.253  0.972±0.182  0.4±0.197  1.23  17.38  150.781±0.015  0.351±0.158  0.018±0.034  1.62  0.13  156.702±0.106  0.619±0.157  0.164±0.13  1.54  4.44  νk  Ak  γk  SNR  AIC  77.346±0.013  0.408±0.117  0.016±0.013  2.23  5.58  80.273±0.016  0.424±0.113  0.02±0.015  2.24  5.35  81.711±0.02  0.888±0.118  0.051±0.018  6.30  59.13  86.586±0.005  0.735±0.174  0.009±0.006  9.09  62.09  89.771±0.036  0.919±0.121  0.107±0.045  3.87  42.59  91.109±0.014  1.323±0.147  0.048±0.015  11.96  178.70  96.065±0.142  1.458±0.182  0.4±0.158  2.08  25.66  96.276±0.192  1.528±0.168  0.4±0.108  2.90  30.72  98.752±0.014  0.426±0.128  0.016±0.014  5.56  3.28  99.406±0.037  1.203±0.201  0.104±0.06  2.52  44.72  99.585±0.011  1.456±0.219  0.027±0.012  6.70  97.67  100.892±0.011  2.43±0.242  0.045±0.012  19.33  782.22  102.424±0.005  0.283±0.071  0.01±NaN  42.01  3.54  102.749±0.009  0.587±0.138  0.015±0.013  1.91  23.72  104.797±0.006  0.538±0.153  0.008±0.006  4.42  19.86  105.311±0.008  0.827±0.168  0.016±0.011  5.56  43.82  105.544±0.008  1.185±0.208  0.017±0.008  8.23  104.40  105.743±0.004  0.802±0.267  0.006±0.006  16.51  31.28  106.11±0.004  1.438±0.428  0.005±0.004    51.76  106.207±0.067  1.531±0.186  0.249±0.112    40.17  106.492±0.014  0.537±0.171  0.013±0.01  5.43  1.67  109.437±0.015  3.392±0.254  0.081±0.016  59.91  1315.23  110.746±0.013  3.926±0.311  0.068±0.014  88.56  1936.95  112.688±0.013  0.999±0.169  0.035±0.029  5.73  77.42  114.824±0.008  1.174±0.195  0.02±0.01  15.76  102.34  115.617±0.016  3.089±0.224  0.092±0.019  45.39  1071.80  116.228±0.058  1.017±0.237  0.094±0.082    2.37  116.237±0.003  1.956±0.608  0.005±0.004    107.30  116.448±0.01  0.55±0.171  0.011±0.01  4.96  4.80  117.112±0.005  0.757±0.208  0.007±0.005  10.38  51.62  117.376±0.007  0.489±0.152  0.006±0.005  5.08  15.39  119.324±0.018  2.807±0.195  0.105±0.021  36.71  846.37  120.609±0.011  3.423±0.309  0.053±0.012  76.99  1464.76  122.528±0.052  1.184±0.122  0.175±0.051  5.35  65.47  125.524±0.015  2.139±0.188  0.068±0.017  26.29  470.58  126.266±0.02  1.623±0.152  0.081±0.024  12.72  204.09  126.997±0.011  0.517±0.133  0.015±0.011  3.65  10.76  127.292±0.009  0.744±0.151  0.015±0.007  7.92  43.37  129.403±0.022  2.303±0.163  0.12±0.025  22.12  487.64  130.681±0.031  2.164±0.141  0.177±0.034  15.18  380.50  132.825±0.022  0.586±0.119  0.038±0.022  2.92  13.20  134.588±0.019  0.585±0.127  0.034±0.023  2.71  12.91  134.786±0.004  0.449±0.169  0.005±0.007  4.06  12.18  135.793±0.03  1.626±0.13  0.145±0.036  9.96  195.09  136.802±0.052  1.155±0.126  0.173±0.054  4.12  56.87  139.227±0.015  0.625±0.132  0.028±0.019  3.32  17.78  139.613±0.018  0.848±0.143  0.045±0.029  5.08  39.21  140.808±0.092  1.452±0.126  0.398±0.103  3.48  89.35  143.357±0.023  0.43±0.129  0.033±0.031  1.50  2.38  146.199±0.043  1.075±0.116  0.143±0.044  5.20  60.98  147.289±0.027  0.713±0.117  0.057±0.025  3.67  23.13  149.464±0.253  0.972±0.182  0.4±0.197  1.23  17.38  150.781±0.015  0.351±0.158  0.018±0.034  1.62  0.13  156.702±0.106  0.619±0.157  0.164±0.13  1.54  4.44  View Large Table B2. Unresolved oscillations for KIC 3744043. νk  Hk  AIC  105.89±0.002  32.931±31.542  9.79  105.951±0.002  69.363±51.538  17.41  106.737±0.002  63.729±46.161  36.35  107.529±0.002  84.342±43.169  82.07  108.382±0.002  37.01±23.872  23.61  113.904±0.001  50.605±40.142  63.95  116.848±0.002  84.986±46.947  54.30  118.102±0.001  31.681±21.444  41.29  124.615±0.003  49.973±56.838  0.73  124.632±0.002  172.94±93.903  133.71  128.467±0.002  52.997±34.592  53.68  νk  Hk  AIC  105.89±0.002  32.931±31.542  9.79  105.951±0.002  69.363±51.538  17.41  106.737±0.002  63.729±46.161  36.35  107.529±0.002  84.342±43.169  82.07  108.382±0.002  37.01±23.872  23.61  113.904±0.001  50.605±40.142  63.95  116.848±0.002  84.986±46.947  54.30  118.102±0.001  31.681±21.444  41.29  124.615±0.003  49.973±56.838  0.73  124.632±0.002  172.94±93.903  133.71  128.467±0.002  52.997±34.592  53.68  View Large Table B3. Resolved oscillations for KIC 6117517. νk  Ak  γk  SNR  AIC  67.114±0.018  0.423±0.148  0.023±0.034  1.52  4.72  68.496±0.038  0.466±0.123  0.053±0.04  1.65  2.85  89.088±0.01  0.718±0.132  0.021±0.011  5.64  44.86  92.379±0.005  0.642±0.18  0.007±0.007  4.85  44.95  93.71±0.038  1.073±0.115  0.124±0.039  5.36  69.58  98.595±0.006  0.929±0.184  0.013±0.007  12.17  103.93  99.073±0.005  0.674±0.174  0.008±0.006  6.96  47.59  99.733±0.004  0.39±0.139  0.005±0.006  3.74  15.38  102.106±0.019  1.342±0.134  0.076±0.025  9.13  162.09  103.439±0.012  1.845±0.191  0.045±0.013  23.81  428.76  105.436±0.011  0.598±0.123  0.019±0.011  4.42  25.05  108.522±0.007  1.747±0.271  0.018±0.007    437.74  111.945±0.008  0.712±0.168  0.012±0.008  7.28  28.29  112.303±0.012  2.027±0.207  0.047±0.013  26.24  467.97  112.587±0.009  0.766±0.193  0.013±0.012  6.69  24.81  113.609±0.007  4.115±0.552  0.023±0.007  170.47  2740.04  115.631±0.016  1.344±0.143  0.053±0.015  12.53  176.63  118.462±0.01  2.416±0.252  0.042±0.013    716.29  122.202±0.028  1.265±0.176  0.081±0.036  8.98  46.52  122.447±0.011  2.094±0.241  0.038±0.012  31.57  286.86  123.638±0.012  3.807±0.304  0.067±0.014  92.30  2062.10  128.819±0.01  2.479±0.3  0.032±0.01  30.28  350.33  128.819±0.212  1.276±0.237  0.4±0.436    7.46  132.603±0.021  2.167±0.166  0.108±0.026  19.06  411.98  133.799±0.018  2.805±0.199  0.098±0.019  36.20  891.32  136.039±0.018  0.869±0.124  0.046±0.019  6.11  51.31  138.681±0.015  1.37±0.152  0.049±0.016  12.82  174.16  139.492±0.017  1.681±0.155  0.067±0.017  16.75  286.89  140.623±0.011  0.725±0.136  0.023±0.013  5.40  38.15  142.105±0.015  0.425±0.128  0.015±0.012  2.95  3.30  142.284±0.01  0.503±0.141  0.013±0.01  3.81  8.00  142.917±0.032  1.534±0.142  0.132±0.039  2.84  148.52  144.119±0.053  1.86±0.131  0.276±0.058  6.88  244.22  148.335±0.007  0.74±0.162  0.012±0.008  7.51  50.23  149.425±0.043  1.357±0.118  0.169±0.041  6.50  113.10  150.532±0.044  0.985±0.121  0.116±0.041  4.69  44.61  152.051±0.006  0.327±0.36  0.005±0.02  1.77  4.66  153.581±0.326  1.423±0.216  0.4±0.231  2.78  74.05  159.132±0.027  0.576±0.118  0.045±0.025  3.15  10.95  160.169±0.056  1.144±0.116  0.192±0.054  4.51  64.30  161.566±0.018  0.515±0.122  0.028±0.021  2.43  9.73  163.604±0.102  1.002±0.131  0.306±0.115  2.65  28.46  170.876±0.064  0.721±0.13  0.127±0.071  2.49  15.07  νk  Ak  γk  SNR  AIC  67.114±0.018  0.423±0.148  0.023±0.034  1.52  4.72  68.496±0.038  0.466±0.123  0.053±0.04  1.65  2.85  89.088±0.01  0.718±0.132  0.021±0.011  5.64  44.86  92.379±0.005  0.642±0.18  0.007±0.007  4.85  44.95  93.71±0.038  1.073±0.115  0.124±0.039  5.36  69.58  98.595±0.006  0.929±0.184  0.013±0.007  12.17  103.93  99.073±0.005  0.674±0.174  0.008±0.006  6.96  47.59  99.733±0.004  0.39±0.139  0.005±0.006  3.74  15.38  102.106±0.019  1.342±0.134  0.076±0.025  9.13  162.09  103.439±0.012  1.845±0.191  0.045±0.013  23.81  428.76  105.436±0.011  0.598±0.123  0.019±0.011  4.42  25.05  108.522±0.007  1.747±0.271  0.018±0.007    437.74  111.945±0.008  0.712±0.168  0.012±0.008  7.28  28.29  112.303±0.012  2.027±0.207  0.047±0.013  26.24  467.97  112.587±0.009  0.766±0.193  0.013±0.012  6.69  24.81  113.609±0.007  4.115±0.552  0.023±0.007  170.47  2740.04  115.631±0.016  1.344±0.143  0.053±0.015  12.53  176.63  118.462±0.01  2.416±0.252  0.042±0.013    716.29  122.202±0.028  1.265±0.176  0.081±0.036  8.98  46.52  122.447±0.011  2.094±0.241  0.038±0.012  31.57  286.86  123.638±0.012  3.807±0.304  0.067±0.014  92.30  2062.10  128.819±0.01  2.479±0.3  0.032±0.01  30.28  350.33  128.819±0.212  1.276±0.237  0.4±0.436    7.46  132.603±0.021  2.167±0.166  0.108±0.026  19.06  411.98  133.799±0.018  2.805±0.199  0.098±0.019  36.20  891.32  136.039±0.018  0.869±0.124  0.046±0.019  6.11  51.31  138.681±0.015  1.37±0.152  0.049±0.016  12.82  174.16  139.492±0.017  1.681±0.155  0.067±0.017  16.75  286.89  140.623±0.011  0.725±0.136  0.023±0.013  5.40  38.15  142.105±0.015  0.425±0.128  0.015±0.012  2.95  3.30  142.284±0.01  0.503±0.141  0.013±0.01  3.81  8.00  142.917±0.032  1.534±0.142  0.132±0.039  2.84  148.52  144.119±0.053  1.86±0.131  0.276±0.058  6.88  244.22  148.335±0.007  0.74±0.162  0.012±0.008  7.51  50.23  149.425±0.043  1.357±0.118  0.169±0.041  6.50  113.10  150.532±0.044  0.985±0.121  0.116±0.041  4.69  44.61  152.051±0.006  0.327±0.36  0.005±0.02  1.77  4.66  153.581±0.326  1.423±0.216  0.4±0.231  2.78  74.05  159.132±0.027  0.576±0.118  0.045±0.025  3.15  10.95  160.169±0.056  1.144±0.116  0.192±0.054  4.51  64.30  161.566±0.018  0.515±0.122  0.028±0.021  2.43  9.73  163.604±0.102  1.002±0.131  0.306±0.115  2.65  28.46  170.876±0.064  0.721±0.13  0.127±0.071  2.49  15.07  View Large Table B4. Unresolved oscillations for KIC 6117517. νk  Hk  AIC  98.05±0.002  117.084±59.233  101.87  107.811±0.002  97.1±51.656  116.98  109.066±0.001  726.76±234.549  398.68  116.561±0.002  31.435±28.226  46.70  117.565±0.001  123.591±84.557  248.40  119.089±0.001  4803.886±1292.46  990.16  119.93±0.001  353.564±164.366  346.94  125.57±0.001  44.673±45.806  65.45  126.763±0.001  73.775±52.275  95.38  127.926±0.001  334.899±146.307  427.70  129.553±0.001  1054.725±327.149  238.62  130.716±0.001  164.473±74.261  172.81  132.286±0.002  81.004±61.568  37.35  133.114±0.002  39.786±38.407  22.61  137.434±0.001  93.987±54.975  120.10  νk  Hk  AIC  98.05±0.002  117.084±59.233  101.87  107.811±0.002  97.1±51.656  116.98  109.066±0.001  726.76±234.549  398.68  116.561±0.002  31.435±28.226  46.70  117.565±0.001  123.591±84.557  248.40  119.089±0.001  4803.886±1292.46  990.16  119.93±0.001  353.564±164.366  346.94  125.57±0.001  44.673±45.806  65.45  126.763±0.001  73.775±52.275  95.38  127.926±0.001  334.899±146.307  427.70  129.553±0.001  1054.725±327.149  238.62  130.716±0.001  164.473±74.261  172.81  132.286±0.002  81.004±61.568  37.35  133.114±0.002  39.786±38.407  22.61  137.434±0.001  93.987±54.975  120.10  View Large Table B5. Resolved oscillations for KIC 6144777. νk  Ak  γk  SNR  AIC  86.174±0.046  0.587±0.159  0.09±0.093  1.94  7.44  96.907±0.042  0.756±0.174  0.116±0.112  2.26  19.81  97.445±0.005  0.314±0.141  0.005±0.011  1.65  1.50  100.483±0.032  0.55±0.118  0.054±0.033  1.92  8.64  101.924±0.014  1.048±0.136  0.042±0.018  8.07  102.77  107.172±0.067  1.132±0.145  0.214±0.081  1.53  45.56  107.612±0.02  1.036±0.144  0.056±0.025  6.15  52.97  107.91±0.011  0.603±0.143  0.017±0.012  4.39  15.87  111.153±0.028  1.547±0.127  0.123±0.03  10.46  196.01  112.615±0.014  1.986±0.179  0.063±0.016  23.50  463.93  117.969±0.011  1.061±0.184  0.024±0.013  9.96  47.57  118.137±0.011  1.617±0.232  0.029±0.013  18.86  142.17  118.285±0.012  1.18±0.197  0.028±0.016  12.04  52.85  118.815±0.004  0.649±0.208  0.005±0.006  4.94  39.34  119.726±0.059  1.103±0.137  0.211±0.084  2.83  46.86  122.216±0.017  2.774±0.197  0.098±0.02  37.63  899.51  123.628±0.007  4.25±0.516  0.027±0.008  160.61  2701.71  125.331±0.006  0.659±0.164  0.009±0.006  7.05  36.59  125.83±0.011  1.42±0.173  0.035±0.012  16.43  223.09  128.871±0.011  1.703±0.223  0.034±0.017  29.02  200.76  129.035±0.007  1.586±0.3  0.015±0.008  29.20  96.59  129.17±0.009  1.438±0.238  0.02±0.009  17.22  109.46  129.768±0.004  1.73±0.385  0.007±0.004  37.64  293.47  131.968±0.006  0.623±0.167  0.007±0.005  7.45  31.43  133.179±0.012  3.737±0.311  0.062±0.013  82.74  1872.07  134.515±0.009  3.849±0.391  0.04±0.01  116.97  2015.89  136.774±0.024  1.091±0.126  0.077±0.027  6.64  77.91  139.183±0.032  1.632±0.135  0.144±0.036  9.48  162.19  140.225±0.018  2.701±0.196  0.096±0.019  34.41  745.62  141.307±0.037  1.361±0.134  0.112±0.028  9.23  98.89  144.243±0.028  2.32±0.151  0.167±0.034  18.45  429.31  145.574±0.021  2.921±0.187  0.126±0.023  34.80  880.38  147.979±0.022  0.772±0.126  0.052±0.027  3.93  31.44  149.441±0.012  0.552±0.136  0.016±0.015  3.74  18.05  150.603±0.021  1.052±0.14  0.056±0.023  5.92  68.21  150.93±0.028  0.789±0.145  0.059±0.033  2.77  19.62  151.712±0.023  1.797±0.139  0.112±0.025  14.07  298.66  153.082±0.003  0.554±0.191  0.005±0.007  3.60  33.31  154.858±0.005  0.383±0.167  0.005±0.012  1.97  4.23  155.544±0.027  1.614±0.132  0.13±0.034  10.20  202.76  156.846±0.035  1.566±0.126  0.17±0.043  8.19  169.60  162.159±0.041  1.329±0.117  0.17±0.043  6.40  110.27  163.337±0.023  0.986±0.122  0.067±0.024  5.93  63.46  167.021±0.145  1.075±0.162  0.325±0.18  1.70  28.86  168.136±0.23  1.026±0.158  0.397±0.208  1.91  20.36  173.367±0.038  0.689±0.186  0.098±0.117  2.28  15.20  νk  Ak  γk  SNR  AIC  86.174±0.046  0.587±0.159  0.09±0.093  1.94  7.44  96.907±0.042  0.756±0.174  0.116±0.112  2.26  19.81  97.445±0.005  0.314±0.141  0.005±0.011  1.65  1.50  100.483±0.032  0.55±0.118  0.054±0.033  1.92  8.64  101.924±0.014  1.048±0.136  0.042±0.018  8.07  102.77  107.172±0.067  1.132±0.145  0.214±0.081  1.53  45.56  107.612±0.02  1.036±0.144  0.056±0.025  6.15  52.97  107.91±0.011  0.603±0.143  0.017±0.012  4.39  15.87  111.153±0.028  1.547±0.127  0.123±0.03  10.46  196.01  112.615±0.014  1.986±0.179  0.063±0.016  23.50  463.93  117.969±0.011  1.061±0.184  0.024±0.013  9.96  47.57  118.137±0.011  1.617±0.232  0.029±0.013  18.86  142.17  118.285±0.012  1.18±0.197  0.028±0.016  12.04  52.85  118.815±0.004  0.649±0.208  0.005±0.006  4.94  39.34  119.726±0.059  1.103±0.137  0.211±0.084  2.83  46.86  122.216±0.017  2.774±0.197  0.098±0.02  37.63  899.51  123.628±0.007  4.25±0.516  0.027±0.008  160.61  2701.71  125.331±0.006  0.659±0.164  0.009±0.006  7.05  36.59  125.83±0.011  1.42±0.173  0.035±0.012  16.43  223.09  128.871±0.011  1.703±0.223  0.034±0.017  29.02  200.76  129.035±0.007  1.586±0.3  0.015±0.008  29.20  96.59  129.17±0.009  1.438±0.238  0.02±0.009  17.22  109.46  129.768±0.004  1.73±0.385  0.007±0.004  37.64  293.47  131.968±0.006  0.623±0.167  0.007±0.005  7.45  31.43  133.179±0.012  3.737±0.311  0.062±0.013  82.74  1872.07  134.515±0.009  3.849±0.391  0.04±0.01  116.97  2015.89  136.774±0.024  1.091±0.126  0.077±0.027  6.64  77.91  139.183±0.032  1.632±0.135  0.144±0.036  9.48  162.19  140.225±0.018  2.701±0.196  0.096±0.019  34.41  745.62  141.307±0.037  1.361±0.134  0.112±0.028  9.23  98.89  144.243±0.028  2.32±0.151  0.167±0.034  18.45  429.31  145.574±0.021  2.921±0.187  0.126±0.023  34.80  880.38  147.979±0.022  0.772±0.126  0.052±0.027  3.93  31.44  149.441±0.012  0.552±0.136  0.016±0.015  3.74  18.05  150.603±0.021  1.052±0.14  0.056±0.023  5.92  68.21  150.93±0.028  0.789±0.145  0.059±0.033  2.77  19.62  151.712±0.023  1.797±0.139  0.112±0.025  14.07  298.66  153.082±0.003  0.554±0.191  0.005±0.007  3.60  33.31  154.858±0.005  0.383±0.167  0.005±0.012  1.97  4.23  155.544±0.027  1.614±0.132  0.13±0.034  10.20  202.76  156.846±0.035  1.566±0.126  0.17±0.043  8.19  169.60  162.159±0.041  1.329±0.117  0.17±0.043  6.40  110.27  163.337±0.023  0.986±0.122  0.067±0.024  5.93  63.46  167.021±0.145  1.075±0.162  0.325±0.18  1.70  28.86  168.136±0.23  1.026±0.158  0.397±0.208  1.91  20.36  173.367±0.038  0.689±0.186  0.098±0.117  2.28  15.20  View Large Table B6. Unresolved oscillations for KIC 6144777. νk  Hk  AIC  117.521±0.002  45.826±26.688  51.09  118.626±0.002  76.016±66.075  4.94  118.648±0.002  161.22±133.998  36.62  118.997±0.002  144.317±73.789  98.76  121.087±0.002  34.776±21.847  31.75  127.034±0.001  65.586±51.112  68.83  127.786±0.002  53.426±38.098  56.73  128.236±0.002  234.129±135.883  109.43  129.594±0.001  1339.976±639.958  158.25  129.955±0.001  625.488±293.78  324.77  130.668±0.001  88.336±54.587  90.59  131.119±0.002  203.146±124.72  160.28  132.738±0.001  132.713±93.244  87.96  134.974±0.002  37.783±28.857  16.03  138.077±0.001  38.795±26.616  43.53  141.021±0.002  101.514±63.871  40.41  νk  Hk  AIC  117.521±0.002  45.826±26.688  51.09  118.626±0.002  76.016±66.075  4.94  118.648±0.002  161.22±133.998  36.62  118.997±0.002  144.317±73.789  98.76  121.087±0.002  34.776±21.847  31.75  127.034±0.001  65.586±51.112  68.83  127.786±0.002  53.426±38.098  56.73  128.236±0.002  234.129±135.883  109.43  129.594±0.001  1339.976±639.958  158.25  129.955±0.001  625.488±293.78  324.77  130.668±0.001  88.336±54.587  90.59  131.119±0.002  203.146±124.72  160.28  132.738±0.001  132.713±93.244  87.96  134.974±0.002  37.783±28.857  16.03  138.077±0.001  38.795±26.616  43.53  141.021±0.002  101.514±63.871  40.41  View Large Table B7. Resolved oscillations for KIC 7060732. νk  Ak  γk  SNR  AIC  90.76±0.015  0.557±0.116  0.027±0.017  2.96  17.64  95.679±0.004  0.364±0.152  0.005±0.011  2.25  6.26  96.285±0.015  0.384±0.113  0.018±0.015  1.88  2.81  101.327±0.03  0.893±0.112  0.077±0.026  5.25  48.70  106.715±0.039  1.158±0.125  0.157±0.059  5.31  78.31  107.157±0.004  0.371±0.156  0.005±0.011  2.95  3.13  107.888±0.008  0.34±0.118  0.007±0.006  2.78  4.48  110.436±0.013  0.893±0.133  0.031±0.013  7.34  70.40  111.853±0.011  1.515±0.173  0.04±0.013  17.48  273.88  116.137±0.005  0.441±0.197  0.005±0.009  3.19  8.62  117.494±0.067  1.845±0.14  0.352±0.085  9.10  166.72  117.882±0.021  0.934±0.22  0.012±0.016  13.54  29.64  121.393±0.016  2.162±0.175  0.077±0.017  26.08  540.98  122.784±0.012  2.296±0.214  0.054±0.014  34.13  657.91  125.012±0.017  0.914±0.127  0.044±0.019  6.33  62.66  127.595±0.011  0.652±0.137  0.019±0.012  4.98  24.11  128.248±0.006  1.769±0.384  0.012±0.007  36.98  135.94  128.52±0.023  2.131±0.177  0.094±0.024  27.33  394.78  128.995±0.005  1.416±0.283  0.011±0.005  28.33  165.37  129.251±0.004  0.852±0.261  0.006±0.005  11.78  55.16  129.535±0.011  0.664±0.13  0.022±0.014  5.41  26.64  130.127±0.004  0.524±0.171  0.005±0.007  2.81  33.41  132.318±0.016  2.794±0.208  0.089±0.019  44.14  907.93  133.652±0.011  3.22±0.304  0.048±0.011  70.62  1387.90  135.871±0.019  0.807±0.129  0.041±0.02  4.79  39.91  136.136±0.013  0.659±0.151  0.023±0.022  3.72  24.39  137.52±0.004  0.428±0.185  0.005±0.01  3.37  18.18  138.536±0.01  0.745±0.151  0.017±0.009  7.14  31.69  138.785±0.005  1.823±0.397  0.01±0.005  47.03  337.63  139.251±0.011  1.78±0.269  0.027±0.012    114.58  139.526±0.019  2.433±0.21  0.082±0.021  32.36  460.81  143.213±0.026  2.199±0.146  0.155±0.034  18.71  454.72  144.571±0.012  2.437±0.224  0.055±0.014  38.33  704.79  147.003±0.019  0.712±0.119  0.041±0.02  4.30  30.12  148.593±0.004  0.361±0.147  0.005±0.008  3.18  5.89  149.517±0.011  0.678±0.155  0.02±0.016  4.96  19.68  149.87±0.036  1.352±0.134  0.139±0.043  4.04  105.37  150.488±0.011  1.274±0.182  0.031±0.016  15.07  117.68  150.855±0.019  1.235±0.151  0.062±0.027  8.74  98.20  151.735±0.014  0.499±0.121  0.02±0.015  3.32  9.93  152.346±0.005  0.494±0.178  0.005±0.008  1.84  20.15  154.316±0.048  1.58±0.127  0.243±0.067  6.40  149.86  155.764±0.042  1.295±0.125  0.161±0.05  5.82  91.76  160.803±0.029  0.597±0.137  0.05±0.038  2.05  9.41  161.138±0.024  0.806±0.126  0.051±0.023  4.13  31.51  161.793±0.042  1.026±0.121  0.123±0.045  3.88  52.31  163.358±0.028  0.464±0.114  0.033±0.02  2.32  4.26  164.954±0.007  0.389±0.144  0.007±0.009  2.77  4.75  165.731±0.08  1.018±0.131  0.23±0.085  2.01  35.84  167.018±0.034  0.825±0.142  0.088±0.053  2.76  33.22  172.043±0.006  0.312±0.131  0.005±0.014  1.82  2.90  177.148±0.051  0.617±0.172  0.089±0.102  1.91  10.00  νk  Ak  γk  SNR  AIC  90.76±0.015  0.557±0.116  0.027±0.017  2.96  17.64  95.679±0.004  0.364±0.152  0.005±0.011  2.25  6.26  96.285±0.015  0.384±0.113  0.018±0.015  1.88  2.81  101.327±0.03  0.893±0.112  0.077±0.026  5.25  48.70  106.715±0.039  1.158±0.125  0.157±0.059  5.31  78.31  107.157±0.004  0.371±0.156  0.005±0.011  2.95  3.13  107.888±0.008  0.34±0.118  0.007±0.006  2.78  4.48  110.436±0.013  0.893±0.133  0.031±0.013  7.34  70.40  111.853±0.011  1.515±0.173  0.04±0.013  17.48  273.88  116.137±0.005  0.441±0.197  0.005±0.009  3.19  8.62  117.494±0.067  1.845±0.14  0.352±0.085  9.10  166.72  117.882±0.021  0.934±0.22  0.012±0.016  13.54  29.64  121.393±0.016  2.162±0.175  0.077±0.017  26.08  540.98  122.784±0.012  2.296±0.214  0.054±0.014  34.13  657.91  125.012±0.017  0.914±0.127  0.044±0.019  6.33  62.66  127.595±0.011  0.652±0.137  0.019±0.012  4.98  24.11  128.248±0.006  1.769±0.384  0.012±0.007  36.98  135.94  128.52±0.023  2.131±0.177  0.094±0.024  27.33  394.78  128.995±0.005  1.416±0.283  0.011±0.005  28.33  165.37  129.251±0.004  0.852±0.261  0.006±0.005  11.78  55.16  129.535±0.011  0.664±0.13  0.022±0.014  5.41  26.64  130.127±0.004  0.524±0.171  0.005±0.007  2.81  33.41  132.318±0.016  2.794±0.208  0.089±0.019  44.14  907.93  133.652±0.011  3.22±0.304  0.048±0.011  70.62  1387.90  135.871±0.019  0.807±0.129  0.041±0.02  4.79  39.91  136.136±0.013  0.659±0.151  0.023±0.022  3.72  24.39  137.52±0.004  0.428±0.185  0.005±0.01  3.37  18.18  138.536±0.01  0.745±0.151  0.017±0.009  7.14  31.69  138.785±0.005  1.823±0.397  0.01±0.005  47.03  337.63  139.251±0.011  1.78±0.269  0.027±0.012    114.58  139.526±0.019  2.433±0.21  0.082±0.021  32.36  460.81  143.213±0.026  2.199±0.146  0.155±0.034  18.71  454.72  144.571±0.012  2.437±0.224  0.055±0.014  38.33  704.79  147.003±0.019  0.712±0.119  0.041±0.02  4.30  30.12  148.593±0.004  0.361±0.147  0.005±0.008  3.18  5.89  149.517±0.011  0.678±0.155  0.02±0.016  4.96  19.68  149.87±0.036  1.352±0.134  0.139±0.043  4.04  105.37  150.488±0.011  1.274±0.182  0.031±0.016  15.07  117.68  150.855±0.019  1.235±0.151  0.062±0.027  8.74  98.20  151.735±0.014  0.499±0.121  0.02±0.015  3.32  9.93  152.346±0.005  0.494±0.178  0.005±0.008  1.84  20.15  154.316±0.048  1.58±0.127  0.243±0.067  6.40  149.86  155.764±0.042  1.295±0.125  0.161±0.05  5.82  91.76  160.803±0.029  0.597±0.137  0.05±0.038  2.05  9.41  161.138±0.024  0.806±0.126  0.051±0.023  4.13  31.51  161.793±0.042  1.026±0.121  0.123±0.045  3.88  52.31  163.358±0.028  0.464±0.114  0.033±0.02  2.32  4.26  164.954±0.007  0.389±0.144  0.007±0.009  2.77  4.75  165.731±0.08  1.018±0.131  0.23±0.085  2.01  35.84  167.018±0.034  0.825±0.142  0.088±0.053  2.76  33.22  172.043±0.006  0.312±0.131  0.005±0.014  1.82  2.90  177.148±0.051  0.617±0.172  0.089±0.102  1.91  10.00  View Large Table B8. Unresolved oscillations for KIC 7060732. νk  Hk  AIC  116.797±0.002  70.56±45.876  40.42  118.225±0.002  96.389±58.644  36.66  126.752±0.001  81.066±49.444  95.79  127.311±0.001  179.366±104.066  258.20  127.852±0.001  211.069±162.818  246.88  130.743±0.002  46.288±47.778  66.49  138.281±0.001  149.966±100.648  132.00  νk  Hk  AIC  116.797±0.002  70.56±45.876  40.42  118.225±0.002  96.389±58.644  36.66  126.752±0.001  81.066±49.444  95.79  127.311±0.001  179.366±104.066  258.20  127.852±0.001  211.069±162.818  246.88  130.743±0.002  46.288±47.778  66.49  138.281±0.001  149.966±100.648  132.00  View Large Table B9. Resolved oscillations for KIC 7619745. νk  Ak  γk  SNR  AIC  122.424±0.049  0.546±0.125  0.075±0.05  1.76  5.99  128.42±0.018  0.486±0.16  0.013±0.022  3.24  14.07  128.725±0.01  0.545±0.123  0.017±0.012  3.18  18.87  129.195±0.023  0.49±0.117  0.029±0.021  2.37  8.58  133.25±0.005  0.472±0.171  0.005±0.007  4.31  15.57  133.449±0.019  0.557±0.126  0.036±0.028  1.87  12.72  135.018±0.028  1.124±0.117  0.095±0.028  6.41  91.69  141.295±0.084  1.143±0.121  0.29±0.085  4.29  49.88  146.186±0.041  1.325±0.116  0.168±0.043  6.53  112.06  147.825±0.012  1.852±0.194  0.044±0.012  24.19  431.27  153.294±0.004  0.395±0.143  0.005±0.011  2.49  14.16  154.06±0.018  1.04±0.129  0.054±0.02  6.46  85.43  154.493±0.006  1.399±0.269  0.012±0.006  26.59  250.01  155.094±0.013  0.964±0.137  0.035±0.015  7.42  81.65  155.672±0.007  0.739±0.155  0.013±0.007  7.81  53.24  159.29±0.015  2.013±0.173  0.071±0.018  23.65  470.41  160.964±0.009  3.199±0.342  0.037±0.009  77.21  1570.93  163.76±0.021  0.743±0.118  0.044±0.02  4.60  34.21  165.374±0.003  0.647±0.221  0.005±0.005  8.32  49.58  166.593±0.005  1.628±0.335  0.01±0.005  39.09  376.12  167.114±0.007  1.591±0.243  0.019±0.007  26.45  307.34  167.782±0.007  2.33±0.318  0.023±0.008  51.18  725.04  168.212±0.005  2.058±0.376  0.012±0.005  56.81  539.12  169.392±0.006  0.868±0.192  0.01±0.006  10.70  87.06  170.086±0.005  0.643±0.184  0.006±0.005  8.34  45.15  172.463±0.013  2.18±0.204  0.056±0.015  25.52  544.96  173.985±0.012  3.397±0.294  0.057±0.012  74.70  1693.49  176.95±0.006  0.672±0.166  0.011±0.011  4.38  43.67  179.236±0.005  0.976±0.225  0.008±0.005    121.27  180.348±0.009  1.98±0.256  0.028±0.01  32.63  337.11  180.681±0.015  1.824±0.182  0.057±0.016  15.76  275.55  181.707±0.007  1.108±0.203  0.014±0.007  15.46  143.52  182.351±0.005  0.77±0.186  0.008±0.005  10.31  68.05  184.819±0.005  0.79±0.198  0.007±0.005  11.71  60.07  185.073±0.008  0.505±0.159  0.009±0.009  4.50  8.32  185.331±0.026  0.946±0.178  0.054±0.031  4.41  17.23  185.594±0.013  2.179±0.211  0.056±0.017  30.68  449.61  187.121±0.016  2.411±0.19  0.079±0.017  29.49  711.35  190.16±0.016  0.736±0.124  0.033±0.018  4.69  37.66  192.104±0.008  0.572±0.136  0.012±0.007  4.79  24.79  192.705±0.015  0.702±0.126  0.032±0.017  4.23  29.33  193.961±0.033  1.716±0.122  0.162±0.032  11.57  248.72  195.675±0.012  0.6±0.126  0.02±0.013  3.98  22.66  198.528±0.013  0.378±0.157  0.012±0.018  1.97  0.74  198.819±0.021  1.098±0.135  0.063±0.023  6.27  76.01  199.204±0.024  0.971±0.135  0.064±0.028  4.76  46.78  200.449±0.048  1.528±0.118  0.226±0.051  6.43  147.40  207.029±0.053  1.218±0.113  0.192±0.048  5.93  81.13  212.401±0.094  1.105±0.139  0.346±0.142  1.36  38.89  220.743±0.014  0.591±0.125  0.03±0.023  2.52  19.90  νk  Ak  γk  SNR  AIC  122.424±0.049  0.546±0.125  0.075±0.05  1.76  5.99  128.42±0.018  0.486±0.16  0.013±0.022  3.24  14.07  128.725±0.01  0.545±0.123  0.017±0.012  3.18  18.87  129.195±0.023  0.49±0.117  0.029±0.021  2.37  8.58  133.25±0.005  0.472±0.171  0.005±0.007  4.31  15.57  133.449±0.019  0.557±0.126  0.036±0.028  1.87  12.72  135.018±0.028  1.124±0.117  0.095±0.028  6.41  91.69  141.295±0.084  1.143±0.121  0.29±0.085  4.29  49.88  146.186±0.041  1.325±0.116  0.168±0.043  6.53  112.06  147.825±0.012  1.852±0.194  0.044±0.012  24.19  431.27  153.294±0.004  0.395±0.143  0.005±0.011  2.49  14.16  154.06±0.018  1.04±0.129  0.054±0.02  6.46  85.43  154.493±0.006  1.399±0.269  0.012±0.006  26.59  250.01  155.094±0.013  0.964±0.137  0.035±0.015  7.42  81.65  155.672±0.007  0.739±0.155  0.013±0.007  7.81  53.24  159.29±0.015  2.013±0.173  0.071±0.018  23.65  470.41  160.964±0.009  3.199±0.342  0.037±0.009  77.21  1570.93  163.76±0.021  0.743±0.118  0.044±0.02  4.60  34.21  165.374±0.003  0.647±0.221  0.005±0.005  8.32  49.58  166.593±0.005  1.628±0.335  0.01±0.005  39.09  376.12  167.114±0.007  1.591±0.243  0.019±0.007  26.45  307.34  167.782±0.007  2.33±0.318  0.023±0.008  51.18  725.04  168.212±0.005  2.058±0.376  0.012±0.005  56.81  539.12  169.392±0.006  0.868±0.192  0.01±0.006  10.70  87.06  170.086±0.005  0.643±0.184  0.006±0.005  8.34  45.15  172.463±0.013  2.18±0.204  0.056±0.015  25.52  544.96  173.985±0.012  3.397±0.294  0.057±0.012  74.70  1693.49  176.95±0.006  0.672±0.166  0.011±0.011  4.38  43.67  179.236±0.005  0.976±0.225  0.008±0.005    121.27  180.348±0.009  1.98±0.256  0.028±0.01  32.63  337.11  180.681±0.015  1.824±0.182  0.057±0.016  15.76  275.55  181.707±0.007  1.108±0.203  0.014±0.007  15.46  143.52  182.351±0.005  0.77±0.186  0.008±0.005  10.31  68.05  184.819±0.005  0.79±0.198  0.007±0.005  11.71  60.07  185.073±0.008  0.505±0.159  0.009±0.009  4.50  8.32  185.331±0.026  0.946±0.178  0.054±0.031  4.41  17.23  185.594±0.013  2.179±0.211  0.056±0.017  30.68  449.61  187.121±0.016  2.411±0.19  0.079±0.017  29.49  711.35  190.16±0.016  0.736±0.124  0.033±0.018  4.69  37.66  192.104±0.008  0.572±0.136  0.012±0.007  4.79  24.79  192.705±0.015  0.702±0.126  0.032±0.017  4.23  29.33  193.961±0.033  1.716±0.122  0.162±0.032  11.57  248.72  195.675±0.012  0.6±0.126  0.02±0.013  3.98  22.66  198.528±0.013  0.378±0.157  0.012±0.018  1.97  0.74  198.819±0.021  1.098±0.135  0.063±0.023  6.27  76.01  199.204±0.024  0.971±0.135  0.064±0.028  4.76  46.78  200.449±0.048  1.528±0.118  0.226±0.051  6.43  147.40  207.029±0.053  1.218±0.113  0.192±0.048  5.93  81.13  212.401±0.094  1.105±0.139  0.346±0.142  1.36  38.89  220.743±0.014  0.591±0.125  0.03±0.023  2.52  19.90  View Large Table B10. Unresolved oscillations for KIC 7619745. νk  Hk  AIC  141.996±0.002  24.341±17.567  21.78  152.631±0.002  21.4±16.24  16.53  156.725±0.001  47.87±30.048  106.10  162.619±0.001  46.85±31.866  90.53  164.669±0.001  85.663±60.049  129.06  171.562±0.001  37.569±29.887  45.63  172.092±0.002  384.319±190.437  105.06  174.598±0.002  26.466±28.051  21.70  178.569±0.002  151.005±59.669  143.22  184.091±0.002  23.311±19.106  23.50  186.726±0.002  58.27±48.794  32.50  νk  Hk  AIC  141.996±0.002  24.341±17.567  21.78  152.631±0.002  21.4±16.24  16.53  156.725±0.001  47.87±30.048  106.10  162.619±0.001  46.85±31.866  90.53  164.669±0.001  85.663±60.049  129.06  171.562±0.001  37.569±29.887  45.63  172.092±0.002  384.319±190.437  105.06  174.598±0.002  26.466±28.051  21.70  178.569±0.002  151.005±59.669  143.22  184.091±0.002  23.311±19.106  23.50  186.726±0.002  58.27±48.794  32.50  View Large Table B11. Resolved oscillations for KIC 8366239. νk  Ak  γk  SNR  AIC  134.935±0.006  0.463±0.135  0.008±0.009  3.21  14.93  135.937±0.01  0.489±0.126  0.014±0.012  3.00  14.83  137.402±0.006  0.322±0.153  0.006±0.009  1.85  3.93  141.663±0.024  0.87±0.115  0.062±0.023  4.92  50.75  147.599±0.01  0.611±0.134  0.018±0.014  3.92  26.44  147.898±0.01  0.739±0.134  0.021±0.011  5.59  46.42  148.643±0.03  0.455±0.121  0.038±0.029  1.68  3.71  148.948±0.009  0.526±0.13  0.015±0.012  3.35  17.65  153.149±0.017  0.984±0.128  0.048±0.018  5.90  76.21  153.37±0.014  0.442±0.142  0.018±0.019  2.27  3.20  154.917±0.021  1.65±0.138  0.092±0.022  13.54  276.48  160.32±0.004  0.752±0.206  0.006±0.005  10.21  69.15  161.737±0.013  1.054±0.153  0.033±0.013  7.55  80.60  161.983±0.013  1.198±0.158  0.036±0.013  9.23  121.53  163.636±0.008  0.507±0.133  0.01±0.006  4.52  18.82  166.88±0.03  1.683±0.126  0.137±0.028  12.09  243.53  168.59±0.015  2.138±0.181  0.07±0.016  25.69  551.54  172.703±0.006  0.435±0.152  0.006±0.006  3.96  13.05  173.14±0.007  0.583±0.145  0.01±0.006  5.52  30.48  174.82±0.008  1.436±0.221  0.02±0.008  20.95  193.31  175.091±0.017  1.284±0.153  0.048±0.016  8.34  130.37  176.135±0.007  1.667±0.257  0.018±0.008    372.31  180.5±0.013  1.923±0.193  0.05±0.014  22.95  420.94  180.863±0.008  1.092±0.19  0.019±0.01  12.39  85.05  182.258±0.012  2.745±0.242  0.057±0.013  46.96  1056.25  185.473±0.039  0.698±0.116  0.079±0.036  3.43  18.36  186.865±0.004  0.865±0.265  0.006±0.004  12.88  89.62  187.28±0.005  0.966±0.216  0.009±0.005  14.79  118.80  188.888±0.007  2.238±0.343  0.019±0.008  52.88  287.39  189.087±0.015  1.892±0.205  0.052±0.016  18.89  212.10  190.631±0.012  0.973±0.141  0.031±0.012  8.11  85.22  191.02±0.004  0.929±0.249  0.006±0.004  15.35  103.04  194.284±0.019  2.607±0.182  0.104±0.02  30.43  768.75  195.871±0.015  2.936±0.221  0.082±0.016  45.09  1077.82  199.198±0.06  0.624±0.119  0.092±0.043  2.45  9.06  200.506±0.005  0.818±0.196  0.008±0.005  11.61  80.58  202.65±0.017  2.466±0.187  0.084±0.016  31.97  774.14  204.365±0.013  0.745±0.129  0.026±0.012  5.49  40.05  204.761±0.005  0.928±0.215  0.009±0.006  13.13  100.55  208.074±0.048  1.476±0.119  0.217±0.051  6.17  129.57  209.681±0.026  1.731±0.133  0.13±0.03  12.03  258.60  215.218±0.06  1.033±0.112  0.156±0.042  5.21  50.70  217.155±0.033  1.375±0.118  0.136±0.033  8.77  138.78  222.143±0.062  0.966±0.139  0.183±0.093  2.41  34.72  223.525±0.095  0.933±0.139  0.269±0.12  1.99  22.39  230.819±0.058  0.784±0.129  0.147±0.075  2.88  19.16  236.197±0.174  0.875±0.164  0.4±0.209  1.49  11.45  νk  Ak  γk  SNR  AIC  134.935±0.006  0.463±0.135  0.008±0.009  3.21  14.93  135.937±0.01  0.489±0.126  0.014±0.012  3.00  14.83  137.402±0.006  0.322±0.153  0.006±0.009  1.85  3.93  141.663±0.024  0.87±0.115  0.062±0.023  4.92  50.75  147.599±0.01  0.611±0.134  0.018±0.014  3.92  26.44  147.898±0.01  0.739±0.134  0.021±0.011  5.59  46.42  148.643±0.03  0.455±0.121  0.038±0.029  1.68  3.71  148.948±0.009  0.526±0.13  0.015±0.012  3.35  17.65  153.149±0.017  0.984±0.128  0.048±0.018  5.90  76.21  153.37±0.014  0.442±0.142  0.018±0.019  2.27  3.20  154.917±0.021  1.65±0.138  0.092±0.022  13.54  276.48  160.32±0.004  0.752±0.206  0.006±0.005  10.21  69.15  161.737±0.013  1.054±0.153  0.033±0.013  7.55  80.60  161.983±0.013  1.198±0.158  0.036±0.013  9.23  121.53  163.636±0.008  0.507±0.133  0.01±0.006  4.52  18.82  166.88±0.03  1.683±0.126  0.137±0.028  12.09  243.53  168.59±0.015  2.138±0.181  0.07±0.016  25.69  551.54  172.703±0.006  0.435±0.152  0.006±0.006  3.96  13.05  173.14±0.007  0.583±0.145  0.01±0.006  5.52  30.48  174.82±0.008  1.436±0.221  0.02±0.008  20.95  193.31  175.091±0.017  1.284±0.153  0.048±0.016  8.34  130.37  176.135±0.007  1.667±0.257  0.018±0.008    372.31  180.5±0.013  1.923±0.193  0.05±0.014  22.95  420.94  180.863±0.008  1.092±0.19  0.019±0.01  12.39  85.05  182.258±0.