TY - JOUR
AU - Mallick,, Subhashis
AB - SUMMARY We developed a multi-objective optimization method for inverting marine controlled source electromagnetic data using a fast-non-dominated sorting genetic algorithm. Deterministic methods for inverting electromagnetic data rely on selecting weighting parameters to balance the data misfit with the model roughness and result in a single solution which do not provide means to assess the non-uniqueness associated with the inversion. Here, we propose a robust stochastic global search method that considers the objective as a two-component vector and simultaneously minimizes both components: data misfit and model roughness. By providing an estimate of the entire set of the Pareto-optimal solutions, the method allows a better assessment of non-uniqueness than deterministic methods. Since the computational expense of the method increases as the number of objectives and model parameters increase, we parallelized our algorithm to speed up the forward modelling calculations. Applying our inversion to noisy synthetic data sets generated from horizontally stratified earth models for both isotropic and anisotropic assumptions and for different measurement configurations, we demonstrate the accuracy of our method. By comparing the results of our inversion with the regularized genetic algorithm, we also demonstrate the necessity of casting this problem as a multi-objective optimization for a better assessment of uncertainty as compared to a scalar objective optimization method. Electromagnetic methods, Inverse theory, Multi-objective optimization INTRODUCTION Controlled-source electromagnetic (CSEM) methods have gained popularity in the oil and gas industry for exploring the hydrocarbon reservoirs in the marine sediments (MacGregor & Sinha 2000; Eidesmo et al.2002, among others). CSEM data are acquired by towing a high-powered electric dipole source transmitting low frequency (0.1–10 Hz) electromagnetic (EM) signal through the earth and recording the resultant electric and magnetic fields using an array of receivers at different transmitter–receiver separations (Constable 2010). Analysis of CSEM data reveals the subsurface resistivity, which, in turn, can be correlated with hydrocarbon-bearing formations or other resistive features. Because of the diffusive nature of the EM signal, the vertical resolution of the CSEM method is low. Secondly, the data recorded by the receivers placed on the sea floor surface are contaminated by noise. Such a low-resolution data, in conjunction with the presence of noise causes non-uniqueness of the solution, the inversion becomes an ill-posed problem, and its interpretation is ambiguous (Christensen & Dodds 2007). The first category of method practiced for the inversion of CSEM data includes deterministic algorithms such as conjugate gradient or Gauss–Newton techniques. Occam's inversion (Constable et al.1987; Key 2009) for example, has been the preferred method for inverting the CSEM data to date. These classic methods such as Occam or conjugate gradients (Newman & Alumbaugh 2000) require the calculation of derivatives—the Jacobian of the model-data mapping—as well as an appropriate initial model. These methods also include a regularization term to limit the variability in the resulting models. These techniques are very efficient in locating minima in the objective function and produce smooth models. It is clear that geology often comprises sharp boundaries, however smooth models are preferred because they contain only those features which are well constrained by the data and ignore false rough features which might lead to incorrect interpretations (Constable et al.1987). However, these methods rely on selecting weighting parameters to balance the data misfit with the model roughness, which in turn biases the inversion towards one area of model space and does not provide the measure of the non-uniqueness associated with the inversion. As an ill-posed problem, the inverse solution for EM data is not limited to one predicted model but is represented by a probability density function (PDF) in the model space (Buland & Kolbjørnsen 2012). This is not well represented in the results obtained from these deterministic methods. The second category of inverse methods, called the probabilistic methods explore the posterior, which is the probability of a certain earth model given the observed data (Trainor-Guitton & Hoversten 2011). These methods are further subdivided into two subcategories, namely stochastic and Bayesian methods. Stochastic methods have been successful in finding the global optima of non-linear inverse problems. These methods iteratively perturb the model realizations to minimize the misfit between the observed and predicted data. Some examples where stochastic methods have been applied include the inversion of resistivity sounding data using simulated annealing (SA, Sen et al.1993), very fast simulated annealing (VFSA) for inverting 2-D dipole–dipole resistivity data (Chunduru et al.1996) and genetic algorithms (GA) to invert the electromagnetic reflection response from transversely isotropic media with vertical symmetry axes (VTI media, Hunziker et al.2016). In a previous study, Ayani et al. (2017, 2018) used GA to invert CSEM data under an isotropic assumption. Besides stochastic methods, Bayesian methods have also been applied for CSEM inversion (Trainor-Guitton & Hoversten 2011; Ray & Key 2012) or for the joint inversion of CSEM and seismic data (Hou et al.2006; Chen et al.2007) and CSEM and magnetotelluric data (Buland & Kolbjørnsen 2012; Blatter et al.2019). These methods use Bayes’ theorem to infer the models which explain the data and the associated uncertainty by treating the model parameters as random variables and calculating the likelihood based on the misfit between the observed and predicted data. The drawback however with these probabilistic (stochastic and Bayesian) methods is their computational costs and therefore they require state-of-the-art high-performance computing to speed up the forward modelling calculations (Schwarzbach et al.2005). The goal of this study is to implement a global multi-objective optimization method called the fast-non-dominated sorting genetic algorithm two (NSGA-II) which considers the objective as a vector and simultaneously optimizes multiple objectives. NSGA-II is an improved version of the original non-dominated sorting genetic algorithm (NSGA) which had some drawbacks such as the lack of elitism and computational complexity of the non-dominated sorting (Deb et al.2002). In general, non-linear and non-unique inverse problems are known to have multiple optimal solutions or the Pareto-optimal solutions, defined as the set of solutions where none are superior to the other (Deb et al.2002; Padhi & Mallick 2014; Li & Mallick 2015). For the CSEM problem considered here, the two objectives to be simultaneously minimized are (1) data misfit and (2) model roughness. Being a stochastic method, the method uses a directed random search and attempts to estimate the shape of the PDF in the model space. Unlike linearized methods, NSGA-II does not depend upon the choice of initial model and is well suited for the CSEM inversion. The advantage of using the NSGA-II compared to the deterministic methods is that it produces an ensemble of the models which gives the measure of the non-uniqueness in the inversion. In addition, these methods are robust in the sense that they assume very little prior information about the reservoir depth or the local geology of the area. Computational cost of the NSGA-II increases with increasing number of objectives and number of model parameters. We have therefore followed the parallelization approach of Li & Mallick (2015) to speed up the forward modelling calculations in our method. NSGA-II have been successfully implemented in seismic inverse problems by Padhi & Mallick (2014) to invert multicomponent prestack data for VTI media. Li & Mallick (2015) used this method to invert multicomponent, multi-azimuth seismic data to invert for orthorhombic elastic properties. In electromagnetic problems, Schwarzbach et al. (2005) used NSGA-II for 2-D inversion of direct current (dc) resistivity data. While the assumption of isotropy can be appropriate in certain geological situations, sediment formations are often observed to be electrically anisotropic at several scales. At the grain scale, anisotropy can be caused by mineral alignment, most often because of compaction. In general, this results in the resistivity in the vertical direction to be higher than that in the horizontal direction (Ramananjaona et al.2011). Fine scale layering in a section can also lead to significant electrical anisotropy. For an anisotropic subsurface, an inversion assuming the subsurface is isotropic results in artefacts and errors in the reconstructed models (Ramananjaona et al.2011). Therefore, accounting for anisotropy during the inversion is important. We tested our proposed method on the synthetic data generated from a VTI earth model and compared the inversion results for different measurement configurations. In the following, we first describe the multi-objective inversion and NSGA-II method used in this study including its parallel implementation, and briefly discuss the regularized GA and forward modelling methods for the isotropic and anisotropic earth models. Next, we demonstrate the applications of our method on noisy synthetic data and discuss results. Finally, we end with some concluding remarks. METHOLOGY Multi-objective inversion The goal of CSEM inversion is to find the resistivity model (or a family of models) which best fits the observed data. The method aims at minimizing the misfit between the observed and predicted data. However, minimizing the data misfit alone produces highly oscillatory models which are geologically meaningless. Therefore, several regularization methods are used to stabilize the inversion. The most preferred such method is Tikhonov regularization (Constable et al.1987; Zhdanov 2002) which focuses on structure minimization and searches for the smoothest model which fits the data. Features not constrained by the data are smoothed over or be entirely absent. The objective functional for such regularized inverse problem is given by $$\begin{eqnarray*} U = \frac{1}{{{N_D}}}{\boldsymbol{||W}}\left( {{\boldsymbol{d}} - F\left( {\boldsymbol{m}} \right)} \right){\boldsymbol{|}}{{\boldsymbol{|}}^2} + \mu {\boldsymbol{||}}\partial {\boldsymbol{m|}}{{\boldsymbol{|}}^2} - {\chi ^*}. \end{eqnarray*}$$ (1) In eq. (1), |${\boldsymbol{d}} = {[ {{d_1},{d_2}, \ldots ,{d_{{N_D}}}} ]^T}$| is the data vector representing the observed CSEM data with ND being the number of data points, |${\boldsymbol{m}} = {[ {{m_1},{m_2}, \ldots ,{m_K}} ]^T}$| is the model vector with K being the number of model parameters, and F is the forward modelling operator which maps the model vector (m) to CSEM data (d). W is a data covariance weighting function and is selected here to be a diagonal matrix with elements corresponding to inverse data standard errors. Finally, |$\partial $| is the smoothing operator, μ is the associated Lagrange multiplier, and |${\chi ^*}$| is the target misfit whose value depends on the data noise and is usually set to 1 for Gaussian distributed noise. Detailed explanation for this choice of target misfit can be found in Constable et al. (1987). For the 1-D anisotropic cases, the smoothing term is decomposed as given by Ramananjaona et al. (2011) $$\begin{eqnarray*} {\left\| {\partial {\boldsymbol{m}}} \right\|^2} = {\left\| {{\partial _v}{{\boldsymbol{\sigma }}_h}} \right\|^2} + {\left\| {{\partial _v}{{\boldsymbol{\sigma }}_v}} \right\|^2}, \end{eqnarray*}$$ (2) where |${\partial _v}$| is the vertical differencing operator and m is a vector of horizontal and vertical conductivities |${{\boldsymbol{\sigma }}_h}$| and |${{\boldsymbol{\sigma }}_v}$|⁠. In this paper we seek models that are smooth in the first derivative sense. Other regularizations are possible, for example second derivative smoothing (Constable et al.1987) or the use of a priori information on structure and stratigraphy allowing sharp boundaries at pre-defined depths (Key 2009; MacGregor 2012) and could be easily incorporated into the formulation presented here. The approach defined above assigns weights to the multiple objectives to form a scalar objective as a linear combination of both objective functions with coefficients 1 and μ (see eq. 1). Several methods have been proposed to find the optimal value of μ, for example the L-curve criterion (Engl et al.2000; Sen & Roy 2003; Padhi et al.2015), varying μ with iterations by treating it as an annealing parameter (Sen & Biswas 2015; Ayani et al.2018), or finding the value of m that minimizes the misfit at each iteration (Constable et al.1987; Key 2009). However, these approaches yield only one solution to the inverse problem which may be a local rather than global minimum. In addition, results may also be biased by the choice of regularization weights. Problems requiring optimization of multiple objectives are non-linear and non-unique with multiple optimal solutions (Pareto-optimal solutions). For these non-linear and non-unique problems, it is advisable that the entire set of Pareto-optimal solutions is estimated as it provides an ensemble of models that are compatible with the data instead of a single model like the above methods provide. To get an assessment of the possible range of solutions or the Pareto-optimal solutions, it is necessary to treat the inversion as a multi-objective optimization and simultaneously optimize all objectives. An objective vector y in the objective space Y is thus defined as $$\begin{eqnarray*} {\boldsymbol{y}} = {\left[ {{y_1},{y_2}, \ldots ,{y_M}} \right]^T}, \end{eqnarray*}$$ (3) where M is the number of objectives to be optimized, which is two in our case. Note that in this formulation we do not require the coefficient μ which in the formulations above balances the misfit and roughness terms. Several multi-objective evolutionary algorithms (MOEA) have been suggested for solving such problems. Since evolutionary algorithms (EAs) work with the population of models, they can be extended to maintain a diverse set of solutions, with an emphasis for moving towards the true Pareto-optimal region, and NSGA-II is known to be one such MOEA (Deb et al.2002). Parallel NSGA-II NSGA-II belongs to a family of MOEA and have been appreciated mainly because of the two features: non-dominated sorting and preservation of diversity. The basic concept of NSGA-II is described in detail by Deb et al. (2002), and its geophysical application has been described by Padhi & Mallick (2014) and Li & Mallick (2015). For completeness, here we just briefly describe the method with reference to the CSEM inversion. The first fundamental concept of NSGA-II is dominance. Consider a multi-objective optimization where M objectives are to be minimized. Also consider two solutions |${{\boldsymbol{x}}^1} = {[ {x_1^1,x_2^1, \ldots ,x_K^1} ]^T}$| and |${{\boldsymbol{x}}^2} = {[ {x_1^2,x_2^2, \ldots ,x_K^2} ]^T}$| in the model space X, mapping, respectively, onto the objective space Y as |${{\boldsymbol{y}}^1} = {[ {y_1^1,y_2^1, \ldots ,y_M^1} ]^T}$| and |${{\boldsymbol{y}}^2} = {[ {y_1^2,y_2^2, \ldots ,y_M^2} ]^T}$|⁠. Then, if |$y_i^1 \le y_i^2$| for all |$i = 1,2, \cdots M$|⁠, and |$y_j^1 < y_j^2$| for at least one value of j where 1 |$\le $| j |$\le $|M, the set |${{\boldsymbol{x}}^1}$| is said to dominate over the set |${{\boldsymbol{x}}^2}$|⁠, and is denoted as $$\begin{eqnarray*} {{\boldsymbol{x}}^1} \succ {{\boldsymbol{x}}^2}. \end{eqnarray*}$$ (4) When the solutions do not dominate each other in the objective space, they belong to the same front termed as Pareto-optimal front (Deb et al.2002). In the context of CSEM inversion problem, a pseudo code for non-dominated sorting when comparing the objective values of two models is described in the Appendix. The method of non-dominated sorting guides the search towards the region in the model space with the misfit values close to the target RMSE (root mean square error) in the initial generations. Once the models have attained the target RMSE, then the smoother models dominate the others. Thus, the inversion proceeds by first guiding its search towards the feasible region in the misfit space followed by sampling the smoothest models in model space. In non-dominated sorting, the entire population is sorted such that the members belonging to rank 1 dominate the rest of the population without dominating each other. Members belonging to rank 2 are dominated by members from rank 1 but they dominate the others. This process is continued until the ranks are assigned to all the members in the population. The second fundamental concept in NSGA-II is the population diversity, defined as the normalized Euclidean distance of a given member of the population from all other members that belong to the same rank (Deb et al.2002). A pseudo code for the crowding-distance computation procedure of all solutions in a non-dominated set ℐ is described in the Appendix. In essence, the higher the crowding distance of a given member of the population, the less crowded or more diverse it is compared with its neighbors. Original implementation of NSGA-II by Deb et al. (2002) computes crowding distance in the objective space to maintain diversity. For multi-objective problems where the total number of objectives are much smaller than the number of model parameters to be inverted, Li & Mallick (2015) observed that computing crowding distance in the objective space only does not necessarily guarantee population diversity and combining crowding distance computed in the objective space with the same computed in the model space is the key to maintain diversity within each generation. Because the number of objectives (two) is much smaller than the total number of model parameters to be inverted for in all of our examples (see below), we also computed crowding distance in both objective and model spaces, which, in turn, makes our NSGA-II implementation slightly different from that of Deb et al. (2002). It will be shown below that the dominance-based ranking, in conjunction with this new crowding distance in selecting new members, allows achieving the most diverse population of models that meets the target RMS misfit of one. Deb et al. (2002) demonstrated that the computational complexity of NSGA-II is |$O( {M{N^2}} )$| per iteration, where M is the number of objectives and N is the population size. Consequently, NSGA-II is computationally expensive for large population sizes, and it is necessary to parallelize the algorithm for compute efficiency. Note that large populations not only require a large number of objective function evaluations per generation, but it also requires a large number of generations to reach convergence (Schwarzbach et al.2005). Fig. 1 shows the schematic flow diagram of our parallel NSGA-II, which is a modified version of the parallel NSGA-II implementation of Li & Mallick (2015). First, a random model population of size N is generated from an initial user-defined depth-dependent search window and distributed to P compute nodes such that each node receives a subpopulation of size N/P. In CSEM inversion, for example, these random models are the resistivity values as a function of depth. Each node performs the forward modelling of its subpopulation and calculates the objective vectors for each and sends it to the master node. The master node performs the non-dominated sorting into ranks and calculates the crowding distance of the members of each rank. This is followed by the standard GA operations of tournament selection (reproduction), crossover and mutation (Goldberg 1989) which produces a population of N child members from N parents. The child population is distributed to the P compute nodes for objective vector evaluation. The master node then receives the combined parent and child population of size 2 N and performs the non-dominated sorting and crowding distance calculation to select the best N models for the next generation. The best N members for the next generation is selected first in preference of their ranks and then their crowding distances. Members belonging to rank 1 are selected first. This follows selecting rank 2, then rank 3 members, and so on until the point where all members of the given rank cannot fill-up the rest of the N spots. At this stage, the rest of the spots are chosen from the remaining members in descending order of their crowding distance. Thus, the member with the highest crowding distance, that is the one which is least crowded from its neighbors is chosen first, followed by the one with the second highest crowding distance and so on until all N members of the new generation of population are filled-up. The entire process is then repeated over the generations (see Fig. 1). Figure 1. Open in new tabDownload slide Schematic flow diagram of parallel NSGA II. Figure 1. Open in new tabDownload slide Schematic flow diagram of parallel NSGA II. To continuously sample the model space, we followed the work of Padhi & Mallick (2014) and Li & Mallick (2015) to implement real coded GA instead of binary coded GA which can only discretely sample the model space. Real coded GA uses simulated binary crossover (SBX) and real parameter mutation (RPM) (Deb & Agrawal 1995, 1999). SBX is controlled by crossover probability |$( {{P_C}} )$| and the crossover distribution index η. |${P_C}$| is used exactly the same way as the binary coded GA. The parameter η on the other hand is used to control how close or far away from the parents the child members are produced. Choosing a high value of η tends to produce children close to while a low value of η tends to produce them away from their parents. RPM is controlled by the probability of mutation |$( {{P_m}} )$| and the mutation distribution index κ. Like SBX, |${P_m}$| has the same meaning as in the binary coded GA. Choice of a small value of κ produces a mutated solution far away from the original solution and a large value produces a solution close to it. For seismic inversions, it has been shown that by varying |${P_C}$|⁠, |${P_m}$|⁠, η and κ over generations provides NSGA-II additional robustness not only in preserving diversity and propagating good solutions over generations, but also in allowing an exhaustive sampling of the model space (Li & Mallick 2015). For inverting CSEM data for the anisotropic case, we also found that varying SBX and RPM parameters over generations produces good results. Parallel regularized GA The regularized GA is based on the single objective inversion with the primary goal to minimize eq. (1). The approach used here is to cast the regularization weight |$\mu $| as temperature like annealing parameter (Sen & Biswas 2015) and can be written as $$\begin{eqnarray*} \mu = 0.9 \times {e^{ - \left( {2.3 \times {{10}^{ - 7}} \times t} \right)}}, \end{eqnarray*}$$ (5) where t is the current generation. Note that we cast this regularized GA as a maximization instead of a minimization problem. So, we define fitness f as f = 1/U, where U is the objective function defined by eq. (1) and attempt to find the regions where f is maximum. To avoid the rapid convergence of GA to local maxima, we follow Goldberg (1989) and Mallick (1995, 1999) to linearly scale the fitness values at each generation. Although Stoffa & Sen (1991) and Sen & Stoffa (1992) preferred exponential scaling over linear scaling, in our case linear scaling gave good results. In addition, we parallelize this regularized GA as shown in Fig. 1, but without the non-dominated sorting and crowding distance calculations. Forward modelling For the isotropic examples shown here, we used the DIPOLE1D code (Key 2009) for forward modelling. DIPOLE1D is generalized for dipole source and observation points can be placed anywhere within the stack of layers, and all components of electric field E and magnetic field B are computed (Key 2009). For the anisotropic forward modelling calculations, we used the EMmod code (Hunziker et al.2015), which can model 3-D electromagnetic field in a 1-D earth for both diffusive methods such as CSEM and for wave methods such as ground penetrating radar. It also allows placement of the source and the receivers anywhere within a stack of VTI layers for any source–receiver configuration and outputs the EM field in the frequency-space domain. As mentioned above, we have implemented the forward modelling in parallel where each compute node calculates the objective vectors of its subpopulation. This parallelization of the forward computation substantially improved the overall computational efficiency. EXAMPLES We present the inversion results for synthetic data sets generated using the forward modelling methods described above. Normally distributed 5 per cent random noise were added to the computed synthetics prior to inversion. A minimum absolute noise level of |${10^{ - 15}}$| V Am–2 was set for electric fields E and data below this minimum level were eliminated. The data were kept in log-amplitude and phase domain since these are smoothest of the available data forms (as compared with the real and imaginary, amplitude and phase, etc.) and do not exhibit flattening at low model resistivities (Wheelock et al.2015). In all examples, we assumed that the water layer is 1 km thick and its resistivity was kept fixed to 0.3 ohm-m. Since resistivity can span many orders of magnitude, we follow the usual approach for EM geophysics and parametrized the code to invert for |$\log{_{10}}$| (conductivity) instead of its linear counterpart (Ray & Key 2012). Example 1: resolution of a thin resistive layer Our first example shows the range of resistivity values obtained when inverting the inline electric field for 0.25-Hz data from the well-studied canonical reservoir model (Constable & Weiss 2006), which consists of a 100 m thick 100 ohm-m reservoir, 1 km below the seafloor, with surrounding 1 ohm-m sediments. The conductive ocean is 1 km deep, and the transmitter is located 25 m above the seafloor. A single receiver is positioned on the seafloor and CSEM responses were computed for transmitters located at 500 m intervals from 0 through 20 km. This example is similar to the one used by Key (2009). However, instead of running the inversion multiple times with different realizations of noise, we have shown the uncertainty present in the given noisy data for a single simulation. We believe that our approach is robust in the uncertainty estimation and it shows the ensemble of models which have achieved the target RMSE value. For inversion, the model was parametrized with 53 layers for which the thickness increased logarithmically with depth. Also, we set the search limits on the log-conductivities in the seafloor to vary between –2.3010 and 1.00 (corresponding to the linear resistivities of 0.1 to 200 ohm-m). The population size and number of generations for both NSGA II and regularized GA inversions were set respectively to 1280 and 2000. The probabilities of crossover and mutation (PC, Pm) were kept fixed at 0.7 and 0.01, respectively. Both the crossover index(η) and the mutation index (κ) were set to 20. The results for the 0.25 Hz data are shown in Fig. 2. Figs 2(a) and (b) compare the ensemble of acceptable models which have their RMSE values in the range 0.9–1.1 generated during the entire course of inversion run (gray) along with the estimated mean (magenta), median (cyan), the maximum likelihood (red) models and the search window used (black dashed lines) from NSGA-II and regularized GA, respectively. Both inversions found a smooth increase in resistivity at the reservoir zone. However, regularized GA sampled 6760 acceptable models whereas NSGA-II sampled 142 972 acceptable models. In Fig. 3, we compare the amplitude and phase residuals for 50 randomly selected acceptable models from two methods. All these models fit the observed data within the error limits and therefore demonstrate their equivalence. In addition, note that the residuals of these acceptable models (Fig. 3, bottom row) are centred around the zero mean, indicating that the inversion is not biased to either the short- or long-offset data, and that the model responses fit the data to within error bar (Key 2009). Figure 2. Open in new tabDownload slide Result of inversion of the inline E of the 0.25 Hz data using (a) NSGA II and (b) regularized GA for example 1. Gray curves are the models with the RMSE values 1. The mean, median, maximum likelihood and true models are shown, respectively, in magenta, cyan, red and blue. The black dashed lines are the search window used. Figure 2. Open in new tabDownload slide Result of inversion of the inline E of the 0.25 Hz data using (a) NSGA II and (b) regularized GA for example 1. Gray curves are the models with the RMSE values 1. The mean, median, maximum likelihood and true models are shown, respectively, in magenta, cyan, red and blue. The black dashed lines are the search window used. Figure 3. Open in new tabDownload slide Amplitude (top row) and phase (middle row) of the data sets generated for 0.25 Hz source signal (blue) are compared with the predicted data from the acceptable models (grey) obtained using NSGA II and regularized GA inversion approach. Normalized residuals (bottom row) of the model fit for log10 (amplitude) (blue dots) and phase (red dots) for all the models with RMSE value 1. Note that the vertical scales are the same of each row of plots. Figure 3. Open in new tabDownload slide Amplitude (top row) and phase (middle row) of the data sets generated for 0.25 Hz source signal (blue) are compared with the predicted data from the acceptable models (grey) obtained using NSGA II and regularized GA inversion approach. Normalized residuals (bottom row) of the model fit for log10 (amplitude) (blue dots) and phase (red dots) for all the models with RMSE value 1. Note that the vertical scales are the same of each row of plots. Fig. 4 shows the misfit versus roughness plot of all the solutions evolving through generations for the NSGA-II inversion. Different coloured dots represent the models belonging to that generation. The solutions are widely distributed in the objective space for the first generation and slowly converge to the most optimal set of solutions over generations. The trade-off between misfit and roughness in the final model ensemble is clear. The diversity preservation mechanism in NSGA-II helps to explore the model space more exhaustively compared to the regularized GA. The transverse resistance of a resistive target layer, defined as the vertically integrated resistivity of the layer (TVR), is the parameter best constrained in a CSEM inversion (MacGregor & Tomlinson 2014). We therefore plotted the TVR values of the target (reservoir) zone and the overburden zone for the ensemble of models obtained by NSGA-II and regularized GA in Fig. 5 to illustrate the variation in the final model ensemble. The diversity of the final model ensembles is considerably wider for the NSGA-II than the regularized GA indicating a wider sampling of model space. The TVR value of the models sampled through NSGA-II (Fig. 5a), have their values close to the true value as compared with the models obtained from regularized GA (Fig. 5b), which does not explore the model space well enough and fails to explore all feasible regions of the model space. Figure 4. Open in new tabDownload slide Evolution of all the solutions obtained using NSGA-II through generations. The dots in different colours represent different generations. Note that there was little change in the result after 900 generations. Figure 4. Open in new tabDownload slide Evolution of all the solutions obtained using NSGA-II through generations. The dots in different colours represent different generations. Note that there was little change in the result after 900 generations. Figure 5. Open in new tabDownload slide TVR plots of the ensemble of models obtained using (a) NSGA II and (b) regularized GA for the overburden layer and the reservoir layer. The red triangle represents the true value. The NSGA-II samples model space more effectively than GA, leading to a wider diversity of acceptable models, which encompass the true result. Figure 5. Open in new tabDownload slide TVR plots of the ensemble of models obtained using (a) NSGA II and (b) regularized GA for the overburden layer and the reservoir layer. The red triangle represents the true value. The NSGA-II samples model space more effectively than GA, leading to a wider diversity of acceptable models, which encompass the true result. Although it is evident that NSGA-II outperformed regularized GA in sampling the model space, it still remains to be addressed whether NSGA-II sampling is exhaustive enough for computing the true PDF. Although continuous sampling of the model space using real-coded GA with proper choices of crossover and mutation probabilities (Pc and Pm), crossover and mutation distribution indices (η and κ) and computation of the crowding distance in both objective and model space ensures that NSGA II exhaustively samples the model space, the method still tends to underestimate the true PDF (Li & Mallick 2015). This is in true for all stochastic optimization methods like GA, simulated annealing (SA), particle swarm optimization (PSO), etc. These methods can avoid the local minima by intelligently sampling the model space. But the models that are sampled away from the global minima by these methods do not usually contribute to the estimation of the posterior uncertainty, and in Bayesian sense they should if they have non-zero likelihood in the prior (Li & Mallick 2015). It may be argued the by increasing the probability of mutation (Pm) would, in theory, allow exhaustive sampling away from the global minima. In previous studies, Mallick (1995, 1999) however found that for finite population sizes, increasing Pm does not necessarily guarantee exhaustive sampling and instead, it may steer the solution away from the global minima. To estimate the true PDF, although it is necessary to employ an exhaustive sampling like the Gibbs sampler, Sen & Stoffa (1996), pointed out that these methods can still estimate the posterior probability density (PPD) function, which approximates the PDF. Additionally, by making several GA runs with different random sets of population and combining them all provide additional robustness to the PPD estimation. These issues are explored more in the later part of the paper. Example 2: uncertainty in the anisotropic inversion In our next example, we study the effects of anisotropy on the performance of NSGA II. We used the same canonical model of example 1, but with a VTI anisotropy in the overburden with a vertical resistivity of 2 ohm-m and horizontal resistivity of 1 ohm-m. Like example 1, we parametrized the inversion with 53 layers, each with two model parameters resulting in a total of 106 parameters. In this case we used the EMmod code (Hunziker et al.2015) for the forward modelling engine. We tested the model for the inline and joint (inline and broadside) configurations under anisotropic assumptions and inverted the electric field data for 0.1, 0.3 and 0.7 Hz. We used a population size of 1800 and maximum generations were set to 2000. Because the number model parameters are doubled compared to example 1, in this example, we followed the work of Li & Mallick (2015) and varied the PC, Pm,|${\rm{\eta }}$| and |${\rm{\kappa }}$| over generations to improve convergence as follows: $$\begin{eqnarray*} {P_C} = 0.7 - 0.1 \times \frac{t}{{{t_{\rm{max}}}}}, \end{eqnarray*}$$ (6) $$\begin{eqnarray*} {P_m} = \frac{1}{n} + \frac{t}{{{t_{\rm{max}}}}} \times \left( {1 - \frac{1}{n}} \right), \end{eqnarray*}$$ (7) $$\begin{eqnarray*} {\rm{\eta }} = 1.0 + 19.0 \times \frac{t}{{{t_{\rm{max}}}}}, \end{eqnarray*}$$ (8) and $$\begin{eqnarray*} {\rm{\kappa }} = 100 + t. \end{eqnarray*}$$ (9) In eqs (6)–(9), t is the current generation number, tmax is the maximum number of generations, and n is the total number of model parameters. Note from eqs (6) and (8), |${P_C}$| decreases and η increases with generation, permitting wide sampling of the model space at the beginning and slowly reduce it as the generation progresses and approaches the global optimum. The generation-dependent Pm and κ, used in eqs (7) and (9) ensures 1 per cent mutation in solutions out of the entire set of population, maintain diversity in the population at early stage, and slowly reduce it as the solutions reach the global optimum (for details, see Deb & Agrawal 1999; Li & Mallick 2015). The inversion results with the acceptable models for example 2 are shown in Figs 6 and 7. Figure 6. Open in new tabDownload slide Results of inverting inline E for the 0.1, 0.3 and 0.7 Hz data using NSGA II and for the Example 2. (a) The vertical resistivity. Different curves are the same as they are in Fig. 2(b). (b) Same as (a) but for the horizontal resistivity. Figure 6. Open in new tabDownload slide Results of inverting inline E for the 0.1, 0.3 and 0.7 Hz data using NSGA II and for the Example 2. (a) The vertical resistivity. Different curves are the same as they are in Fig. 2(b). (b) Same as (a) but for the horizontal resistivity. Figure 7. Open in new tabDownload slide Results of inverting inline and broadside E for the 0.1, 0.3 and 0.7 Hz data using NSGA II and for the Example 2. (a) The vertical resistivity. Different curves are the same as they are in Fig. 6. (b) Same as (a) but for the horizontal resistivity. Figure 7. Open in new tabDownload slide Results of inverting inline and broadside E for the 0.1, 0.3 and 0.7 Hz data using NSGA II and for the Example 2. (a) The vertical resistivity. Different curves are the same as they are in Fig. 6. (b) Same as (a) but for the horizontal resistivity. The inline measurements are dominated by the transverse magnetic (TM) mode component of the electric field and as a result, data are primarily sensitive to the vertical resistivity of the earth (Ramananjaona et al.2011). This is evident from Fig. 6(a) which shows that our inversion was successful in recovering the presence of an increase in resistivity associated with the reservoir layer, albeit a smoothed representation of this feature. Horizontal resistivity, shown in Fig. 6(b) shows a slight increase in the resistivity at the reservoir depth. This inability to resolve the horizontal resistivity within the reservoir is not surprising. Because the current density in the resistive reservoir layer is vertically polarized with little horizontal flow, sensitivity to the horizontal resistivity within such features is very low (Brown et al.2012). Next, we jointly inverted the data from inline and broadside configurations, results of which are shown in Fig. 7. The vertical resistivity profile (Fig. 7a) shows the presence of the reservoir layer, but again is smoothed out in the unconstrained inversion approach as expected given the regularized nature of the inversion. In common with many regularized inversions approaches the depth of the peak in resistivity is also slightly shallow (Key 2009; MacGregor & Tomlinson 2014). The background horizontal resistivity (Fig. 7b) is now better resolved than when only inline data are considered. Broadside data are more sensitive to the horizontal resistivity, and so their inclusion stabilizes the inversion, allowing both the target reservoir (in the vertical resistivity) and background structure (in the horizontal resistivity) to be resolved. For the isotropic inversion, using 8 nodes with 16 cores per node of Intel Broadwell @ 2.1 GHz with 128 Mb of memory, our implementation took 42 min. For the anisotropic inversion it took 17.7 hr. Computation times can however be substantially reduced by using more nodes. DISCUSSION In the examples considered above, we studied the effect of non-uniqueness associated with the single run of the CSEM inversion. As noted previously, GA estimates the PPD, which is not the true PDF, but an ensemble of acceptable models that explain the data equally well. Because models away from global minimum are not sampled as closely as they are near the global minimum, which in strict Bayesian sense they should be if they have a non-zero likelihood in the prior (Li & Mallick 2015), PPD is an approximation to the true PDF (Sen & Stoffa 1996). Additionally, genetic drift—a result of using finite population sizes (Goldberg 1989; Sen & Stoffa 1992), causes the models in the population to converge near a single minimum. This problem is more severe for the regularized GAs than NSGA-II (Fig. 5). Thus, to get a better estimate of non-uniqueness, we ran inversion several times with different random starting population. We consider the model to be the same as in Example 1 but with the reservoir layer of a resistivity of 20 ohm-m. The inversion was run 30 times with different starting populations and the mean models from three runs are shown in Fig. 8(a). The three models show smooth increase in resistivity at the reservoir depth. However, the extent of the higher resistivity zone is observed at different depth ranges in each result. We also applied Occam inversion (Constable et al.1987) in which the model shows the smooth resistivity peak corresponding to the reservoir depth. Fig. 8(b) shows the amplitude and phase responses of the mean and Occam inverted model along with the observed data. The predicted data from the models satisfactorily match the observed data suggesting that they are all equally valid. Fig. 9 shows the histogram for the depth at which the maximum resistivity value was obtained for the acceptable models from different NSGA-II runs. These results highlight the inherent poor depth resolution of the method and show the trade-off in resistivity values in the target layers. Figure 8. Open in new tabDownload slide (a) Mean resistivity models (red, cyan and magenta) obtained by running inversion three times with different starting population along with Occam inverted model (black) and true model (blue). (b) Amplitude and phase data of the models shown in (a). Figure 8. Open in new tabDownload slide (a) Mean resistivity models (red, cyan and magenta) obtained by running inversion three times with different starting population along with Occam inverted model (black) and true model (blue). (b) Amplitude and phase data of the models shown in (a). Figure 9. Open in new tabDownload slide Histogram showing the depth ranges at which the peak resistivity value was obtained for the acceptable models during the different runs of NSGA II. The true depth value is shown by the vertical red line. Figure 9. Open in new tabDownload slide Histogram showing the depth ranges at which the peak resistivity value was obtained for the acceptable models during the different runs of NSGA II. The true depth value is shown by the vertical red line. CONCLUSIONS The inversion algorithm proposed here is a global method for inverting marine CSEM data using genetic algorithm to estimate the subsurface resistivity. This inverse problem is non-linear with non-unique solutions and therefore, we have cast the problem as a multi-objective optimization to get an assessment of the uncertainty. Since all MOEA methods require large population sizes while dealing with such complex non-linear problems shown, the parallelization step suggested here is of crucial importance. We examined our proposed method on the synthetic noisy data sets generated from horizontally stratified earth models under different assumptions. We first tested it on the isotropic models to study the capability of the method to resolve the thin resistive layers representative of offshore hydrocarbons. We next tested it on models having anisotropy in the overburden and studied its effect on the inversion under different configurations. Our results confirm with the previous studies showing that both TE and TM mode data are required to recover the vertical and horizontal resistivities and reduce the uncertainty in the inversion. All examples discussed here gives the range of possible inverted models which are compatible with the given data set under given noisy conditions and considering only one out of the possible solutions that traditional regularized methods provide can lead to misleading interpretation. Furthermore, NSGA-II works with the random population of models whose model parameters are defined within a search window as compared to the traditional methods which requires defining the initial models. In all our examples, we used wide search windows, which, in turn, makes our proposed method a powerful tool when there is no or very little a priori information available. Therefore, the method proposed here can be used to generate the initial models for the subsequent 2-D/3-D inversions. Despite all the advantages, a major drawback of NSGA-II is its computational cost. To get a good assessment of uncertainty, the inversion should be run multiple times with different starting random population and the acceptable models obtained from all these inversions should be used to give a robust estimate of the subsurface resistivity. Although the parallelization used here makes application of the method feasible for 1-D problems, we still think that further investigations are necessary to improve the computational efficiency of our methodology to be feasible for 2-D/3-D applications. ACKNOWLEDGEMENTS We thank Dr Kerry Key and Dr Jürg Hunziker for making their codes DIPOLE1D and EMmod available under the open source. We also thank the School of Energy Resources and the Department of Geology and Geophysics of University of Wyoming for financial support. We acknowledge the use of computational resources (doi:10.5065/D6RX99HX) at the NCAR-Wyoming Supercomputing Center provided by the National Science Foundation and the State of Wyoming and supported by NCAR's Computational and Information Systems Laboratory. Finally, we thank the Editor Dr Ute Weckmann, Daniel Blatter and one anonymous reviewer whose constructive criticisms greatly improved the quality of the originally submitted manuscript. 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TI - Inversion of marine controlled source electromagnetic data using a parallel non-dominated sorting genetic algorithm
JF - Geophysical Journal International
DO - 10.1093/gji/ggz501
DA - 2020-02-01
UR - https://www.deepdyve.com/lp/oxford-university-press/inversion-of-marine-controlled-source-electromagnetic-data-using-a-In1Y01y2Aq
SP - 1066
VL - 220
IS - 2
DP - DeepDyve
ER -