With the recent increase in numbers of small and versatile low-power meteor radars, the opportunity exists to benefit from simultaneous application of multiple systems spaced by only a few hundred km and less. Transmissions from one site can be recorded at adjacent receiving sites using various degrees of forward scatter, potentially allowing atmospheric conditions in the mesopause regions between stations to be diagnosed. This can allow a better spatial overview of the atmospheric conditions at any time. Such studies have been carried out using a small version of such so-called multistatic meteor radars, e.g. Chau et al. (Radio Sci 52:811–828, 2017, https ://doi.org/10.1002/2016r s0062 25). These authors were able to also make measurements of vorticity and divergence. However, measurement uncertainties arise which need to be considered in any application of such techniques. Some errors are so severe that they prohibit useful application of the technique in certain locations, particularly for zones at the midpoints of the radars sites. In this paper, software is developed to allow these errors to be determined, and examples of typical errors involved are discussed. The software should be of value to others who wish to optimize their own MMR systems. Keywords: Radar, Meteor, Errors, Multistatic, Specular, Reflection, Scatter Furthermore, remote receiving sites can be established Introduction relatively easily, so that one transmitter can be used by Monostatic interferometric meteor radars have been in several receivers. GPS technology allows locking of the existence for many decades, and their basic principles phases between the transmitter and receiver sites. Chau are well described in various texts (e.g. Hocking et al. et al. (2017) has demonstrated the application of such 2001; 2016, Chapter 10). A substantial increase in meteor techniques. detection efficiency occurred in the late 1990s and early The radars usually transmit on a broad beam and 2000s following the development of better techniques, receive on a cluster of receiver antennas—often 5 (e.g. spurring an increase in the deployment of such radars. Jones et al. 1998). Each receiver records separate signals, These radars rely on the fact that meteor trails are very and by cross-correlating the complex signal measured on effective radio-wave scatterers, and so radars of mod - each receiver, interferometry may be used to determine est power (6–30 kW peak pulsed) can be used to detect echo location angles. Combined with range information, them. As a result, there are many such radars worldwide. this allows complete location of the meteor trail. The In some cases, such as in Europe and Scandinavia, there signal may be further interrogated to determine decay are a significant number of such radars within relatively times, atmospheric temperatures, atmospheric wind close proximity (a few hundred to 1000 km or so sepa- speed and other significant atmospheric parameters per - ration) and so it has been proposed that these radars taining to the height region 75–100 km altitude. can be used in concert to study regions of the atmos- However, care is always needed, as the signals are quite phere between the radars, in addition to studies of the short-lived and can easily be confused with other impul- meteor field in the immediate proximity of each radar. sive signals like lightning and man-made ignition sys- tems. Great care is needed to distinguish meteors from *Correspondence: whocking@uwo.ca other short-lived phenomena, as discussed by Hocking Department of Physics and Astronomy, University of Western Ontario, et al. (2001). 1151 Richmond St. North, London, ON N6A 3K7, Canada © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creat iveco mmons .org/licen ses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Hocking E arth, Planets and Space (2018) 70:93 Page 2 of 13 Most interferometric meteors working at HF, MF Height resolution and lower VHF frequencies (the most common fre- Height resolution involves two separate aspects of the quencies for these types of radars) rely on so-called radar: (1) the pulse length and (2) the angular resolution. specular reflections from the meteor trail. For mono - For meteors close to overhead of a monostatic radar, only static systems, this means that the trail must be ori- (1) matters, but as soon as the meteors occur at significant entated perpendicularly to the vector from the radar off-zenith angles, angular effects contribute to the height to the midpoint of the trail, so that the signal reflects resolution. For example, a meteor at 45° from overhead back as if being bounced off a mirror. For bistatic sys - has vertical resolution of approximately √{((Δz/√2) + (r tems, in which the transmitter and the receiver are not Δθ/√2) }, Δz being the pulse resolution and Δθ being the co-located, the angle of incidence of the wave from angular resolution. If the angular resolution Δθ is ~ 1°–2° the transmitter relative to the meteor trail must equal (as is typical in many such systems), then the contribu- the angle of reflection of the wave back towards the tion of the angular component for a 1° angular resolution receiver location. We will develop our theory using is of the order of 1.5 km. Meteor radars commonly use the bistatic case: the monostatic case is simply a limit- a pulse length of a similar value—typically 2 km—since ing case of the bistatic case, where the transmitter and there is no real improvement in resolution by using receiver become coincident. shorter pulses. (The angular effect often dominates at Resolution is always an issue, since many meteors angles where meteors are most easily detected.) Wider are detected at significant angles from zenith (up to 60° pulses also require a narrower frequency allocation band from zenith), and resolution degrades at lower angles. and allow the use of narrower-band receiver filters (hence With applications involving bistatic and multistatic reducing noise and interference). systems, errors become even more of an issue. In this For a monostatic radar, the delay time between trans- paper, we address these different errors and develop mission of the pulse by the transmitter and reception software to study them. The code is written in the by the co-located receiver is used to determine the python language and is presented in the Appendix. It range of the target through the relation r = cΔt/2. Often may be used freely by the reader, provided it is properly receiver systems are even calibrated in terms of “range” acknowledged in any correspondence and publications. by using this formula. For bistatic systems, the situation In the following discussions, we refer to the software is a little more complex. Figure 1 shows the delay time frequently. associated with various paths. Any signal that moves Our main interest here is in determination of errors from the transmitter to a target and back to the receiver involving (1) height resolution and (2) wind-velocity has the same time delay as long as the target is located accuracy. These are discussed in the following "Height on a common ellipse with foci at the transmitter and resolution" and "Velocity measurements" sections, fol- receiver. In Fig. 1, two closely located ellipses with the lowed by some discussion in the "Discussion" section same foci are shown. If the targets move, then they and finally conclusions in the "Conclusions " section. may follow the arrows indicated at scatterers A, B and Fig. 1 Sketch showing the nature of scattering in bistatic mode, showing a vertical cut through two surfaces of constant “range” delay Hocking E arth, Planets and Space (2018) 70:93 Page 3 of 13 C. For meteor targets, each of the meteor trails must With regard to range resolution, turn to Fig. 2. We be aligned with an orientation perpendicular to a line concentrate on the vertical resolution due to the pulse which bisects the position vectors from the transmitter length. to the target and the target to the receiver. (The case for a non-specular scatterer is discussed in Hocking et al. Pulse‑length effect on Δz 2016, Fig. 3.20.) A meteor trail is shown orientated perpendicular to the In Fig. 2, scatterer A crosses from the outer ellipse thick solid arrow, where the arrow bisects the lines from to the inner one, and so sees a decrease in path length the transmitter and the receiver. The elevation of the as time progresses. The time-rate-of-change of path meteor trail from the perspectives of the transmitter and length appears as a Doppler shift in the received signal the receiver is β and β , respectively. The thick arrow is 1 2 and may be used to calculate a component of the wind at an elevation of ZA = (β + β )/2 and so bisects the lines 1 2 strength. Scatterer B moves almost parallel to the inner from the transmitter and the receiver (thereby ensuring ellipse and so shows no rate-of-change of path length, that the meteor is specularly reflecting). Note that mete - and so appears stationary to the receiver (no Doppler ors at the positions A, B and C in Fig. 1 will all have dif- shift). No useful horizontal velocity component can be ferent orientations in order to satisfy specular reflection. measured here, although the radar will be quite sensi- [It is conceivable that when multiple meteor transmitters tive to vertical movement. Scatterer C moves from the and receivers are used, the locations of the transmitters inner to the outer ellipse and so shows a change in path and receivers could be arranged so that under some cir- length which can be translated to velocity component. cumstances a single meteor might reflect signals along Velocity sensitivity will be discussed in the next section. different ray paths, thereby enabling two measurements of the Doppler shift and hence determination of a full Fig. 2 Orientation of a meteor trail, and path lines of ray vectors, for a general meteor. The radio pulse travels from the transmitter to the meteor trail and back to the receiver on the left. Note that only one receiver is drawn—normally there are more (typically 5) at separations of only one or two wavelengths from each other Hocking E arth, Planets and Space (2018) 70:93 Page 4 of 13 vector at one place. As a warning, this does not work. Even if two reflections occur from the same trail, it will generally be from different portions of the trail, and tur - bulence effects will result in the different portions mov - ing at different speeds. Such a procedure can be used to estimate turbulent strengths, however (e.g. Roper 1966).] Now turn to the inset of Fig. 2. The meteor trail passes through P. We consider the instant at which the centre of the transmitted pulse rests at P. The leading half of the pulse has already passed P and is now at B, where the dis- tance BP is cΔt/2. The trailing half of the pulse is in the region between A and P, which also has a length cΔt/2. The distance along the direction MP is therefore cΔt/2 cos (ZAR) where ZAR is the bisection angle indicated in the figure and so equals (β − β )/2. The vertical resolu - 2 1 tion is then Δz as shown in the figure, which is just MP sin(ZA). Then, the resolution due to the pulse length is Fig. 3 a A simple 2-antenna interferometer, for illustrative purposes. b Spatial resolution corresponding to an angular resolution δθ �z = c�t/2 cos (ZAR) sin (ZA) p (1) The notations ZAR and ZA match the coding in the Appendix. Note that for a scatterer like B in Fig. 1, the multiplicity of possibilities exist due to angular ambigui- transverse separation between A and B is quite large and ties. Up to 6 and even 10 possible locations occur. Then, could be considered to some extent as a contribution to the fifth (central) antenna is used to determine which of the angular resolution uncertainty in the case of volume the ambiguous positions is the correct one. Having found scatter. However, because the meteor is a discrete target, this correct position, a simulation is performed in real there is no such contribution here. time in the radar controller to determine what the phase If the reader runs the code in the Appendix, the first differences between all the antenna pairs should be, and two graphs produced will show the variation of Δz as this is compared with the actual phase differences. Invar - a function of position of the meteor. At an altitude of iably the differences are not zero. The phase differences 90 km, the resolution varies from cΔt/2 at the overhead can vary from the simulation by several tens of degrees. point to a small fraction of cΔt/2 at low elevations. We This is a result of noise, interference and imperfections will not plot the graph here in order to save space. in the assumption of specular reflection. The spread in phase errors limits the angular accuracy with which the target can be located. Angular vertical resolution With the SKiYMET system, this phase difference varia - Figure 3 shows two antennas of an interferometer. Radar tion is generally limited to 35°; meteors with larger maxi- signals enter from the right, as shown by the two arrows. mum phase errors are discarded. This value has been set We assume that the target is far enough away that the empirically—large values allow acceptance of more mete- two rays can be considered as parallel. The two rays will ors—even doubling the counts—but also allows many be in phase along the perpendicular line, since they both other false targets, and degrades the height resolution were reflected coherently from the same target, and orig - noticeably. Limits of less than 35° restrict the acceptable inated from the same pulse. The phase difference at the meteors too severely. two receivers is therefore 2πξ/λ. So assuming that each meteor recorded has this maxi- In application of interferometric techniques in 3D, mum phase error, we may place some limits on the angu- there are at least 3 and generally 4 or 5 receivers. The lar resolution. This will be an upper limit, since often the receivers are placed at least one wavelength apart, to maximum phase difference is less than our prescribed reduce antenna–antenna coupling, but this introduces limit. redundancies into the possible directions. Therefore, Then, from Fig. 3, the true phase delay can be written there are usually more antennas than might be consid- as Δφ = 2πξ/λ = 2π Dcosθ/λ. However, we seek the possi- ered necessary, in order to resolve ambiguities. The SKi - ble errors in Δφ, which we will denote δ(Δφ). Differentia - YMET system is a typical example (Hocking et al. 2001) tion produces and uses 5 antennas in the form of a cross (e.g. see Hock- ing et al. 1997, Fig. 1). The four outermost antennas are δ(�φ) =[(−2πD sin θ)/]δθ . (2) used to determine the possible meteor locations, but a Hocking E arth, Planets and Space (2018) 70:93 Page 5 of 13 Total resolution Let the distance to the target meteor be r, where we con- The program in the Appendix plots δz as a function of sider this to be about the same for each antenna, since position, but we will not show it here in order to save the target is over 90 km away and the antennas spacing space. Rather, we combine the effects of Eqs. (1) and (3) is only a few metres. If the meteor is at height z, then by plotting the normalized total error, which is found by sinθ = z/r, so (2) can be rewritten as adding the squares of (1) and (3) (after each is normal- δ(�φ)r ized relative to cΔt/2), and taking the square root. The δθ =− 2πDz result is shown in Fig. 4, for the case of a transmitter at x = 350 km and the receiver at x = 650 km. Only values up to 3 times cΔt/2 are plotted—poorer resolutions are From Fig. 2b, δz is clearly (rδθ) cosθ, so that of no value to us and appear as a brown/dark-red colour. δ(�φ)r (At large distances, the angular effect of the system reso - δz =− cos θ (3) 2πDz lution clearly translates to very large vertical resolutions.) Clearly only the region immediately above the receiver array produces data with suitable resolution. Meteors In the program in the Appendix, D/λ is just the vari- further away (including over the transmitter) offer little able “antrx”, and we take δ(Δφ) = 35° (converted to radi- useful information. ans). Note that r is the distance from the centre of the The fact that the transmitter is 300 km from the receiv - receiver array and is unrelated to the transmitter posi- ers also obviously will result in significant loss of useful tion. This introduces asymmetries that will appear later. power before the radar signal reaches the meteors in the The variable δz occurs in the Appendix as the variable vicinity of the receiver, and in addition the transmitter “DTANG” and is then normalized relative to cΔt/2. Fig. 4 Vertical resolution of the system relative to the “vertical backscatter resolution”. Only the data between 75 and 110 km altitude are of interest here. See text for details Hocking E arth, Planets and Space (2018) 70:93 Page 6 of 13 pulses that reach these meteors will need to be trans- mitted at low elevations (in this case atan(300/90) = 73° from zenith). This will further reduce the signal arriving overhead because the signal will be transmitted at angles where the polar diagram of the transmitter antenna has weak values. Velocity measurements Now we turn to the topic of measurements of the drift speed of the trail, which may be used to determine the wind velocities over the radar when combined with measurements with other radars. We will look at the Doppler shift of the reflected signal as it arrives at the receiver. For a monostatic system, this is referred to as Fig. 5 Doppler-shifted velocity components measured by the the “radial velocity”; while the term is not really pertinent radar as a function of horizontal displacement at 90 km range, for a here, we will use it in a loose sense at times. transmitter at x = 350 km and a receiver at x = 650 km. The speeds are To do this, we return to Fig. 1. We will consider a gen- normalized by division by the true horizontal velocity, so approach eralized point representing any of the scatters A, B or C unity at the edges. The radar-measured parameter is referred to as v , by analogy with the backscatter case, but it is not truly a radial in the figure and calculate the distance from the transmit - rad velocity. Lines are drawn horizontally near the zero-point on the ter to the scattering point and on to the receiver. Then, abscissa—values within these lines will have measured speeds less we will allow the scattering point to move horizontally a than 0.05 of times the true horizontal speed, making inversion of the distance vδt, where δt is a small time interval (typically data difficult and potentially unreliable 0.1 s), and v is the velocity of interest. Then, the change in distance is divided by the time interval to give the rate of change of distance with time. This will result in a Dop - Discussion pler shift of the radiowaves, but because the meteor trail The results of Fig. 5 suggest that reliable velocities are acts like a mirror, the measured Doppler shift will appear likely to the right of x = 550 km. The results of Fig. 4 as twice the speed of the meteor. We therefore need to suggest that the resolution is unreliable at values of x divide by 2 to get the true “radial” velocity. to the left of 550 km and to the right of about 750 km. The results at 90 km altitude for a transmitter–receiver It would therefore appear that the bistatic radar is of 300 km separation are seen in Fig. 5. Results are nor- only really useful above the receiver system, between malized relative to the true horizontal velocity. The trans - x = 550 and 750 km, i.e. within ± 100 km of the receiver mitter was at x = 350 km and the receiver at x = 650 km. system. Within this region, the ratio of the measured As seen, at low elevations (x = 0 and x = 1000) the radar-determined component relative to the true speed measured velocity approaches that of the true horizon- changes rapidly, potentially introducing more errors. tal velocity, confirming that the correct normalization It should also be noted that the calculations of Fig. 5 has been used. Some extra lines have been added to the do not include height error considerations explicitly, graph, as discussed in the figure caption. Data between so uncertainties in meteor trail location at low zeniths x = 450 and x = 550 km give very small Doppler-shifted could add further errors. velocities and could have substantial errors in inversion. In the "Pulse-length effect on Δz " section, it was The Doppler-shifted values also change quite sharply as a noted that the meteor trail is a specular reflector, while function of distance beyond 550 km. our height-error determinations were done on the basis of an assumed volume scatter. Because the meteor Hocking E arth, Planets and Space (2018) 70:93 Page 7 of 13 trail is discrete, it is possible to essentially decon- velocity inversion limitations under various circum- volve the pulse received at the receiver and determine stances. Preliminary results suggest that it is not possi- the true trail range to better accuracy than the theory ble to use meteors with confidence over some areas of suggests, i.e. the position of the peak as a function of the sky, while the region immediately above the receiver “range” will be a good representation of the true posi- produces the most reliable data. However, software is tion. This could help reduce the height uncertainty provided to allow users to probe their own particular but will, however, require that the signal is digitized situations more carefully and perhaps fine-tune their sys - at higher resolution than 2 km. However, at distances tems for optimum performance. more than 100 km to the left or right of the receiver, the Authors’ contributions dominant cause of worsening resolution is the angular The author read and approved the final manuscript. effect (" Angular vertical resolution" section), so the cor- rections for range achieved by deconvoution offer only Acknowledgements limited opportunity for improvement. Support for this work was provided by Natural Sciences and Engineering Finally, we recall that the losses in effective power Research Council of Canada, Grant # 121401-2013RGPIN, and by Mardoc Inc., Canada. from the transmitter due to the long ranges involved and the low elevations required for the transmitter Competing interests pulse passage will result in power losses in excess of The author declares that he has no competing interests. 10 and up to 15 dB. Procedures might be developed to Availability of data and materials direct the transmitter signal more strongly at low eleva- No data are presented—the work is theory and simulations. The software used tions, but such considerations are beyond the scope of to determine the results is provided as the Appendix. this paper. Ethics approval and consent to participate Not applicable. Conclusions Funding This paper sets limits on the capabilities of bistatic Work is supported by the Natural Sciences and Engineering Research Council meteor radars and provides software that may be used of Canada, Grant # 121401-2013RGPIN. to investigate range-height errors, angular errors and Hocking E arth, Planets and Space (2018) 70:93 Page 8 of 13 Appendix Python program to determine errors and Doppler-shifted velocities for a bistatic meteor radar. # -*-coding: utf-8 -*- """ Created on Sun May 29 2016 @author: W.K. Hocking. Copyright May 31 2016. """ import numpy as np import matplotlib import numpy as np import matplotlib.cm as cm import matplotlib.mlab as mlab import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D matplotlib.rcParams['xtick.direction'] = 'out' matplotlib.rcParams['ytick.direction'] = 'out' # Tx nd Rx stations, spacing, step sizes etc # (may have xspace = 0 -monostatic) # xspace is the distance between the transmitter # and receiver (km). A superspace around the transmitter # and receiver of length 1000 km is allowed for plotting. xspace=300.0 xmid = 1000.0/2.0 # Tx x1=xmid-xspace/2.0 # Rx x2=x1+xspace # Assume a cross-structure for the receiver antennas. # spacing between extreme Rx antennas in wavelengths antrx = 4.5 # steps in km in horizontal and vertical directions deltax = 5.0 deltaz = 1.0 x = np.arange(0.0, 1000.0, deltax) z = np.arange(50.0, 110.0, deltaz) # maximum allowed phase difference between receivers in degrees phdifd = 35.0 phdif = phdifd/180.0*np.pi Hocking E arth, Planets and Space (2018) 70:93 Page 9 of 13 # Resolution analysis # pulse length (km) = 4.0, resolution = 2.0 km (ct/2) ct=4.0 # create a mesh X, Z = np.meshgrid(x,z) # direction of perpendicular to meteor alignment # ZA is angle anticlockwise from horizontal ZA = (np.arctan2(Z,(X-x1)) + np.arctan2(Z,(X-x2)))/2.0 # angle between incoming and reflected rays # at the meteor (absolute only) ZAR = np.abs(((np.arctan2(Z,(X-x2))) - np.arctan2(Z,(X-x1)))) # divide by 2 to get half-angle ZAR = ZAR/2.0 # vertical resolution due to pulse is 0.5*abs(ct/cos(zar)*sin(za)) # normalize relative to ct/2.0 ZRES = (0.5*ct/(np.cos(ZAR)) * np.sin(ZA))/(0.5*ct) plt.figure() cmap = cm.get_cmap() #CS = "contour set" # FILLED colour contour plot, 20 colours -use contourf CS = plt.contourf(X, Z, ZRES, 20,cmap=cmap) # Other options used in tests -plot ZA or ZAR separately #CS = plt.contourf(X, Z, ZR, 20,cmap=cmap) #CS = plt.contourf(X, Z, ZAR, 20,cmap=cmap) #CB = Colour Bar CB = plt.colorbar(CS) CB.set_label('Vertical Resolution relative to case of backscatter') plt.xlabel('Horizontal Distance') plt.ylabel('Altitude') plt.show() # Now do angular resolution effect.. Hocking E arth, Planets and Space (2018) 70:93 Page 10 of 13 # add angular effect -- receiver at x2 # sin(theta) = zdivided by radial distance # Then mutiply by radial distance to get tangential resolution # -hence range-effect appears as squared DTANG = np.abs(phdif/(2.0*antrx*np.pi*(Z)) * (((X-x2)**2 + Z*Z)) ) # multiply by cosine of elevation to convert to a vertical # resolution -max effect when elevation = 0 degrees, no vertical # effect overhead (all effect is in horizontal when overhead). DTANG = DTANG * np.abs((X-x2)/(np.sqrt((X-x2)**2 + Z*Z))) # normalize relative to ct/2 DTANG = DTANG/(0.5*ct) # graph plt.figure() cmap = cm.get_cmap() #CS = "contour set" # FILLED colour contour plot, 20 colours -use contourf CS = plt.contourf(X, Z, DTANG, 20,cmap=cmap) CB = plt.colorbar(CS) CB.set_label('Vertical Resolution -angular effect relative to case of backscatter') plt.xlabel('Horizontal Distance') plt.ylabel('Altitude') plt.show() # replot contours with limits between 0 and 3 plt.figure() cmap = cm.get_cmap() #CS = "contour set" # FILLED colour contour plot, 10 colours -use contourf # set contour levels so only goes to specified max rather than use default levels = np.arange(0.0, 3.25, 0.25) CS = plt.contourf(X, Z, DTANG,levels,cmap=cmap,extend="both") # set any value over (>) maximum specified level to brown. CS.cmap.set_over('brown') Hocking E arth, Planets and Space (2018) 70:93 Page 11 of 13 CB = plt.colorbar(CS) CB.set_label('Vertical Resolution -angular effect relative to case of backscatter') plt.xlabel('Horizontal Distance') plt.ylabel('Altitude') plt.show() # combine resolutions DRCOMB = np.sqrt(DTANG*DTANG + ZRES*ZRES) plt.figure() cmap = cm.get_cmap() # set contour levels so only goes to specified max rather than use default levels = np.arange(0.0, 3.25, 0.25) #CS = "contour set" CS = plt.contourf(X, Z, DRCOMB, levels, cmap=cmap, extend="both") CS.cmap.set_over('brown') CB = plt.colorbar(CS) CB.set_label('Vertical Resolution -combined effect relative to case of backscatter') plt.xlabel('Horizontal Distance') plt.ylabel('Altitude') plt.show() # =============================== # Radial velocity analysis #*** Note at this time, error in LOCATING the scattering point is not included # - should be included for full analysis -see del-elevation discussion above # -and only keep angular parts. # create a mesh X, Z = np.meshgrid(x,z) # horizontal speed, m/s v0=10.0 deltat=0.1 Hocking E arth, Planets and Space (2018) 70:93 Page 12 of 13 # total distance covered by radiowave from Tx to Rx at t=0 Z1 = np.sqrt((X-x1)*(X-x1) + Z*Z) + np.sqrt((X-x2)*(X-x2) + Z*Z) # total distance covered by radiowave from Tx to Rx at t=deltat (particle has moved # horizontally by v0*deltat to right) Z2 = np.sqrt((X+v0*deltat-x1)*(X+v0*deltat-x1) + Z*Z) Z2 = Z2 + np.sqrt((X+v0*deltat-x2)*(X+v0*deltat-x2) + Z*Z) # rate of change of total distance with time -divide by 2 because a reflection # is involved (i.e. imagemoves at twice the speed of the mirror (meteor trail). VR = (Z1-Z2)/deltat/2.0 # normalize VR = VR/v0 # line graph at 90 km.. z1=90.0 z1d1 = np.sqrt((x-x1)*(x-x1) + z1*z1) + np.sqrt((x-x2)*(x-x2) + z1*z1) z1d2 = np.sqrt((x+v0*deltat-x1)*(x+v0*deltat-x1) + z1*z1) + np.sqrt((x+v0*deltat- x2)*(x+v0*deltat-x2) + z1*z1) # rate of change of total distance with time -divide by 2 because a # reflection is involved. zz = (z1d1-z1d2)/deltat/2.0 # normalize zz = zz/v0 # plots... line graph and contour # also plot staions and # cutoffs approx where v_rad = 0.05 v0. xlim=[0.0,1000.0] zlim1=[-0.05,-0.05] zlim2=[0.05,0.05] xstat1 = [x1,x1] xstat2 = [x2,x2] zstat1 = [-1.0,1.0] xstat3 = [x1+0.3*xspace,x1+0.3*xspace] xstat4 = [x2-0.3*xspace,x2-0.3*xspace] plt.figure() # 1D line plot at z1 km... Hocking Earth, Planets and Space (2018) 70:93 Page 13 of 13 plt.plot(x,zz) plt.plot(xlim,zlim1) plt.plot(xlim,zlim2) plt.plot(xstat1,zstat1) plt.plot(xstat2,zstat1) plt.plot(xstat3,zstat1) plt.plot(xstat4,zstat1) plt.xlabel('Horiz. distance (km) at 90 km ** no scatterer location error added**') plt.ylabel('Normalized v_{rad}') plt.show() plt.figure() cmap = cm.get_cmap() #CS = "contour set" # FILLED colour contour plot, 10 colours -use contourf CS = plt.contourf(X, Z, VR, 20,cmap=cmap) CB = plt.colorbar(CS) CB.set_label('Normalized radial velocity') plt.xlabel('Horizontal Distance') plt.ylabel('Altitude') Jones J, Webster AR, Hocking WK (1998) An improved interferometer design for use with meteor radars. Radio Sci 33:55–65 Publisher’s Note Roper RG (1966) Atmospheric turbulence in the meteor region. J Geophys Res Springer Nature remains neutral with regard to jurisdictional claims in pub- 71:5785–5792 lished maps and institutional affiliations. Received: 28 February 2018 Accepted: 15 May 2018 References Chau JL, Stober G, Hall CM, Tsutsumi M, Laskar FI, Hoffmann P (2017) Polar mesospheric horizontal divergence and relative vorticity measurements using multiple specular meteor radars. Radio Sci 52:811–828. https ://doi. org/10.1002/2016R S0062 25 Hocking WK, Thayaparan T, Jones J (1997) Meteor decay times and their use in determining a diagnostic mesospheric temperature–pressure parameter: methodology and one year of data. Geophys Res Lett 24:2977–2980 Hocking WK, Fuller B, Vandepeer B (2001) Real-time determination of meteor- related parameters utilizing modern digital technology. J Atmos Solar Terr Phys 63:155–169 Hocking WK, Röttger J, Palmer RD, Sato T, Chilson PB (2016) Atmospheric radar: application and science of MST radars in the earth’s mesosphere, stratosphere, troposphere, and weakly ionized regions. Cambridge University Press, Cambridge. https ://doi.org/10.1017/97813 16556 115. ISBN 9781316556115
Earth, Planets and Space – Springer Journals
Published: Jun 5, 2018
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