An Algorithm for Retrieving the 2-D Distribution of Moderate Rain by X-SAR

Abstract

Synthetic aperture radar (SAR) can detect ground information with high precision, which provides another opportunity for the retrieval of rain. Rainfall intensities in East Asia are mainly moderate. The current retrieval algorithms have high accuracy in rainstorms, but they overestimate the rainfall intensity greatly in moderate rain. Therefore, it is very important to reduce the retrieval error of SAR in moderate rain. After analyzing the scattering model of precipitation, this paper proposes an algorithm for retrieving 2-D moderate rain distribution (MRA). Since the 2-D distribution of rain is related to the vertical and horizontal distributions, MRA combines the empirical regression equation with the directional model of rain rates at different levels to retrieve the vertical distribution of precipitation. Compared with the model-oriented statistical (MOS) algorithm, MRA reduces the root mean square error when retrieving the surface rain rate from 2.6 to 0.1. In addition, based on the high-precision rain parameters retrieved by MRA, the horizontal distribution is retrieved through the likelihood distance. This horizontal distribution retrieval method not only has less amount of calculation but also avoids the difficulties of mathematical analysis.

1. Introduction

Precipitation is a key component of the global thermal cycle and drives the atmosphere and surface water cycle [1,2,3]. The accurate measurement of precipitation can provide scientific guidance for the geochemical cycle, industrial and agricultural production, and freshwater supply. Precipitation varies greatly in different regions [4]. In East Asia, the average value of the annual ground rain rate ranges from 1.7 mm/h to 10.2 mm/h [5]. Even in areas with high precipitation, such as Sanya, the time probability of rain rate above 21.9 mm/h is only 0.01 [6], and the common rain rates are between 1 mm/h and 15 mm/h. Therefore, it is very important to consider the distribution of moderate rain (surface rain rates between 1 mm/h and 15 mm/h).

At present, the precipitation observation system constructed by meteorological radars and extensive rain gauge networks can accurately estimate rain in conventional terrain, but it cannot accurately estimate high variability differential precipitation in complex terrain [7]. As a high-altitude detection technology, satellite remote sensing can effectively avoid the defects of rain gauges and ground-based weather radars limited by geographical location, and provide a larger spatial range of precipitation distribution information. Satellite remote sensing measures precipitation in two ways: passive and active. Passive remote sensing mainly refers to spaceborne visible infrared radiometer and spaceborne microwave radiometer, while active remote sensing mainly refers to spaceborne precipitation radar, such as precipitation radar (PR) and dual-frequency precipitation radar (DPR) [8]. The emergence of spaceborne PR measurement largely offset the tropical and marine area precipitation data, but it may ignore or underestimate the precipitation intensity for significant shallow precipitation [9]. At the same time, due to the high costs and the difficult procurement of components, only a few countries have on-orbit spaceborne precipitation radar.

Synthetic aperture radar (SAR) is an active microwave remote sensing technology, which can obtain surface information stably and continuously with high resolution at all times [10,11]. With the development of SAR technology, these high-resolution ground data are gradually applied in various fields, such as the detection of water areas, management of watersheds, supplementation of optical images, object detection, and identification of terrain [12,13,14,15]. Since 1988, China has started research on spaceborne SAR and has launched more than 10 spaceborne SAR satellites of various types. The working frequency bands cover L, S, C, and X, ranging from low-resolution discovery and identification to high-resolution targets [16,17]. For example, the X-band HY-2 satellite, the S-band HJ-1C satellite, and the C-band GF-3 03 satellite were launched in 2011, 2012, and 2022, respectively [18,19,20]. X-band (10 GHz) is very close to Ku-band (14 GHz) and is also sensitive to precipitation. Many scholars have shown that X-band SAR (X-SAR) can detect the distribution of surface precipitation with a horizontal distance of less than 100 m [21,22]. At the same time, X-SAR can not only reduce the errors caused by cirrus clouds in precipitation retrieval [23], but its combination with data of microwave radiation can also solve the problems caused by uneven beam congestion [24]. Therefore, the precipitation retrieval using X-SAR echo data can supplement the precipitation radar and improve the current problems of the small number of precipitation satellites in orbit and the uneven distribution of global precipitation retrieval.

In 1991, Pichugin et al. set up a rainfall model assuming that the vertical distribution is homogeneous and solved the rainfall attenuation expression through the Volterra integral equation (VIE) of the second kind [25]. However, when this method is applied to vertical inhomogeneous models, the computational complexity of the analytical solution increases significantly. In 2006, Weinman et al. established a model-oriented statistical (MOS) algorithm, which solved the rain rate by analyzing the contour lines in the contoured frequency by altitude diagrams (CFAD) [26,27]. MOS has a good effect on rainstorms, but it is poor in moderate rain and requires a large number of empirical parameters. At the same time, MOS needs to construct a suitable simulation database for retrieving the horizontal distribution, which makes the calculation redundant. In 2016, Xie et al. proposed the model-oriented statistical and Volterra integration (MOSVI) algorithm [28]. It uses a mathematically analytic method to reduce the amount of calculation when retrieving the horizontal distribution in a single-layer precipitation model, but it is difficult to solve the analytical solution in multi-layer precipitation models. Furthermore, MOSVI will still overestimate the rain rate in moderate rain.

Combined with the above analysis, this paper proposes MRA, an algorithm for retrieving the 2-D distribution of moderate rain. MRA aims to reduce the overestimation error when retrieving moderate rain rate and avoid the problem of the complex calculation when retrieving the horizontal distribution. In terms of retrieving the vertical distribution, MRA proposes an empirical regression equation between the surface rain rate and the radar cross-section of the precipitation area to retrieve the surface rain rate. The relative error of the retrieval surface rain rate obtained by the equation is between 0.1 and 0.28. Compared with MOS, MRA reduces the root mean square error from 2.6 to 0.1. At the same time, MRA also combines the empirical regression equation with the direct model of rain rates at different levels to retrieve the vertical distribution of precipitation. In terms of retrieving horizontal distribution, MRA proposes a mathematical-statistical method based on the idea of maximum likelihood classification. This method not only avoids the complex problem in VIE but also solves the problem of computational redundancy in MOS. Compared with MOS, MRA reduces the calculation amount of retrieving precipitation horizontal distribution by 97%. Finally, the high-precision two-dimensional distribution of precipitation in moderate rain is obtained by combining the horizontal distribution and vertical distribution.

2. Normalized Radar Cross Section of Precipitation

Attenuation occurs when microwave energy travels along the propagation path in the precipitation area. This is because when microwaves are projected onto precipitation particles, part of the energy is scattered, and another part of the energy is absorbed, thereby weakening the microwaves. The scattering characteristics of precipitation particles are the basis for the measurement of precipitation by radar. Scattering is mainly represented by the radar cross section (RCS). In 1994, a large number of SAR images in X, C, and I bands were acquired from the Space Shuttle “Endeavour”. It is found that, on the X-band images, the precipitation area can be distinguished [29]. This is because the RCS of the precipitation area is lower than the RCS of the rainless area. The size, distribution, and dielectric constant of the precipitation particles will cause the RCS to change [30]. Since freezing makes highly polar molecules in liquid water bound in the crystalline lattice, the dielectric constant of snow is much lower than that of rain. The effects of rain and snow on the RCS are also different [31]. Considering both snow particles and rain particles in this paper, the two-layer precipitation model (ignoring the melt layer) is shown in Figure 1.

In this model, x is the cross-track coordinate, z is the height coordinate, z_t is the top height of the snow layer, z₀ is the top height of the rain layer and $\theta$ is the slant off-nadir angle of X-SAR. $x_{l }$ is the starting point of the precipitation area, $x_{min}$ is the ending point of the precipitation area, and $x_{r }$ is the farthest cross-track distance affected by the precipitation area. Microwave pulses emitted by X-SAR are approximated as plane wavefront slices, shown as a pair of lines with a width of $\mathsf{\Delta }r$ in Figure 1. The error caused by this approximation is about 0.2 km [27], which is within the acceptable range.

According to the cross-section model of precipitation, the normalized radar cross section (NRCS) of precipitation area can be obtained [32], which will be denoted as ${\sigma }_{SAR}$ in this paper. ${\sigma }_{SAR}$ of the precipitation area is the sum of the surface scattering cross-section ${\sigma }_{srf}$ and the volume scattering cross section ${\sigma }_{vol}$:

$${\sigma }_{SAR}\left(x\right)={\sigma }_{srf}\left(x\right)+{\sigma }_{vol}\left(x\right)$$

(1)

$${\sigma }_{srf}\left(x\right)={\sigma }^{0}exp\left[-2{{\displaystyle \int }}_{z_{s1}}^{z_{s2}}k\left(x,z\right)dz/\cos \theta \right]$$

(2)

$${\sigma }_{vol}\left(x\right)=\sin \theta {{\displaystyle \int }}_{z_{r1}}^{z_{r2}}\eta \left(x,z\right)exp\left[-2{{\displaystyle \int }}_{z_{r3}}^{z_{r4}}k\left(x,z^{\prime }\right)d{z}^{\prime }/\cos \theta \right]dz/\cos \theta$$

(3)

where $z_{s1}$, $z_{s2}$, $z_{r3}$, and $z_{r4}$ are the upper and lower limits of the integration of the path projection of the radar beam in the precipitation area onto the z-axis. The $z_{r1}$ and $z_{r2}$ are the upper and lower limits of the integration of the path projection of the reflected microwaves via the ground in the precipitation area onto the z-axis. ${\sigma }^0$ is the RCS of no rain area. Concerning the typical value of the average RCS of vertically polarized radiation incident on land at 30°, ${\sigma }^0$ is about −7 dB [33]. $k\left(x,z\right)$ is the attenuation coefficient and $\eta \left(x,z\right)$ is the coefficient of radar volume scattering, both of which can be obtained according to Equation (4) [26] and Equation (5) [27]:

$$k\left(x,z\right)=a{R}^b\left(x,z\right)$$

(4)

$$\eta \left(x,z\right)=\frac{{\pi }^5{\left|K\right|}^2}{{\lambda }^4}c{R}^d\left(x,z\right)$$

(5)

where ${\left|K\right|}^2$ is an expression of a negative refractive index with a value of 0.93 for rain and a value of 0.19 for snow [26,27]. $\lambda$ is the wavelength of the radar incident microwave, which is about 3.1 cm for X-SAR. For rain, the empirical values of a, b, c, and d are 2.6 × 10⁻³, 1.11, 300, and 1.85; for snow, the empirical values of a, b, c, and d are $5.6\times {10}^{-5}$, 1.6, 182, and 1.6 [34].

As the incident microwave moves along the cross-track in the range of $0\le x\le \left(z_t-z_0\right)$, ${\sigma }_{SAR}$ will increase above ${\sigma }^0$ due to the existence of the ${\sigma }_{vol}$ of snow particles. As the microwave moves in the range of $\left(z_t-z_0\right)/\tan \theta \le x\le z_t/\tan \theta$, the ${\sigma }_{vol}$ of rain particles is enhanced and ${\sigma }_{SAR}$ still increases. However, because the ${\sigma }_{vol}$ of rain particles is less than snow particles, the growth rate of ${\sigma }_{SAR}$ in this phase decreases compared to the previous phase. As the microwave moves in the range of $x_l\le x\le x_{min}$, the ${\sigma }_{srf}$ is greatly reduced by the attenuation of the precipitation area in the bidirectional path of the microwave transmission. Although the ${\sigma }_{vol}$ of snow and rain particles is enhancing, the increase is small. So, the ${\sigma }_{SAR}$ continues to decrease to its lowest point, that is the ${\sigma }_{SAR}\left(x_{min}\right)$. When the microwave moves out of the precipitation area in the range of $x_{min}\le x\le x_{min}+z_t\cdot \tan \theta$, the attenuation decreases and the ${\sigma }_{SAR}$ starts to increase until it returns to ${\sigma }^0$. The simulation of NRCS is shown in Figure 2.

3. Retrieval of Vertical Distribution

It is assumed that the two-dimensional precipitation distribution $R\left(x,z\right)$ can be divided into vertical distribution $v\left(z\right)$ and horizontal distribution $H\left(x\right)$, and the two are uncorrelated and can be retrieved separately.

$$R\left(x,z\right)=v\left(z\right)\cdot H\left(x\right)$$

(6)

There are 18 existing $H\left(x\right)$, and the three most commonly used: rectangular distribution, triangular distribution and trapezoidal distribution will be used in this paper.

3.1. Retrieval of $v\left(z\right)$ by MOS Algorithm

The retrieval of vertical distribution in MOS requires the help of CFAD. Through the approximation of contour lines, the rain rates can be obtained [35]:

$$v\left(0\right)=1.13{{\displaystyle \int }}_{x_l}^{x_{min}}\left[{\sigma }_{dB}^0-{\sigma }_{SARdB}\left(x\right)\right]dx-21.62{{\displaystyle \int }}_0^{x_l}\left[{\sigma }_{SAR}\left(x\right)-{\sigma }^0\right]dx-2.58w+23.3$$

(7)

$$v\left(z\right)=v\left(0\right)\left[0.85+0.15{\left(\frac{z_0-z}{z_0}\right)}^{0.62}\right](0<z\le z_0)$$

(8)

$$v\left(z\right)=v\left(z_0\right){\left(\frac{z_t-z}{z_t-z_0}\right)}^g(z_0<z\le z_t)$$

(9)

where $g$ is the freezing coefficient. A larger $g$ indicates heavier snow, 1.13, 21.62, 2.58, and 23.3 are empirical values, which are calculated under $\theta$ = 30°, $z_t$ = 13 km and z₀ = 4.5 km. $w$ is the width of the precipitation area. For horizontal distributions of the rectangle, triangle and trapezoid, $w$ can be retrieved by the following equations [27,28]:

$$w=0.97\left(x_{min}-x_l\right)$$

(10)

$$w=1.61{\left(x_{min}-x_l\right)}^{0.93}$$

(11)

$$w=\left[0.97\left(x_{min}-x_l\right)+1.61{\left(x_{min}-x_l\right)}^{0.93}\right]/2$$

(12)

From Equations (8) and (9), it is found that the rain rates of the rain layer and snow layer depend on the surface rain rate $v\left(0\right)$, so the retrieval accuracy of $v\left(0\right)$ largely determines the retrieval accuracy of $v\left(z\right)$.

For ease of reading, variables designated with circumflexes are used below to denote the values retrieved from the algorithm, and variables without circumflexes to denote the real or given values. For example, $v\left(0\right)$ represents the given surface rain rate, and $\widehat{v}\left(0\right)$ represents the surface rain rate retrieved by the algorithm. Weinman et al. evaluated the error of $\widehat{v}\left(0\right)$ of MOS in rainstorms and torrential rains. For the 18 sets of data with surface rain rates ranging from 15 mm/h to 160 mm/h, the root mean square error between $\widehat{v}\left(0\right)$ and $v\left(0\right)$ is evaluated by Equation (13) and is 0.1 [26]. From the experimental results, the retrieval accuracy of MOS was higher in rainstorms.

$$RM{S}_m=\sqrt{\frac{1}{m}{\displaystyle \sum }_{i=1}^m{\left[\frac{\widehat{v}{\left(0\right)}_i-v{\left(0\right)}_i}{v{\left(0\right)}_i}\right]}^2}$$

(13)

where m represents the number of data sets.

However, a rainstorm of more than 15 mm/h is not always the case, so this paper tests the retrieval results of MOS in moderate rain. For simulated NRCS data with surface rain rates ranging from 1 mm/h to 15 mm/h, $\widehat{v}\left(0\right)$ retrieved by MOS are shown in Figure 3. The horizontal distribution of precipitation used in the following section is a rectangle unless otherwise noted. Figure 3 shows $\widehat{v}\left(0\right)$ is overestimated with an $RM{S}_n$ of about 2.6. Therefore, the retrieval error of MOS is large in the case of moderate rain.

3.2. Retrieval of $v\left(z\right)$ by MRA Algorithm

To reduce the retrieval error of $v\left(z\right)$ and simplify the retrieval equation, MRA proposed an empirical regression equation between $v\left(0\right)$ and ${\sigma }_{SAR}$. The equation is mainly based on the fitting relationship between NRCS and surface rain distribution $R\left(x,0\right)$ [36]:

$$R\left(x,0\right)=a_e{\left[\Delta {\sigma }_{SARdB}\left(x\right)\right]}^{b_e}$$

(14)

where $a_e$ and $b_e$ are empirical values,

$$\Delta {\sigma }_{SARdB}\left(x\right)={\sigma }_{dB}^0-{\sigma }_{SARdB}\left(x\right)$$

(15)

Equation (14) is obtained by fitting TerraSAR-X (TSX) and next-generation weather radar (NEXRAD) data at the same time and in the same place. Take the surface rain distribution $R\left(x, 0\right)$ retrieved from data of NEXRAD as the true value. By combining the relationship between radar reflectivity factor $Z_e$ and $R\left(x, 0\right)$ in NEXRAD and the relationship between NRCS and $Z_e$ in TSX, it is obtained that the power-law relationship of the NRCS measured by TSX and the true value $R\left(x, 0\right)$ as shown in Equation (14). Using a large number of TSX data in moderate rain for fitting, $a_e$ is 2.84 and $b_e$ is 1.83.

Equation (14) is obtained by fitting the data, but it can also be derived theoretically. By simulating the NRCS in moderate rain, the relationships between ${\sigma }_{srf}$, ${\sigma }_{vol}$, and ${\sigma }_{SAR}$ in different surface rain rates are shown in Figure 4. Figure 4a–c show NRCS, ${\sigma }_{srf}$ and ${\sigma }_{vol}$, respectively, when the surface rainfall rate is 15 mm/h. It is clearly shown in Figure 4c that ${\sigma }_{vol}$ is smaller than ${\sigma }_{srf}$ by more than one order of magnitude in the precipitation of 15 mm/h. Furthermore, the proportion of ${\sigma }_{vol}$ decreases as $v\left(0\right)$ decreases as shown in Figure 4e. Therefore, ${\sigma }_{vol}$ can be neglected relative to ${\sigma }_{srf}$ in moderate rain, and then ${\sigma }_{SAR}$ in Equation (1) can be expressed as:

$${\sigma }_{SAR}\left(x\right)\approx {\sigma }_{srf}\left(x\right)={\sigma }^{0}exp\left[-2{{\displaystyle \int }}_{x_l}^{x_{min}}k\left(x,x/\tan \theta \right)dx/\sin \theta \right]$$

(16)

$R\left(x,0\right)$ can be expressed by effective path-integrated rain rate $I$ [36]:

$$R\left(x,0\right)=\{\begin{array}{c}{I}^p{\left(\frac{x}{w/2}\right)}^q 0<x<w/2\\ I^p{\left(\frac{w-x}{w/2}\right)}^q w/2<x<w\end{array}$$

(17)

where $p$ is the scale factor and $q$ is the horizontal distribution shape exponent.

The integral of $k\left(x,z\right)$ can be obtained by combining Equations (4) and (17):

$${{\displaystyle \int }}_0^{w}k\left(x,x/\tan \theta \right)dx=a{I}^{bp}\frac{w}{bq+1}$$

(18)

Bring Equation (18) into Equation (16):

$${\sigma }_{SAR}\left(x\right)={\sigma }^{0}exp\left[-\frac{2aw{I}^{bp}}{\sin \theta \left(bq+1\right)}\right]$$

(19)

After taking the logarithm and changing the base on both sides simultaneously, $I$ can be written as:

$$I={\left[\frac{\sin \theta \left(bq+1\right)}{20awlg\left(e\right)}\right]}^{\frac{1}{bp}}\cdot {\left[{\sigma }_{dB}^0-{\sigma }_{SARdB}\left(x\right)\right]}^{\frac{1}{bp}}=a_m{\left[\Delta {\sigma }_{SARdB}\left(x\right)\right]}^{b_m}$$

(20)

This form is the same as Equation (14), so it is proved from both theoretical derivation and data fitting that surface rain distribution can be expressed in the exponential form of ${\sigma }_{SAR}$.

It is assumed that $R\left(x,0\right)$ consists of a horizontal distribution $H\left(x\right)$ and a surface rain rate $v\left(0\right)$:

$$R\left(x,0\right)=H\left(x\right)\cdot v\left(0\right)$$

(21)

In Appendix A, a simple test of the effect of Equation (14) on retrieving $R\left(x,0\right)$ is performed. The test results show that Equation (14) has a small error in retrieving the surface rain rate, but a large error in retrieving the horizontal distribution. To reduce the error introduced by the empirical regression equation in retrieving the horizontal distribution, only $v\left(0\right)$ is retained as the retrieval result.

$$v\left(0\right)=max\left\{a_e{\left[\Delta {\sigma }_{SARdB}\left(x\right)\right]}^{b_e}\right\}$$

(22)

MRA uses Equation (22) to retrieve the surface rain rate in moderate rain. Substituting $v\left(0\right)$ into Equations (8) and (9) to obtain the rain rate $v\left(z\right)$ of different heights. Then the horizontal distribution algorithm in Section 4 is used to invert $H\left(x\right)$, and the two-dimensional distribution of moderate rain can be output. The process of MRA is shown in Figure 5. The retrieval methods of ${\widehat{x}}_l$, ${\widehat{x}}_{min}$ and $\widehat{g}$ are described in Appendix B.

4. Retrieval of Horizontal Distribution

The three most commonly used horizontal distributions can be shown as:

$$H\left(x\right)=\{\begin{array}{cc}\hfill 0,& 0\le x\le z_t\hfill \\ \hfill x/d-z_t/(d\tan \theta ),& z_t/\tan \theta \le x\le d+z_t/\tan \theta \hfill \\ \hfill 1,& d+z_t/\tan \theta \le x\le w-d+z_t/\tan \theta \hfill \\ \hfill \left(w-x\right)/d+z_t/(d\tan \theta ),& w-d+z_t/\tan \theta \le x\le w+z_t/\tan \theta \hfill \\ \hfill 0,& x\ge w+z_t/\tan \theta \hfill \end{array}$$

(23)

When $d=0$ and $d=w/2$, it represents rectangular and triangular horizontal distributions, respectively. When $0<d<w/2$, it represents trapezoidal horizontal distribution.

The central idea of the retrieval horizontal distribution is to use the idea of maximum likelihood classification to find a set of simulated data that are most similar to the data to be retrieved. The specific approach is to measure some simulation data with known horizontal distributions and the data to be retrieved one by one and calculate the likelihood distance between the two sets of data. The $H\left(x\right)$ used in the set of simulation data with the smallest likelihood distance is the $\widehat{H}\left(x\right)$. Equations (24)–(27) extract the distribution characteristics of each set of data in terms of four aspects: mean $\mu$, variance ${\sigma }^2$, skewness $\alpha$, and kurtosis $\kappa$. There are positive and negative $\Delta {\sigma }_{SAR}$ in a set of data. By calculating the positive and negative separately, eight statistical parameters can be obtained.

$$\mu =\frac{1}{n}{\displaystyle \sum }_{i=1}^n\Delta {\sigma }_{SAR}\left(i\right)=\frac{1}{n}{\displaystyle \sum }_{i=1}^n\left[{\sigma }_{SAR}\left(i\right)-{\sigma }^0\right]$$

(24)

$${\sigma }^2=\sqrt{\frac{1}{n-1}{\displaystyle \sum }_{i=1}^n{\left[\Delta {\sigma }_{SAR}\left(i\right)-\mu \right]}^2}$$

(25)

$$\alpha =\frac{1}{n-1}{\displaystyle \sum }_{i=1}^n{\left[\Delta {\sigma }_{SAR}\left(i\right)-\mu \right]}^3\frac{1}{{\sigma }^3}$$

(26)

$$\kappa =\frac{1}{n-1}{\displaystyle \sum }_{i=1}^n{\left[\Delta {\sigma }_{SAR}\left(i\right)-\mu \right]}^4\frac{1}{{\sigma }^4}$$

(27)

where $n$ is the amount of a set of data.

In addition to the above 8 statistical values, it is also necessary to focus on the gradient near the precipitation starting point. In this paper, the ${\sigma }_{SAR}$ gradient values at $x_l$, $x_l+1.5$ km, and $x_l-1.5$ km are examined. Together with the first 8 statistical values, a set of data yields 11 measurements, which can form an 11-dimensional statistical vector.

The statistical vector of a set of simulation data is denoted as $X_{SAR}$, and the statistical vector of the data to be retrieved is denoted as $M_{SAR}$. The likelihood distance $d_H$ of the two sets of data can be expressed as:

$$d_H={\left(X_{SAR}-M_{SAR}\right)}^{\mathrm{T}}{C}_{SAR}\left(X_{SAR}-M_{SAR}\right)$$

(28)

where $C_{SAR}$ is the covariance matrix with a dimension of 11 × 11.

As $d_H$ is not only affected by the difference between the horizontal distributions of the two sets of data but also affected by the difference in other parameters such as $z_t$. Therefore, the simulation data should keep parameters except for $H \left(x\right)$ as consistent as possible with the data to be retrieved. Through the retrieval process of MRA, it is found that ${\widehat{x}}_l$, ${\widehat{x}}_{min}$, $\widehat{v}\left(0\right)$ and other parameters can be obtained with high accuracy before the retrieval of $\widehat{H}\left(x\right)$. Using these parameters, $\widehat{v}\left(z\right)$ can be calculated according to Equations (8) and (9). Assuming a kind of $H \left(x\right)$, a simulated rain distribution $R\left(x,z\right)$ can be obtained. As can be seen from Section 2, a set of ${\sigma }_{SAR}$ can be obtained according to Equations (2) and (3) when the $R\left(x,z\right)$ is known. The likelihood distance calculated by ${\sigma }_{SAR}$ and the data to be retrieved only represents the difference between the simulated horizontal distribution and the real horizontal distribution. In the case of considering only the three common horizontal distributions, it is necessary to construct only 3 sets of simulation data and calculate 44 statistics.

In MOS, retrieving the horizontal distribution of rain requires 50 sets of simulation NRCS, and a total of about 1650 statistical values need to be calculated. Compared with MOS, if only three horizontal distributions are considered, this method only needs three sets of simulation data, and the amount of calculation is greatly reduced. Compared with the inversion method of VIE, this method avoids the solving steps of analytical solution and is simpler and more intuitive.

5. Simulation and Discussion

In this section, the MRA will be simulated to retrieve the surface rain rate, horizontal distribution and two-dimensional distribution of precipitation. For X-SAR with a slant off-nadir angle of 30°, set $z_t=13$ km, $z_0=4.5$ km and $g=0.5$. Considering the resolution of SAR and the cross-track distance affected by precipitation, each set of NRCS data consists of 200 data points. Parameters with different values in different sets of data, such as $v\left(0\right)$, $w$ and $H\left(x\right)$, will be described in detail before each simulation.

5.1. Simulation of Surface Rain Rate

In cases 1–3, the surface rain rates of rectangular, triangular, and trapezoidal rain clouds were retrieved using MRA, respectively. Fifteen sets of NRCS data were used in each case. The parameters for the 15 sets of data are: $w=6$ km and surface rain rates between 1 mm/h and 15 mm/h. The relative error (RE) is defined as Equation (29):

$$RE=\left|\frac{\widehat{v}\left(0\right)-v\left(0\right)}{v\left(0\right)}\right|$$

(29)

Case 1: Assume that the horizontal distribution of all NRCS data is a rectangle and the retrieval results are shown in Figure 6. The RE of MRA ranges from 0.01 to 0.28 and the RE of MOS ranges from 0.23 to 8.32.

Case 2: Assume that the horizontal distribution of all NRCS data is the triangle and the retrieval results are shown in Figure 7. The RE of MRA ranges from 0.03 to 0.19 and the RE of MOS ranges from 0.32 to 8.44.

Case 3: Assume that the horizontal distribution of all NRCS data is trapezoid and the retrieval results are shown in Figure 8. The RE of MRA ranges from 0.02 to 0.17 and the RE of MOS ranges from 0.29 to 9.76.

Overall, the RE of MRA is much smaller than MOS in moderate rain. Figure 6, Figure 7 and Figure 8 show the error of MRA increases with the increase of $v\left(0\right)$, which is due to the increase of ${\sigma }_{vol}$ of rain particles when the rain is heavier. At this time, the error caused by ignoring ${\sigma }_{vol}$ in Equation (22) increases.

Table 1. Root mean square error of MRA and MOS.

	MRA	MOS
case 1	0.1433	2.5791
case 2	0.1445	2.6006
case 3	0.1002	3.0359

shows the root mean square error of MRA and MOS in cases 1–3. The $RM{S}_{15}$ of MRA under the rain of three horizontal distributions are very close to each other, all around 0.15, but the $RM{S}_{15}$ of MOS is around 2.6. The experiment proves that MRA can effectively reduce the retrieval error of the surface rain rate in moderate rain.

5.2. Simulation of Horizontal Distribution

There are four sets of data to be retrieved to verify the effectiveness of this method in cases 4–7. $v\left(0\right)$ of the four sets of data is 15 mm/h. The values of parameters such as horizontal distribution in each set of data are shown in

Table 2. The horizontal distribution of data to be retrieved in cases 4–7.

	$\mathit{w}$ (km)	$\mathit{d}$ (km)	$\mathit{H}\left(\mathit{x}\right)$
case 4	10	0	rectangle
case 5	10	5	triangle
case 6	10	3	trapezoid
case 7	6	0	rectangle

.

In each case, it is necessary to construct three sets of simulation data with different horizontal distributions. The horizontal distributions used by the three sets of simulation data are rectangle ($d=0$), triangle ($d=\widehat{w}/2$) and trapezoid ($d=\widehat{w}/3$), where $\widehat{w}$ represents the rain width obtained from the data to be retrieved in each case. Then calculate the likelihood distance $d_H$ between simulation data and data to be retrieved. The $H\left(x\right)$ used in the set of simulation data with the minimum likelihood distance $d_H$ is selected as $\widehat{H}\left(x\right)$.

Table 3. Likelihood distance between simulation data and data to be retrieved.

	Simulation Data 1: d = 0	Simulation Data 2: $\mathit{d}=\widehat{\mathit{w}}/2$	Simulation Data 2: $\mathit{d}=\widehat{\mathit{w}}/3$	$\widehat{\mathit{w}}$ (km)	$\widehat{\mathit{d}}$ (km)	$\widehat{\mathit{H}}\left(\mathit{x}\right)$
case 4	9.3129	46.9394	53.1824	9.3640	0	rectangle
case 5	4.5994	0.4979	0.5224	9.0663	4.5331	triangle
case 6	16.8403	0.0619	0.0394	10.6853	3.5617	trapezoid
case 7	6.4534	15.8121	23.5484	5.9028	0	rectangle

shows the $d_H$ and the retrieval of horizontal distribution in cases 4–7.

The correct retrieval results of the $H\left(x\right)$ were obtained for three horizontal distributions of the retrieved data as shown in Table 3.

5.3. Simulation of 2-D Distribution of Moderate Rain

The retrieval effect of MRA on the two-dimensional distribution of moderate rain is shown in cases 8–10. The parameters of the data used in the three cases are: $v\left(0\right)=5$ mm/h, and $w=10$ km.

Case 8: Set the rain type as rectangular distribution ($d=0$ km). The given two-dimensional distribution of rain, the two-dimensional distribution of rain retrieved by MRA and the absolute error diagram are shown in Figure 9. The surface rain rate retrieved by MRA is 4.9 mm/h and the relative error is 0.02.

Case 9: Set the rain type as triangular distribution ($d=5$ km). The given two-dimensional distribution of rain, the two-dimensional distribution of rain retrieved by MRA and the absolute error diagram are shown in Figure 10. The surface rain rate retrieved by MRA is 4.3 mm/h and the relative error is 0.14.

Case 10: Set the rain type as trapezoidal ($d=3$ km). The given two-dimensional rainfall distribution, the two-dimensional rainfall distribution retrieved by MRA and the absolute error diagram are shown in Figure 11. The surface rain rate retrieved by MRA is 4.8 mm/h and the relative error is 0.04.

Figure 9c, Figure 10c and Figure 11c show that the absolute error between the retrieved two-dimensional distribution of moderate rain and the given two-dimensional distribution of rain is very small, especially in the rainfall layer. The locations with larger absolute errors occur at the edges of the precipitation area, especially in the rain of rectangular distribution as shown in Figure 9c. This is caused by the error of retrieving precipitation width. The relative error of retrieval width is about 0.03 in cases 8–10, which is within an acceptable range.

At different heights, the height at which the absolute error suddenly increases is the junction height of the snowfall layer and the rainfall layer. This is due to the overestimation phenomenon in the retrieval of the freezing coefficient $g$. The accurate retrieval of $g$ is also a direction for subsequent research.

6. Conclusions

This paper introduces a two-layer model for X-SAR precipitation measurement and simulates the NRCS of precipitation. After analyzing the proportional relationship between the components of NRCS, the MRA is proposed to improve the retrieval accuracy in moderate rain. MRA can effectively retrieve the vertical distribution together with the horizontal distribution and output the two-dimensional distribution of moderate rain. The retrieval effect of MRA in moderate rain is verified using simulated echo data under different horizontal distributions. The results show that the root mean square error of MRA is around 0.15. Compared with MOS, MRA reduces the root mean square error in moderate rain by 96%. In addition, MRA also uses mathematical statistics to retrieve the horizontal distribution. The method of mathematical statistics avoids the difficulty of solving analytical solutions in VIE and computational redundancy in MOS.

Obviously, through the above analysis, MRA can be used for daily precipitation detection. At the same time, MRA also has certain utility in preventing natural disasters with precipitation changes as the main driving factor, such as drought and mudslides. The occurrence of drought is due to the fact that the water expenditure of the region is much greater than the water income in a certain period of time. For traditional droughts, MRA can measure precipitation with high accuracy, which can be combined with evapotranspiration data to contribute to drought prevention. For mudslides, the latest research has proved that the prone areas of mudslides can be predicted through the annual precipitation [37]. MRA can provide a reliable estimate of annual precipitation through long-term observations of remote sensing satellites, thereby preventing mudslides. In addition to annual precipitation, estimates of individual precipitation may also be useful in analyzing mudslides. For example, in Indian Himalayas, a mudslide caused by avalanches and rock avalanches occurred. The rapid flow of the mudslide has resulted in huge human casualties and loss of property [38]. In addition to the melting of glaciers separated in the avalanche, which increases the velocity of the mudslide, the precipitation which occurs before the avalanche is also a cause of the rapid movement of the mudslide. MRA can roughly determine the proportion of precipitation in the mudslide by accurately estimating the amount of precipitation, so as to further determine the proportion of the separated glacier meltwater in the mudslide. This could contribute to the study of how avalanches affect mudslides.

The focus of this paper is on the retrieval algorithm, so some parameters are simplified. For example, the influence of rainfall on the RCS of the sea surface is ignored, and this is also the direction that should be worked on in the future. Moreover, the vertical distributions of rain in different regions and types also deserve further study.