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CN112083453B - Troposphere chromatography method related to water vapor space-time parameters - Google Patents

Troposphere chromatography method related to water vapor space-time parameters Download PDF

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CN112083453B
CN112083453B CN202010964909.4A CN202010964909A CN112083453B CN 112083453 B CN112083453 B CN 112083453B CN 202010964909 A CN202010964909 A CN 202010964909A CN 112083453 B CN112083453 B CN 112083453B
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陈必焰
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Abstract

The invention provides a troposphere chromatography method related to water vapor space-time parameters, wherein the wet refractive index changes along with the space position, the value of the wet refractive index is obtained by eight node weighted summation interpolation of the voxel, the horizontal direction adopts inverse distance weighted interpolation, the vertical direction adopts exponential interpolation, and the parameters required by the interpolation method are obtained by real-time estimation according to a chromatography wet refractive index field. On the basis of the existing parameterized chromatographic model, the invention improves the chromatographic modeling precision by a refined parameterized chromatographic model interpolation method. Particularly, when the gridding division is large or the change of the water vapor space is severe, the improved parameterization method provided by the invention can be used for constructing the troposphere chromatographic model more accurately.

Description

Troposphere chromatography method related to water vapor space-time parameters
Technical Field
The invention relates to the field of geophysical, in particular to a troposphere chromatography method relating to water vapor space-time parameters.
Background
The troposphere is the layer of atmosphere closest to the earth's surface and the most dense layer in the earth's atmosphere, containing approximately 75% by mass of the entire atmosphere, as well as almost all water vapor and aerosols. The troposphere chromatography is to carry out inversion calculation according to the wet delay information obtained by ray scanning and reconstruct a three-dimensional image of the distribution rule of the wet refractive index in the range of the measured troposphere. The wet delay refers to that when an electromagnetic wave signal passes through the troposphere, the signal is refracted by water vapor and is bent, the propagation path is longer than the geometric distance, and the propagation speed is accordingly slower, which is called as the delay of the atmospheric water vapor on the electromagnetic wave signal.
The water vapor is an important component of the earth atmosphere and plays a vital role in various fields such as a natural gas power system, atmospheric environment science, surveying and mapping science and technology, hydrology and the like. Although the water vapor content in the atmosphere only accounts for 0.1-3 percent of the total amount, the water vapor is the most active component in the atmosphere. Many weather changes and natural disasters are directly related to the content and migration of water vapor in the atmosphere, so that the atmospheric water vapor plays a critical role in weather forecast and weather changes. The water vapor in the atmosphere is mainly concentrated in the convection layer, and the high dynamic and rapid change characteristics of the water vapor make the accurate and timely acquisition of the water vapor distribution with high spatial and temporal resolution a very difficult task. The high-precision atmospheric water vapor field can be reconstructed by a ground-based GNSS (Global Navigation Satellite System) chromatography technology, and the method has the advantages that a plurality of traditional water vapor detection means have incomparable advantages.
The regional three-dimensional water vapor field can be reconstructed by utilizing the SWD (slope Wet Delay) data densely interwoven in the upper space of the region and combining with the chromatography technology. The commonly used tomographic modeling method is to take a voxel as a minimum cutting unit, discretize an inversion region into a small grid, and then use SWD data to calculate the wet refractive index in each grid. In traditional chromatographic model researches, the uniform and unchanged water vapor distribution in a single grid is assumed, but when the water vapor is strongly changed in space or the grid range is large, the assumption is not reasonable, and a large model error is caused. In contrast, some researchers have proposed a numerical integration parameterization method, which uses different interpolation algorithms to interpolate the wet refractive index of any point in the grid, and uses a numerical integration algorithm to discretize the wet delay of the inclined path. The parameters to be solved in the method are the wet refractive indexes of the vertexes of the grids, the unreasonable assumption that the water vapor distribution in the unit grids is uniform and unchanged in the traditional chromatographic modeling is overcome, and the number of the parameters to be solved is not obviously increased. The numerical integration parameterization method is the key for constructing the high-spatial-resolution chromatographic model, and the method can better invert the inverse increment of the water vapor profile, so that the mathematical model is more reasonable. However, the bilinear/spline interpolation adopted in the existing parameterized model does not take the physical characteristics of the water vapor space-time change into account, so that the water vapor interpolation method is not accurate enough, and the method needs to be researched and improved deeply.
That is, existing tropospheric tomography modeling methods can be basically divided into two categories: one type is a non-parametric model in which the water vapor in each voxel during the analysis period is considered to be uniformly distributed. The other is a parameterized model in which the water vapor in each voxel varies with spatial position over the analysis period.
Wherein the non-parametric model spatially discretizes the troposphere into a plurality of voxels and assumes a constant and uniform distribution of the wet refractive index within each voxel. The method has the advantages of relatively few parameters involved in modeling and simple operation. But when the grid partitioning is large or extreme weather conditions occur, modeling errors that cannot be ignored result.
The parameterized model also discretizes the troposphere into a plurality of voxels, but the wet refractive index in each voxel is changed according to the spatial position, and the wet refractive index at any point is formed by eight nodes (such as N in FIG. 1) of the voxel1～N8These eight nodes) are obtained via horizontal and vertical interpolation. Adverse effects caused by unreasonable grid division can be solved, and meanwhile, the calculation efficiency is ensured.
However, in practice, moisture varies greatly spatially, and in particular in the vertical direction or in extreme weather conditions, this can lead to model errors that cannot be ignored for non-parametric models. Finer mesh partitioning can mitigate the adverse effects of this unreasonable assumption, but it increases computational efficiency and inter-voxel constraints can also impact the results. The parameterized model can solve adverse effects caused by unreasonable grid division, and meanwhile, the calculation efficiency is guaranteed. The wet refractive index of any point in the parameterized model is obtained by performing horizontal and vertical interpolation on the eight nodes of the voxel. However, the existing interpolation methods (such as linear interpolation and cubic spline interpolation) do not take into account that the water vapor space-time parameters cause insufficient modeling precision. When the grid division is large or the change of the water vapor space is severe, the inversion error is large, and the application of troposphere chromatography is limited to a certain extent.
In order to solve the above problems and to build a more accurate parameterized tropospheric tomography model, there is a need in the art for an improved tropospheric tomography method.
Disclosure of Invention
The invention is based on a parameterized troposphere tomographic model, and develops an improved troposphere tomographic parameterization method aiming at the problem of insufficient accuracy of an interpolation method thereof. The defect that water vapor space-time parameters are not considered in the traditional modeling is overcome, and the parameterized troposphere chromatographic model is refined.
That is, the invention provides a troposphere tomography method related to water vapor space-time parameters, in the method, the wet refractive index changes along with the space position, the value of the wet refractive index is obtained by eight node weighted summation interpolation of the voxel, the horizontal direction adopts inverse distance weighted interpolation, the vertical direction adopts exponential interpolation, and the parameters required by the interpolation method are obtained by real-time estimation according to the chromatography wet refractive index field.
The present invention relates to the space-time parameters of water vapor, namely the time parameters and the space parameters thereof, wherein the space parameters of water vapor comprise the interpolation of the horizontal direction and the vertical direction thereof, and the time parameters of water vapor are dynamically different estimated values obtained according to the time transformation.
In a specific embodiment, during tropospheric tomography, vertical interpolation is performed followed by horizontal interpolation or horizontal interpolation is performed followed by vertical interpolation, preferably horizontal interpolation is performed followed by vertical interpolation. This may make the calculation more simplified.
In a particular embodiment of the method of the present invention,
SWD and wet refractive index N along the GNSS satellite to receiver ray pathwThe relationship between can be expressed as:
SWD＝∫l Nwdl (1)
discretizing the reconstructed space into a plurality of voxels in the tomographic model, wherein l is an oblique path in the SWD; the vertical layer division method considering the water vapor distribution is as follows:
Figure BDA0002681914740000031
wherein h isiThe ith layer top height; n is the total number of vertical layers; h isminIs the height of the lowest layer of the chromatographic study area; h ismaxIs the top level of the chromatographic study area; alpha is a vertical variation of water vapourThe number can be obtained by fitting historical sounding profile data through a formula 6;
in the parameterized model, the wet refractive index in each voxel is changed along with the position, and the wet refractive index at any point in the voxel is formed by 8 nodes N of the voxel1To N8Such that SWD is expressed as an integral of the wet refractive index at the voxel node, the integral being solved, in particular, using newton's method; five equidistant points P along the inclined path 11-P5Wet refractive index NWThe integral can be expressed in the following way:
Figure BDA0002681914740000032
wherein DP1P5Is P1To P5Intercept of, P1，P2，P3，P4，P5Is five equidistant points, point PkWet refractive index of N1To N8The wet refractive index values of the 8 voxel nodes are determined by interpolation;
parameterized modeling method considering water vapor space-time distribution, see formula 5, PkCan be measured by point V1And V2Vertical interpolation:
Figure BDA0002681914740000033
wherein V1Is equal to N1～N4Points on the same level, V2Is equal to N5～N8Points on the same level, and V1、V2And PkThe points are positioned on the same vertical line; k is any integer selected from 1 to 5, hPkIs PkThe height of the point;
wherein the parameter α can be estimated from:
Figure BDA0002681914740000034
wherein h is0Refers to the height of the voxel lower surface, hiThe height of any point in a voxel is referred to, and the exponential change characteristic of water vapor along with the vertical height is considered in formula 6;
in order to take the change of water vapor time into consideration, the chromatographic profile of the previous time interval is used for estimating alpha of each voxel of the current time interval;
in addition, V1And V2Is calculated using inverse distance weighted interpolation:
Figure BDA0002681914740000041
wherein i is 1 or 2, wherein djIs ViAnd NjA distance between, ViAnd NjOn the same height plane; u is a power of the distance, NjIs referred to as V i4 surrounding nodes of the voxel surface; n is a radical ofw(Nj) Is VjWet refractive index values of 4 surrounding nodes of the surface of the voxel at which it is located;
in order to improve the modeling precision, the u value of each vertical layer is estimated in each time interval by using the chromatography result of the previous time interval.
In one specific embodiment, all SWDs during modeling are collected, establishing a linear system between the SWD and the wet index field:
y＝Ax (8)
where y is the vector of SWD observations, x is the unknown parameter vector containing the wet refractive index of all voxel nodes, and A is the design matrix composed of the SWD's x contributions.
In one specific embodiment, since equation 8 generally has an ill-defined problem, horizontal and vertical constraints are added to make the equation full-rank, and then a least squares method is used to solve for x.
The invention has the advantages that:
on the basis of the existing parameterized chromatographic model, the invention improves the chromatographic modeling precision by a refined parameterized chromatographic model interpolation method. The traditional parameterized troposphere chromatographic model interpolation method usually adopts linear interpolation or spline interpolation, does not consider water vapor space-time distribution characteristics, and particularly has larger inversion error when grid division is larger or water vapor space change is severe, but the improved parameterized method provided by the invention can more accurately construct a troposphere chromatographic model, and for example, compared with a nonparametric method and a traditional parameterized method, the inversion accuracy of a new parameterized method provided by the invention is respectively improved by 54% and 10% in practical case application. Based on a large number of comparative analyses, the parameterized chromatographic model considering the water vapor space-time distribution provided by the invention can obviously improve the chromatographic resolution precision compared with the traditional model. Therefore, the method can effectively make up for the defects of the traditional model and determine the wet refractive index field structure more accurately.
Drawings
Fig. 1 is a schematic diagram of the discretization of a troposphere tomography model provided by the invention.
FIG. 2 is a graph of RMS error (graph a) versus relative RMS error (graph b) for a non-parametric model (Tomo-I), a conventional parametric model (Tomo-II), and a parametric model provided by the present invention (Tomo-III).
FIG. 3 is a RMS distribution plot of ERA5 versus a non-parametric model (FIG. 3a), a conventional parametric model (FIG. 3b), and an improved parametric model provided by the present invention (FIG. 3 c).
Detailed Description
According to the method, according to the water vapor space change characteristic, the wet refractive index of any point in the troposphere space is obtained by weighting and summing eight nodes of a voxel, the reverse distance weighted interpolation is adopted in the horizontal direction, the exponential interpolation is adopted in the vertical direction, and the required parameters in the interpolation method are obtained according to the chromatography wet refractive index field dynamic estimation, so that the defect that water vapor space-time parameters are not considered in the past modeling is overcome, and the parameterized troposphere chromatography model is refined. The principle and the process are as follows:
SWD and wet refractive index N along the GNSS satellite to receiver ray pathwThe relationship between can be expressed as:
SWD＝∫l Nwdl (1)
as shown in fig. 1, i is the diagonal path in the SWD. The tomographic technique can invert the spatial distribution of the wet refractive index based on the spatially interleaved SWD from each direction over the tomographic period. Typically, the tomographic model has to discretize the reconstruction space into a plurality of voxels (see fig. 1). In the prior art, vertical layering is artificially determined during grid division, and the vertical distribution of water vapor is not considered. The invention provides a vertical layer dividing method considering water vapor distribution, which comprises the following steps:
Figure BDA0002681914740000051
wherein h isiThe ith layer top height; n is the total number of vertical layers; h isminIs the height of the lowest layer of the chromatographic study area; h ismaxIs the top level of the chromatographic study area; alpha is a water vapor vertical variation parameter and can be obtained by fitting historical sounding profile data through a formula 6. Historical sounding profile data is public data that can be readily obtained.
In the traditional non-parametric approach, it is assumed that the water vapor within each voxel is constant and evenly distributed over the modeling period. Thus, each SWD can be viewed as the sum of all segments that pass through those voxels along the ray path, so equation (1) can be approximated as:
Figure BDA0002681914740000054
where i refers to voxel i, n is the number of voxels traversed by the SWD,
Figure BDA0002681914740000052
refers to the wet refractive index, d, in voxel iiIs the intercept of the SWD ray path for voxel i.
In a parametric model (including the prior art and the present invention), the wet refractive index within each voxel is position dependent. The wet refractive index at any point within a voxel is determined by a weighted sum of the wet refractive index values of the 8 nodes of the voxel, such that SWD is expressed as the integral of the wet refractive index at the voxel node. Since an analytical solution for the integration cannot be obtained in most cases, newton's method is used to solve for the integration. As shown in FIG. 1, the wet refractive index (i.e., N) along P1-P5W) The integral can be expressed in the following way:
Figure BDA0002681914740000053
wherein DP1P5Is P1To P5Intercept of, P1，P2，P3，P4，P5Are five equidistant points. Point PkHas a wet refractive index of 8 voxel nodes (i.e., N)1，N2，…，N8) And (4) determining interpolation.
In the past research, the interpolation method usually adopts linear interpolation or spline interpolation, and does not consider the water vapor space-time distribution characteristic. When the grid division is large or the change of the water vapor space is severe, the inversion error is large, and the application of troposphere chromatography is limited to a certain extent.
The invention provides a parameterized modeling method considering water vapor space-time distribution. With P3For example, the wet refractive index may be represented by point V1And V2Vertical interpolation:
Figure BDA0002681914740000061
wherein the parameter α can be estimated from:
Figure BDA0002681914740000062
wherein h is0Refers to the height of the voxel lower surface, hiRefers to the height of any point within the voxel, and equation (6) takes into account the characteristic of the exponential change of water vapor with vertical height. To account for moisture time variations, the tomographic profile of the previous session is used to estimate α for each voxel of the current session. In addition, V1And V2Is calculated using inverse distance weighted interpolation:
Figure BDA0002681914740000063
wherein i is 1 or 2, ViI.e. V in formula 51And V2. Wherein d isjIs ViAnd NjA distance between, ViAnd NjOn the same plane. u is a power of the distance, NjIs referred to as V i4 surrounding nodes of the surface of the voxel. N is a radical ofw(Nj) (j-1, 2, 3, 4) is ViWet refractive index values of 4 surrounding nodes of the surface of the voxel at which it is located. In order to improve the modeling precision, the u value of each vertical layer is estimated in each time interval by using the chromatography result of the previous time interval.
Collecting all SWDs during modeling, a linear system between the SWD and the wet index field can be established:
y＝Ax (8)
where y is the vector of SWD observations, x is the unknown parameter vector containing the wet refractive index of all voxel nodes, and A is the design matrix composed of the SWD's x contributions. Because the equation (8) usually has an ill-defined problem, horizontal and vertical constraints are required to be added to make the equation of full rank, and then the least square method is used to solve x.
The case of fig. 2 shows that the improved parameterized model provided by the invention can significantly improve the accuracy of the chromatographic water vapor field. FIG. 2 shows a graph of RMS error versus relative RMS error as a function of height for a non-parametric model (Tomo-I), a conventional parametric model (Tomo-II), and an improved parametric model of the present invention (Tomo-III). Compared with the other two models, the improved parameterized model can obviously improve the chromatographic accuracy, and the RMS error of the improved parameterized model is gradually reduced along with the height. The improved parameterized model increases from 8% at the lowest level to 443% at the highest level in terms of relative RMS error. Compared with a non-parametric model (blue) and a traditional parametric model (pink), the accuracy of the wet refractive index value of the improved parametric model inversion is greatly improved. In fig. 2a and 2b, curves of the non-parametric model (Tomo-I), the conventional parametric model (Tomo-II) and the improved parametric model of the present invention (Tomo-III) are shown from right to left, respectively.
FIG. 3 shows the comparison of chromatographic wet refractive index with the ERA5 reanalyzed data. FIG. 3(a) shows the results of ERA5 compared to a non-parametric model, FIG. 3(b) shows the results of ERA5 compared to a conventional parametric model, and FIG. 3(c) shows the results of ERA5 compared to a modified parametric model. It is clear that both the non-parametric model and the conventional parametric model have large deviations from the data reanalyzed by ERA 5. The RMS error variation ranges of the three methods Tomo-I, Tomo-II and Tomo-III are respectively 7.0-16.8 mm/km, 5.9-15.8 mm/km and 6.0-11.0 mm/km. For the improved parameterized model of the invention, most regional RMS error values are less than 10mm/km, and the data is analyzed by ERA5 to show higher consistency. On the whole, the accuracy of the steam field inverted by the improved parameterized model is improved by 17% and 8% respectively compared with that of a nonparametric model and a traditional parametric model.
The foregoing is a more detailed description of the invention in connection with specific preferred embodiments and it is not intended that the invention be limited to these specific details. For those skilled in the art to which the invention pertains, several simple deductions and substitutions can be made without departing from the spirit of the invention, and all shall be considered as belonging to the protection scope of the invention.

Claims (3)

1.一种涉及水汽时空参数的对流层层析方法，所述方法中湿折射率随空间位置变化且其值由所在体素的八个节点加权求和内插得到，水平方向采用反距离加权内插，垂直方向采用指数内插，且内插方法所需的参数根据层析湿折射率场实时估计而获取；且所述层析方法具体包括如下步骤：1. A tropospheric tomography method related to water vapor spatiotemporal parameters, in the method, the wet refractive index varies with spatial position and its value is obtained by the weighted sum and interpolation of eight nodes of the voxel where it is located, and the horizontal direction adopts the inverse distance weighted inner The vertical direction adopts exponential interpolation, and the parameters required by the interpolation method are obtained according to the real-time estimation of the tomographic wet refractive index field; and the tomographic method specifically includes the following steps: 沿着GNSS卫星到接收机的射线路径，SWD和湿折射率Nw之间的关系可表达为：Along the ray path from the GNSS satellite to the receiver, the relationship between SWD and the wet refractive index N w can be expressed as: SWD＝∫lNwdl (1)SWD=∫ l N w dl (1) 层析模型中将重建空间离散为多个体素，其中l为SWD中的斜路径；顾及水汽分布的垂直层划分方法如式2：In the tomographic model, the reconstruction space is discretized into multiple voxels, where l is the oblique path in the SWD; the vertical layer division method considering the water vapor distribution is as shown in Equation 2:
Figure FDA0003462203930000011
Figure FDA0003462203930000011
 其中hi第i层层顶高度；n是总的垂直层数；hmin是层析研究区域的最底层高度；hmax是层析研究区域的顶层高度；α是水汽垂直变化参数，可由历史探空廓线数据经公式6拟合得到；where h i is the height of the top layer of the i-th layer; n is the total number of vertical layers; h min is the height of the bottom layer of the tomographic study area; h max is the height of the top layer of the tomographic study area; The sounding profile data is obtained by fitting with Equation 6; 在参数化模型中，每个体素内的湿折射率是随位置变化的，体素内任一点的湿折射率由该体素8个节点N1到N8的湿折射率值的加权和确定，使得SWD表示为体素节点处的湿折射率的积分，具体采用牛顿克茨方法来求解积分；沿斜路径l上的五个等距点P1-P5的湿折射率Nw积分可以通过以下方式表达：In the parametric model, the wet index of refraction in each voxel varies with position, and the wet index of refraction at any point in the voxel is determined by the weighted sum of the wet index values of the eight nodes N 1 to N 8 of the voxel , so that SWD is expressed as the integral of the wet index of refraction at the voxel node. Specifically, the Newton- Kertz method is used to solve the integral; the integral of the wet index of refraction Nw along the five equally spaced points P 1 -P 5 on the oblique path l can be Expressed in the following way:
Figure FDA0003462203930000012
Figure FDA0003462203930000012
 其中DP1P5是P1到P5的截距，P1，P2，P3，P4，P5是五个等距点，点Pk的湿折射率由N1到N8这8个体素节点的湿折射率值内插确定；where D P1P5 is the intercept of P 1 to P 5 , P 1 , P 2 , P 3 , P 4 , P 5 are five equidistant points, and the wet refractive index of point P k is composed of 8 individuals from N 1 to N 8 The wet refractive index value of the prime node is determined by interpolation; 顾及水汽时空分布的参数化建模方法见式5，Pk的湿折射率可以用点V1和V2垂直插值：The parametric modeling method considering the spatiotemporal distribution of water vapor is shown in Equation 5, and the wet refractive index of P k can be vertically interpolated with points V 1 and V 2 :
Figure FDA0003462203930000013
Figure FDA0003462203930000013
 其中V1为与N1～N4处在同一高度面上的点，V2为与N5～N8处在同一高度面上的点，且V1、V2与Pk点位于同一条竖线上；k为选自1～5中的任意整数，hPk为Pk点的高度；Among them, V 1 is a point on the same height plane as N 1 -N 4 , V 2 is a point on the same height plane as N 5 -N 8 , and V 1 , V 2 and P k are on the same line On the vertical line; k is any integer selected from 1 to 5, and h Pk is the height of the Pk point; 其中参数α可从下式估计得到：where the parameter α can be estimated from the following equation:
Figure FDA0003462203930000014
Figure FDA0003462203930000014
 其中h0指的是体素下表面的高度，hi指的是体素内任一点的高度，式6考虑到水汽随垂直高度的指数变化特征；where h 0 refers to the height of the lower surface of the voxel, and hi refers to the height of any point in the voxel. Equation 6 takes into account the exponential variation of water vapor with vertical height; 为顾及水汽时间变化，使用上一时段层析廓线来估算当前时段每个体素的α；To take into account the temporal variation of water vapor, use the tomographic profile of the previous period to estimate the α of each voxel in the current period; 此外，V1和V2的湿折射率值使用反距离加权插值计算：Additionally, the wet refractive index values for V 1 and V 2 are calculated using inverse distance-weighted interpolation:
Figure FDA0003462203930000021
Figure FDA0003462203930000021
 其中i为1或2，其中dj是Vi与Nj间的距离，Vi与Nj处在同一高度面上；u是距离的幂数，Nj是指Vi所处体素表面的4个周围节点；Nw(Nj)是Vj所处体素表面的4个周围节点的湿折射率值；where i is 1 or 2, where d j is the distance between Vi and N j , and Vi and N j are on the same height plane; u is the power of the distance, and N j is the voxel surface where Vi is located The 4 surrounding nodes of ; N w (N j ) is the wet refractive index value of the 4 surrounding nodes on the surface of the voxel where V j is located; 为提高建模精度，每个时段利用上一时段层析结果估计出每一垂直层的u值。In order to improve the modeling accuracy, the u value of each vertical layer is estimated in each period using the tomographic results of the previous period. 2.根据权利要求1所述的对流层层析方法，其特征在于，收集建模期间所有SWD，建立SWD和湿折射率场之间的线性系统：2. The tropospheric tomography method according to claim 1, wherein all SWDs during modeling are collected to establish a linear system between the SWD and the wet refractive index field: y＝Ax (8)y=Ax (8) 其中y是SWD观测量的矢量，x是包含所有体素节点的湿折射率的未知参数矢量，A是由SWD的x贡献组成的设计矩阵。where y is the vector of SWD observations, x is the unknown parameter vector containing the wet refractive index of all voxel nodes, and A is the design matrix consisting of the x contributions of the SWD. 3.根据权利要求2所述的对流层层析方法，其特征在于，由于方程8通常存在不适定问题，则附加水平和垂直约束来使方程满秩，再利用最小二乘方法求解x。3. The tropospheric tomography method according to claim 2, characterized in that, since equation 8 usually has an ill-posed problem, horizontal and vertical constraints are added to make the equation full rank, and then the least squares method is used to solve x.
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