Error analyses of a multistatic meteor radar system to obtain a 3-dimensional spatial resolution distribution

： In recent years, the concept of multistatic meteor radar systems has attracted the attention of the atmospheric radar community, focusing on the mesosphere and lower thermosphere (MLT). Recently, there have been some notable experiments using multistatic meteor radar systems (Chau et al., 2019; Spargo et al., 2019; Stober and Chau, 2015; Stober et al., 2018). 15 Good spatial resolution is vital for meteor radars because nearly all parameter inversion processes rely on the accurate location of the meteor trail reflecting points. It is timely then for a careful discussion focussed on the error distribution of multistatic meteor radar systems. In this study, we discuss the measurement errors that affect the spatial resolution and obtain the resolution distribution in 3-dimensional space for the first time. The spatial resolution distribution can both help design a multistatic meteor radar system and improve the performance of existing radar systems. Moreover, the spatial resolution 20 distribution allows the accuracy of retrieved parameters such as the wind field to be determined.


Introduction
The mesosphere and lower thermosphere (MLT) is a transition region from the neutral to the partially ionized atmosphere. It is dominated by the effects of atmospheric waves, including planetary waves, tides and gravity waves. It is also a relatively poorly sampled part of the Earth's atmosphere by ground based instruments. One widely used approach is the meteor radar 25 technique. The ablation of incoming meteors in the MLT region, i.e., ~80 -110 km, creates layers of metal atoms, which can be observed from the ground by photometry or lidar (Jia et al., 2016;Xue et al., 2013). During meteor ablation, the trails caused by small meteor particles provide a strong atmospheric tracer within the MLT region that can be continuously detected by meteor radar regardless of weather conditions. Consequently, the meteor radar technique has been a powerful tool for studying MLT for decades (Hocking et al., 2001;Holdsworth et al., 2004;Yi et al., 2018). Most modern meteor radars are 30 monostatic and this has two main limitations in retrieving the complete wind field. Firstly, limited meteor rates and relatively https://doi.org/10.5194/amt-2020-353 Preprint. Discussion started: 14 September 2020 c Author(s) 2020. CC BY 4.0 License. low measurement accuracies necessitate that all measurements in the same height range be processed to calculate a "mean" wind. Secondly, traditional monostatic radars retrieve wind fields based on the assumption of a homogenous wind field in the horizontal direction and a zero wind in the vertical direction.
The latter conditions can be relaxed if the count rates are high and the detections are distributed through a representative range 35 of azimuths. If this is the case, a version of a Velocity Azimuth Display (VAD) analysis as first applied to scanning weather radars for longer period motions can be applied by expanding the zonal and meridional winds using a truncated Taylor expansion (Browning and Wexler, 1968). This is because each valid meteor detection yields a radial velocity in a particular look direction of the radar. The radar is effectively a multi-beam Doppler radar where the "beams" are determined by the meteor detections. If there are enough suitably distributed detections in azimuth in a given observing period, the Taylor 40 expansion approach using cartesian coordinates yields the mean zonal and meridional wind components ( 0 , 0 ), and the stretching ( − ) and shearing ( + ) deformations of the wind field from an analysis of the radial velocities. If a measure of the vertical wind is available, then the horizontal divergence ( + ) can also be obtained (assuming a uniform vertical wind over the observing volume). Generally, meteor radars do not provide a reliable measure of the vertical wind component. In addition, because the radar can only retrieve the wind projection in the radial direction as measured from the 45 radar, the vorticity ( − ) of the wind field is not available. This is common to all monostatic radar systems and a discussion of measurable parameters in the context of multiple fixed beam upper atmosphere Doppler radars is given by (Reid, 1987). Even by releasing the assumption of a homogeneous wind field and using the more advanced Volume Velocity Processing (VVP) (Philippe and Corbin, 1979) to retrieve the wind field, the horizontal gradients of the wind field cannot be recovered due to the lack of vorticity information. To obtain a better understanding of the spatial variation of the MLT region 50 wind field, larger area observations (and hence higher meteor count rates) and measurements of the non-homogenous wind field are needed. An extension of the traditional monostatic meteor technique is required to satisfy these needs.
To resolve the limitations outlined above, the concept of multistatic meteor radar systems, such as MMARA (multi-static and multi-frequency agile radar for investigations of the atmosphere) (Stober and Chau, 2015) and SIMO (single input multiple output) (Spargo et al., 2019), MIMO (multiple input multiple output radar)  have been designed and 55 implemented (Stober et al., 2018). Multistatic systems can utilize the forward scatter of meteor trails, thus providing another perspective for observing the MLT. Multistatic meteor radar systems have many advantages over traditional monostatic meteor radars, such as obtaining higher-order wind field information and covering wider observation areas. There have been some particularly innovative studies using multistatic meteor radar systems in recent years. For example, by combining MMARA and the continuous wave multistatic radar technique (Vierinen et al., 2016), Stober and Chau et al. built a 5-station total 7-link 60 multistatic radar network covering an approximately 600 km×600 km region in Germany to retrieve an arbitrary nonhomogenous wind field with a 30 km×30 km horizontal resolution (Stober et al., 2018). Stober et al. used two adjacent traditional monostatic specular meteor radars in northern Norway to obtain horizontal divergence and vorticity (Chau et al., https://doi.org/10.5194/amt-2020-353 Preprint. Discussion started: 14 September 2020 c Author(s) 2020. CC BY 4.0 License. 2017). Other approaches, such as the novel multistatic meteor radar data processing method  and the compressed sense method in MIMO sparse signal recovery , are described in the references in these papers. 65 Analysing spatial resolution in regions of interest is a fundamental but difficult topic for meteor radar systems. Meteor radar systems transmit radio waves and then receive radio waves using a cluster of receiver antennas; commonly five antennas as in the Jones et al. configuration (Jones et al., 1998). By analysing the cross correlation of received signals, we can determine the angle of arrival (AoA), that is, the zenith angle θ and azimuth angle ϕ. By measuring the wave propagation time, we can obtain the range. Most meteor radar systems rely on specular reflections from meteor trails. Thus, by combining the AoA and 70 the range information and then using geometric analysis, we can determine the location of meteor trails. Accurately locating meteor trail reflection points is important since atmospheric parameter retrieval (such as the wind field or the temperature) depends on the location information of meteor trails. The location accuracy, namely, the spatial resolution, determines the reliability of the retrieved parameters. For multistatic meteor radar systems that can relax the assumption of a homogenous horizontal wind field, the resolution distribution becomes a more important issue because the horizontal spatial resolution 75 affects the accuracy of the retrieved horizontal wind field gradient.
There are a number of studies that discuss AoA measuring errors (Kang, 2008;Vaudrin et al., 2018;Younger and Reid, 2017).
However, those error analyses emphasize the receiving antennas and seldom discuss the influence of a multistatic configuration on the spatial resolutions. Moreover, those analyses consider the error propagation starting from the original signals received by the cluster of antennas; therefore, the results of AoA errors become too intricate to utilize in further resolution analyses. 80 Hocking developed a vertical resolution analysis model in a 2-dimensional baseline vertical plane (Hocking, 2018), which simplifies the error propagation process in receiver antennas to emphasize the bistatic configuration. However, Hocking's model (HM hereafter) has nothing to do with the distance between the transmitter and the receiver which will be explained in this paper. Moreover, for practical purposes, the 3-dimensional spatial distribution of both horizontal resolution and vertical resolution should be considered. 85 Although multistatic meteor radar systems have developed well experimentally in recent years, the reliability of retrieved atmospheric parameters lacks discussion. To better understand the reliability of the obtained atmospheric parameters, quantitative error analyses are necessary. In this paper, we analyse the multistatic meteor radar resolution distribution in a three-dimensional space for both vertical and horizontal resolutions for the first time. This work allows for the analysis of the reliability of retrieved atmospheric parameters. 90

Analytical Method
In the HM, measuring errors that affect vertical resolution can be classified into two types (Hocking, 2018) : those caused by the zenith angle measuring error and those caused by the pulse-length effect on vertical resolution. In the HM, the receiver is reduced to a simple antenna pair that is collinear to the baseline (see figure 1) by considering only a two-dimensional vertical plane. The antenna pair in the HM is equal to one antenna pair in the Jones configuration, which is comprised of three collinear antennas and is usually in a 2 \2.5 configuration. The radio wave phase difference between antenna pairs is denoted as ΔΨ.
In meteor radar systems, there are acceptable phase difference measuring errors (ΔΨ). A higher value of (ΔΨ) means that more signals will be judged as meteor trails and that there will be more misidentifications as well. (ΔΨ) is usually set to approximately 30° (Hocking, 2018;Younger and Reid, 2017) in meteor radar systems. In the HM, the AoA error is due to (ΔΨ), which is a constant. Therefore, the error propagation in the receiver is very simple, and is inversely proportional 100 to the cosine of the zenith.
Our model considers a multistatic system with multiple transmitters and one receiving array in 3-dimensional space (figure 2).
The receiving array is in the Jones configuration, which in a plan view layout can be "cross-shaped", "T-shaped" or "L-shaped".
The five receive antennas should be in the same horizontal plane and constitute two orthogonal antenna arms. To avoid a complex AoA error analyses and to place emphasis on multistatic configurations, we treat the two receiver phase difference 105 measuring errors ( (ΔΨ 1 ) and (ΔΨ 1 )) in the two orthogonal antenna arrays as constants. Therefore, the AoA measuring errors (including zenith and azimuth angle measuring errors , ) can simply be a function of zenith and azimuth angle.
The distance between the meteor trail reflection points and the receiver is denoted as . can be determined by combining the AoA, baseline length d, and radio wave propagating distance R (Stober and Chau, 2015) as: where is the angle between the baseline (axis-′ ) and the line from the receiver to the meteor trail reflection point. In eq.
(1), the multistatic configuration will influence the accuracy of (denoted as ). In our model, we consider the error term which is ignored in the HM. is a function of the AoA measuring errors ( and ) and the wave propagation distance measuring error (denoted as ). is caused by the measuring error of the wave propagation time , which is approximately 21 (Kang, 2008). Thus, can be set as a constant and = = 6.3 . It is worth noting that the 115 maximum unambiguous range for pulse meteor radars is determined by the pulse repetition frequency (PRF) (Hocking et al., 2001;Holdsworth et al., 2004). In our model, for the region where R is more than c/PRF (where c is the speed of light), is set to positive infinity.
To better depict the multistatic system configuration, we need to establish appropriate coordinate systems (figure 3). The spatial configuration of the receiver horizontal plane is determined by the local topography and the antenna configuration. We 120 establish a left-hand coordinate system XYZ to depict the receiver horizontal plane. XYZ is fixed on the receiver and thus will rotate with the 5-antenna horizontal plane. The coordinate origin of XYZ is on the receiver. Axis-Z is collinear with the antenna boresight and perpendicular to the horizontal plane. Axis-X and axis-Y are in the horizontal plane and collinear with the arms of the two orthogonal antenna arrays. Therefore, the zenith angle and azimuth angle are conveniently represented in the XYZ coordinate system. For different transmitters , the baseline direction and distance between and the receiver are different. 125 It is convenient to analyse the range information in the plane that goes through the baseline and meteor trail reflection points (figure 4). Thus, we establish a class of coordinate systems ′ ′ ′ for each . The coordinate origins of ′ ′ ′ are https://doi.org/10.5194/amt-2020-353 Preprint. Discussion started: 14 September 2020 c Author(s) 2020. CC BY 4.0 License.
all on the receiver. We stipulate that axis-′ points to transmitter (T i ). Axis-′ and axis-′ need to satisfy the right-hand corkscrew rule with axis-′ . Each transmitter, , and the receiver constitute a radar link, which is referred to as . We will deal with the range information for each in ′ ′ ′ . Spatial resolution distributions for every need to be compared 130 in the same coordinate system, and this coordinate system needs to be convenient for retrieving wind fields. Therefore, we establish a local WNU (west-north-up) coordinate system 0 ′ is on the receiver with axis-X pointing to the west, axis-Y to the north, and axis-Z pointing up. All spatial resolution distributions for each will ultimately be converted to 0 We specify that clockwise rotation is in the right-hand corkscrew rule. By rotating clockwise in order of ψ X,i , ,i and ,i 135 about axis-, and , respectively, we can transform XYZ to It is worth mentioning that ′ ′ ′ is nonunique because any rotation about axis-′ can obtain one ′ ′ ′ . Hence, ψ X,1 can be set to any values. Similarly, by rotating clockwise in order of ψ i,0 , ,0 and ,0 about axis-, and , respectively, we can transform ′ ′ ′ to To realize the coordinate transformation between those three coordinate systems, we now introduce coordinate rotation matrix ( , , ). Using , we can transform the coordinate point or vector presentation from one system to 140 another. The details of the coordinate rotation matrix ( , , ) can be seen in Appendix (A.1).
The analytical method of the spatial resolution of each radar link is the same. The difference between those radar links is only the value of the six coordinates rotation angle (ψ X,i , ,i and ,i ; ψ i,0 , ,0 and ,0 ) and baseline distance d. In the following, we analyse the spatial resolution of one radar link, as an example. The measurement errors, which affect the spatial resolution, cause a location bias in the specular reflection point. These errors can be classified into two types: 1 is 145 caused by measurement errors in the receiver, and 2 is due to the pulse length. These two errors are mutually independent.
Hence, the total error ( ) in the form of the mean square error (MSE) can be expressed as: 1 is related to three indirect measuring errors, , and , which are zenith, azimuth and radial distance measuring errors, respectively. In XYZ, 1 can be decomposed into three orthogonal error vectors using , and (figure 4(c)) 150 and are the functions of phase difference measuring errors (ΔΨ 1 ) and (ΔΨ 2 ) . ΔΨ 1 and ΔΨ 2 are the two orthogonal antenna arrays' phase differences. Those phase differences measuring errors (ΔΨ 1 ) and (ΔΨ 2 ) are treated as two independent measuring errors. Through derivation, the true error of and can be expressed as: where is the wavelength used in the radar system. and are the zenith angle and azimuth angle, respectively. The details can be seen in Appendix (A.2). It is worth noting that and are not independent. The expectation value ( • ) is not identical to zero unless ( 2 (ΔΨ 1 )) 1 2 equals.
( 2 (ΔΨ 2 )) 2 2 . The true error of can be expressed as a function of , and as: 160  c)). These three vectors can be expressed in XYZ as: 165 S is half pulse length and 1 = − . 2 = 1 − 1 . d is the straight-line distance between the receiver and (baseline length).
Here, we introduced two types of errors in different coordinate systems, and we now need to transform them into local 175 coordinates 0 ′ 0 ′ 0 ′, which is convenient for analysing wind fields. The true error vectors ⃗⃗⃗⃗⃗⃗⃗ , ⃗⃗⃗⃗⃗⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ need two coordinate transformations, that is, from to ′ ′ ′ and then to 0 ′ 0 ′ 0 ′ . By deducing, the true error of 1 can be expressed as vector We denote the first term in the right formula as the error projection matrix, which transforms ⃗⃗⃗⃗⃗⃗⃗ , ⃗⃗⃗⃗⃗⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ in 180 XYZ to axis 0 ′ , 0 ′ 0 ′ . The second matrix term is referred to as the error weight matrix, which can assemble , (ΔΨ 1 ) and (ΔΨ 2 ). Using eq. (3) and (4) ) We denote the first term on the right as the base transformation matrix. We denote the dot product of the error projection matrix, error weight matrix and base transformation matrix as . We refer to as the error propagation matrix. is a 3 × 3 matrix, and we denote the element in it as . Then, we define = 2 . Thus, 1 in the form of MSE square can be expressed as vector ( (1) 2 0 For 2 , true error vector ⃗⃗⃗⃗⃗ needs transformation from ′ ′ ′ to 0

Results and Discussion
We wrote a program to study the method above. The code is written in the python language and is president in supplement.
To calculate a special configuration of a multistatic radar system, we initially set six coordinate transformation angles (ψ X,i , Our output data go through the same process as in Hocking's work, and the result is shown in figure 5(a). Our results are the same as those in Hocking's work (Hocking, 2018). The distributions of the error terms ⃗⃗⃗⃗⃗⃗⃗⃗⃗ and 2 are shown in the second and third subplots in figure 5(a) from left to right, respectively. In most cases, 2 is an order of magnitude smaller than ⃗⃗⃗⃗⃗⃗⃗⃗⃗ . 215 Only in the region directly above the receiver does 2 have the same magnitude as ⃗⃗⃗⃗⃗⃗⃗⃗⃗ . In other words, only in the region directly above the receiver can 2 influence the total resolution. E 2 is related to the bistatic configuration, but ⃗⃗⃗⃗⃗⃗⃗⃗⃗ is not. Therefore, in the HM, the distribution of the total resolution is nearly unchanged with varying d (figure 5(b) and 5(c)). After adding the error term ⃗⃗⃗⃗⃗⃗⃗ , which is related to the bistatic configuration, the normalized vertical spatial resolution distribution is shown in figure 6. The vertical spatial resolution in most vertical planes surpasses 4 km (corresponding to 1 km in normalized 220 spatial resolution). With increasing d, the resolution distribution is holistically compressed and is opposite to the transmitter. Now, we analyse spatial resolutions in 3-dimensional space in our model. We set (ΔΨ 2 ) and 2 the same as (ΔΨ 1 ) and 1 ( (ΔΨ 1 ) = (ΔΨ 2 ) = 35° , 1 = 2 = 4.5). Taking normalized spatial resolution distributions in the 0 ′ 0 ′ plane at 90 km as an example, see figure 7, the baseline length is 300 km, and the angle between the baseline and 0 ′ is 45°. Other parameter settings can be seen in Table 1. In the horizontal plane, E 1 is an order of magnitude larger than E 2 in the 225 0 ′ , 0 ′ , 0 ′ direction in most regions. Therefore, the total spatial resolution in the 0 ′ , 0 ′ , 0 ′ direction is mainly controlled by E 1 . The horizontal spatial resolutions distributions will split into two parts in a relatively long baseline length due to the error term " ⃗⃗⃗⃗⃗⃗⃗ ". Because we observe the errors in 0 ′ 0 ′ 0 ′ , the horizontal resolution distributions are not symmetric about the baseline. The area of vertical spatial resolution less than 12 km (corresponding to 3 km in normalized spatial resolution) is much smaller than the area of horizontal resolution less than 12 km. The vertical resolution distribution becomes compressed and tends to move opposite to the transmitter. This tendency can also be seen from the perspective of the vertical plane ( figure   6). Figure 8 shows the normalized spatial resolution distributions in other parameter settings. The parameter setting details are shown in Table 1. Traditional monostatic meteor radar is a special case of a multistatic meteor radar system whose baseline length is zero. Figure 8(a) shows the normalized spatial resolution distributions of monostatic meteor radar. The spatial 235 resolution distributions are highly symmetrical and correspond to the real characteristics of monostatic meteor radar.
Comparing figure 8(b) and figure 8(c), the slight tilting of the receiver horizontal plane (ψ X,i = ,i = 5°) will cause horizontal spatial distributions to apparently change. In practical applications, the Earth's curvature, local topography or receiver horizontal plane calibration error all lead to the receiver horizontal plane tilting. Thus, this kind of slant should also be taken into account for multistatic meteor radar systems. 240 As mentioned above, the AoA error analysis can be complex. Hence, we have greatly simplified the error propagation process to emphasize the multistatic configuration. If analysing AoA errors starts from the original voltage signals, the error propagation process will change with the specific receiver interferometer configuration and specific signal processing method.
In practical situations for an unusual receiver antenna configuration or new original signal processing algorithm, we can establish an error propagation process based on specific circumstances. The distribution of and can be obtained using 245 the established error propagation process. Substituting (ΔΨ 1 ) and (ΔΨ 2 ) into other mutually independent direct measuring errors in a practical situation, and then establishing a new base transformation matrix in eq. (11) using the error propagation analysis above, we can obtain a new eq. (11). Substituting eq. (11) into the new eq. (11), our model will still work.

Conclusion
In this study, we presented the preliminary results of our error analysis model. Our model can calculate the resolutions in the 250 X 0 ′ , Y 0 ′ , Z 0 ′ direction for an arbitrary configuration in three-dimensional space. The true location of a detected meteor trail reflection point can be in equal probability distributed within the spatial resolution region around the measured location. Higher values of spatial resolution mean that this region needs more meteor counts or averaging to obtain a reliable accuracy. Our model shows that the spatial configuration of a multistatic system will greatly influence the resolution distribution in local coordinates X 0 ′ Y 0 ′ Z 0 ′ and thus will in turn influence the retrieval accuracy of atmospheric parameters such as wind fields. The 255 multistatic meteor radar system's spatial resolution analysis is a key point in analysing the accuracy of retrieved wind and other parameters. We will calculate retrieval error analyses on real data in future work. Multistatic radar systems come in many types, and our work in this paper considers only single-input (single-antenna transmitter in each ) and multi-output (5antenna interferometric receiver) pulse radar systems. Although single-input multi-output (SIMO) pulse meteor radar is a classic meteor radar system, other meteor radar systems, such as continuous wave radar systems and MISO (multiple-antenna 260 transmitter and single-antenna receiver), show good experimental results and have some advantages over SIMO systems. Using different types of meteor radar systems to constitute the meteor radar network is the future trend and we will add the spatial https://doi.org/10.5194/amt-2020-353 Preprint. Discussion started: 14 September 2020 c Author(s) 2020. CC BY 4.0 License. resolution analyses of other system to the frame of our model in this paper. We will validate and apply the error analyses of spatial resolution in horizontal wind determination in a multistatic meteor radar system, which will be built soon in China.

265
Code availability. The program to calculate the 3D spatial resolution distributions are available in supplement.
Author contributions：W.Z, X.X, W.Y designed the study. W.Z deduced the formulas and wrote the program. W.Z wrote the paper for the first version. I.R supervised the work and provided valuable comments. I.R revised the paper. All of the authors discussed the results and commented on the paper.

A.1 Coordinates rotation matrix
For a left-handed rectangular coordinate system , we rotate clockwise Ψ about axis-x to obtain a new coordinate ′ .
We specify that clockwise rotation is in the right-hand screw rule. A vector in , denoted as ( , , ) , is represented as The relationship between ( , , ) and ( ′ , ′ , ′ ) is: Similarly, we rotate clockwise Ψ is about axis-y to obtain a new coordinate. The presentation for a vector in new coordinates and origins can be linked by a matrix, We call ( , , ) as the coordinates rotation matrix.

A.2 AoA measuring errors
In coordinate , AoA includes zenith angle and azimuth angle . The AoA is determined by two phase differences ΔΨ 1 and ΔΨ 2 . ΔΨ 1 and ΔΨ 2 are caused by the antenna array length 1 and 2 in the Jones configuration. Taking one 350 antenna array as an example, the radio wave is at angle with an antenna array ( figure A.2), and there is a phase difference between the two ends of the antenna array. Hence, ΔΨ 1 and ΔΨ 2 can be expressed as: The relationship between , and 1 , 2 can be expressed as: 355

A.3 Radial distance measuring error 370
Use the Taylor expansions of , and to first order for eq. (1) and then subtract eq. (1) to obtain δR t as a function of δR and δ(cosα): is the angle between and axis-′ . In coordinate-′ ′ ′ , we denote the zenith and azimuth angles as ′ and ′, respectively. The relationship between and ′, ′ is 375 Using coordinates rotation matrix (ψ X,i , ,i , ,i ) and denoting the 3x3 matrix 's elements as , we can express  figure 4 (b), side AC adding side CB represents the pulse width. Side AC equals side CB, which equals half the pulse width, S. In ′ ′ ′ , the presentation of point A is (x 0 , y 0 , z 0 ), the receiver is (0,0,0) and is (d,0,0). The distance between and A is 1 = − . We denote that the coordinate of B is ( , , ) and C is ( , , ). We use vector collinear to establish equations for B and C. Therefore, we can obtain the coordinates of B and C by the following equations: (x D , y D , z D ) = 1 2 (x B + x c , y B + y c , z b + z c ) = 1 2 ((a 1 + a 2 )x 0 − a 2 d + d, (a 1 + a 2 )y 0 , (a 1 + a 2 )z 0 ) (A4.3) We denote 1 = − . 2 = 1 − 1 . Finally, we can obtain the true error of E 2 as vector ⃗⃗⃗⃗⃗ in ′ ′ ′ : 400
Therefore, in 0 ′ 0 ′ 0 ′ , the true error of 1 can be expressed as vector ( (1)          " (corresponding 520 to the HM). In (a) and (b), d equals 300, 150, and 50 km. For each row from left to right, they are "total vertical resolution distribution", "vertical resolution distribution related to error term ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ " and "vertical resolution distribution related to ". Only the normalized vertical spatial resolutions less than 3 km are shown. The normalized spatial resolution is one quarter of the spatial resolution. Details are provided in the text.

Figure 7
： Normalized three-dimensional spatial resolution distributions in three-dimensional space. From (a) to (c) is the normalized spatial resolution in the ′ , ′ , ′ direction. For each row from left to right, they are "total resolution", " related", and " related" distributions. The black dot represents transmitter , and the black right-angle arrow represents receiver R. d is 300 km; details are provided in the text.