CN111090130B - Improved algorithm for radar-rain gauge joint precipitation estimation based on minimum functional boundary condition acquisition - Google Patents
Improved algorithm for radar-rain gauge joint precipitation estimation based on minimum functional boundary condition acquisition Download PDFInfo
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Abstract
The invention discloses an improved algorithm for radar-rain gauge joint precipitation estimation based on minimum functional acquisition boundary conditions, which solves the complete boundary conditions of a radar scanning circular area by using a large amount of known rain gauge observation data in a radar scanning area and the minimum functional solution in a variational equation form, further solves a variational method, avoids errors introduced by supposing the boundary conditions, and has more perfect theory.
Description
Technical Field
The invention belongs to the technical field of weather forecast, and particularly relates to a method for estimating precipitation in an area by combining a weather radar with a ground rain gauge for observation.
Background
The combination of a weather radar and a ground rain gauge for ground precipitation estimation is a common problem in meteorological and hydrological services. In the prior art, the rainfall of each point in a radar scanning range is obtained by fitting on the basis of the statistical relationship between the radar reflectivity intensity and the rainfall intensity; and (3) calibrating the radar fitting precipitation result by using a certain mathematical method (generally a variational method) and the observation result of the ground rain gauge, thereby obtaining a calibrated large-range precipitation result.
In the case of the joint radar-rain gauge fitting to precipitation by means of variational methods, it is necessary to provide complete region boundary values, i.e. boundary conditions in the mathematical definition. In practical situations, complete boundary conditions are impossible to obtain, and assumed boundary conditions are generally adopted to set the boundary to be a certain constant or a certain fitting value. This method introduces errors, resulting in a less accurate precipitation estimation result.
Disclosure of Invention
The purpose of the invention is as follows: in view of the existing problems and disadvantages, the invention aims to provide an improved algorithm for radar-rain gauge joint precipitation estimation, which obtains boundary conditions based on minimum functional, and solves the complete boundary conditions of a radar scanning circular area by using the known large amount of rain gauge observation data in a radar scanning area and the minimum functional of a variational equation form solution, so as to solve a variational method, avoid errors caused by the assumed boundary conditions, and obtain a more accurate rainfall estimation value.
The technical scheme is as follows: in order to realize the purpose, the invention adopts the following technical scheme: an improved algorithm for radar-rain gauge joint precipitation estimation based on minimum functional boundary condition acquisition comprises the following steps:
(1) in a circular area covered by a weather radar, a polar coordinate system is established by taking the center of the weather radar as the center of a circle, and according to the statistical relationship Z between the radar emissivity Z and the rainfall intensity I, AIb(generally, in China, A is 296, b is 1.24), the rainfall u (r, theta) of each point of the region is obtained through the formula (1),
in the formula, a represents the radius of the circular area, (r, theta) represents the radius and azimuth angle of the polar coordinate of the boundary point of the circular area, (r ', theta ') represents the radius and azimuth angle of the polar coordinate of the internal point of the circular area, and g (a, theta ') represents the rainfall of the boundary point, which is called boundary condition; mu-muR+μGIn which μGAnd muRAre all weight coefficients and correspond to different observation error terms, K1An infinite series of approximations for a finite term number of 1, G (r, theta; r ', theta ') being a Green's function, f (r ', theta ') being a function containing muGAnd muRA known term of (a);
(2) the real rainfall u is directly observed by a rain gauge in a circular area0(r, θ) is substituted as a known term on the left side in formula (1) to give formula (2),
carrying out inversion fitting on the formula (2) to obtain a boundary condition g (a, theta '), and then using the boundary condition g (a, theta') as a known term to replace an acquisition area of the rainfall u (r, theta) equation (1)A corrected value of the rainfall u (r, theta) at each point of the domain; and K in the Green function G (r, theta; r ', theta')kAn infinite series of finite term approximations,
in the above formula, j is the serial number of the finite term, k is the term number of the finite term, m and n are natural numbers, C is a constant of the Euler equation, C is approximately equal to 0.5772157, and x is the independent variable of the finite term series;
(3) stability condition determination for inversion boundary condition g (a, θ
Setting the rainfall u (r, theta) in the equation (1), and only the boundary condition g (a, theta ') is an unknown condition, the matrix coefficient A of g (a, theta') can be obtained,
in the formula (4), the coordinate number (i, k) belongs to uG，uGJ is the serial number of the jth point of the boundary condition, j is 1,2,3 … n, and n is a matrix dimension; a is the radius of the boundary point, r is the radius of the inner point, and further the equivalent independent equation number MI is obtained,
in formula (5), σjIs the singular value of A (X) coefficient matrix, j is the serial number of the j point of the boundary condition, j is 1,2,3 … n, n is the matrix dimension;
when MI is larger than or equal to L, judging that the equation (2) is stable, and obtaining boundary conditions g (a, theta') through inversion fitting of the equation (2) so as to further obtain the rainfall at each point of the circular area; otherwise equation (2) is unstable.
Further, the number of boundary values of the circular area covered by the weather radar is 360.
Has the advantages that: compared with the prior art, the invention develops a new algorithm without assuming the boundary condition aiming at the problem of assuming the boundary condition in the weather radar-rain gauge combination by using the variational method, the method solves the complete boundary condition of the radar scanning circular area by using the minimum functional of the observation data of a large amount of known rain gauges in the radar scanning area and the form solution of the variational equation, and then solves the variational method, thereby avoiding the error introduced by the assumed boundary condition and obtaining a more accurate rainfall estimation value.
Drawings
FIG. 1 is a schematic flow chart of the improved algorithm of the present invention;
FIG. 2 is a diagram illustrating a distribution of radar reflectivity and a rain gauge during a certain rainfall in an embodiment of the present invention.
Detailed Description
The present invention is further illustrated by the following figures and specific examples, which are to be understood as illustrative only and not as limiting the scope of the invention, which is to be given the full breadth of the appended claims and any and all equivalent modifications thereof which may occur to those skilled in the art upon reading the present specification.
With reference to the flow of fig. 1, the technical idea and flow of the method of the present invention are described in detail as follows:
in a circular area with the center of the radar as the center of a circle, according to the statistical relationship Z (AI) between the reflectivity Z of the classical radar and the rainfall intensity Ib(in China, A is 296, b is 1.24), and radar fitted rainfall at each point is obtained. Simultaneously, precipitation obtained by direct observation of a rain gauge in the area is recorded as uG. Establishing an extreme value problem that the correction precipitation u (x, y) on each point meets the following functional according to the smoothness of the problem field and the dependency on the observation data
Wherein Ω { (x, y) | x2+y2≤a2Denotes the circular area of the radar scan, a denotes the radius of the circular area, x and y are the coordinates of the east-west and north-south directions, respectively, and u is the estimated rainfall at each point, i.e. eventuallyThe value, u, obtainedRIs the radar inversion rainfall, u, obtained from the Z-I relationship at each pointGIs a rain gauge observation. min indicates that the equation on the right is to reach a minimum. Mu.sGAnd muRAre weighting coefficients corresponding to different observation error terms. Corresponding Euler equation (2) derived from the variational problem (1) and having boundary conditions (3)
u(x,y)|Γ＝g(x,y) (3)
Wherein Γ { (x, y) | x2+y2＝a2Denotes a boundary, g (x, y) is a rainfall of the boundary point,
μ＝μR+μG。
the form of equation (2) is solved as Green function of Klein-Gordon equation for circular regions
Wherein (r, theta) and (r ', theta') are the radius and azimuth angle of the polar coordinates of any point in the circular region. f (r ', θ') is including uRAnd uGG (a, θ') is the polar coordinate of the boundary point rainfall; wherein green function
G(r,θ；r',θ')
the invention obtains the real rainfall u by directly observing the rain gauge in the circular area0And (r, theta) is used as a known term to be substituted on the left side in the formula (4), then the formula (4) is subjected to inversion fitting to obtain a boundary condition g (a, theta '), and the boundary condition g (a, theta') is used as a known term to be substituted back into the rainfall u (r, theta) equation (4), so that the corrected value of the rainfall u (r, theta) on each point of the area is obtained.
2. Inversion boundary conditions and their stability conditions
Since the inverse fitting may cause a situation that diffusion cannot be approximated in the process of performing the inverse fitting on equation (4) to obtain the boundary condition g (a, θ'), it is necessary to determine a stability condition of the above method to confirm the adaptation range:
to solve the boundary condition g (a, θ'), an inverse problem can be obtained according to equation (4): in equation 4, the matrix coefficient of g (a, θ ') is a by setting only g (a, θ') as unknown condition,
wherein the coordinate number (i, k) is e.uGL is the number of boundary points, a is the boundary point radius, and r is the interior point radius. To solve the stability problem with this inverse problem method, the coefficient matrix of the left side g (a, θ') of equation (5) is A,
defining equivalent independent equation number
σjIs the singular value of the A coefficient matrix, and n is the matrix dimension. When MI.gtoreq.L, equation (5) is stable and the boundary condition g (a, θ') can be found. j is the serial number of the j point of the boundary condition, j is 1,2,3 … n, and n is a matrix dimension; a is the boundary point radius and r is the interior point radius.
Finally, the calculated g (a, θ') is substituted into equation (4), and the rainfall at each point can be obtained.
In the actual radar quantitative estimation precipitation problem, the number of boundary values of a circular area is 360 in terms of radar scanning radial number; the observation number of the ground rain gauges in the scanning area is far more than 360, the stability condition of an equation (5) is met, the equation can be solved, and the complete boundary condition is obtained
And then the rain amount of any point in the circular area can be obtained in the step (4).
Fig. 2 is a diagram showing the radar reflectivity and the rain gauge distribution obtained by applying the present invention to a place during a certain rainfall. Table 1 shows the estimated rainfall for some boundary points obtained by the algorithm (Pos is the polar coordinate of the point, Rad is the radar reflectivity intensity, Rtr is the Z-I relationship fitting precipitation, Obs is the rain gauge observed value, Joi is the algorithm estimated precipitation); table 2 estimates the rainfall for the part of interior points obtained by the present algorithm.
TABLE 1
TABLE 2
| No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Pos | (25,183) | (41,107) | (50,160) | (55,332) | (62,131) | (78,125) | (84,120) | (93,294) |
| |
40 | 28 | 41 | 39 | 42 | 43 | 44.5 | 41.5 |
| Rtr | 1.4245 | 0.1534 | 1.7151 | 1.1830 | 2.0651 | 2.4865 | 3.2851 | 1.8819 |
| Obs | 0.5 | 0.2 | 1.2 | 0.2 | 0.7 | 3.4 | 3.5 | 0.2 |
| Joi | 0.5000 | 0.1900 | 0.9000 | 0.2000 | 0.725 | 2.3533 | 2.4286 | 0.1692 |
Tables 1 and 2 are comparisons of precipitation results obtained for different solutions at the border and inner part points of the circular area, respectively. Where Pos is the polar coordinates of the point, Rad is the radar reflectivity observation, Rtr represents the 5 minute estimated precipitation (in mm) using the Z-I relationship, Obs is the actual 5 minute rain gauge observation (in mm), and Joi is the 5 minute estimated rain (in mm) obtained by this method. Obs were used as standards in all comparisons. It is clear from table 1 that at point 1, both Rtr and Joi are almost identical; at point 3 Rtr is greater than Obs 0.16, Joi is less than Obs 0.3, Joi method is slightly inferior; in point 5, the difference between the Rtr and Joi methods and the Obs is not great; the Joi method results are much improved over Rtr at points 2, 4, 6, 7, and 8. The root mean square error of the deviation of Rtr from Obs is 7.46, and the root mean square error of Joi from Obs is 0.06. The results for the points in Table 2 except for Jor at point 7 were inferior to that for Rtr, and Joi at all 7 points were superior to that for Rtr. The root mean square error of the deviation of Rtr from Obs is 0.669, and the root mean square error of the deviation of Joi from Obs is 0.199.
Claims (2)
1. An improved algorithm for radar-rain gauge joint precipitation estimation based on minimum functional boundary condition acquisition comprises the following steps:
(1) in a circular area covered by a weather radar, a polar coordinate system is established by taking the center of the weather radar as the center of a circle, and according to the statistical relationship Z between the radar emissivity Z and the rainfall intensity I, AIbThe rainfall u (r, theta) at each point of the region is obtained by the formula (1),
in the formula, a represents the radius of the circular area, (r, theta) represents the radius and azimuth angle of the polar coordinate of the boundary point of the circular area, (r ', theta ') represents the radius and azimuth angle of the polar coordinate of the internal point of the circular area, and g (a, theta ') represents the rainfall of the boundary point, which is called boundary condition; mu-muR+μGIn which μGAnd muRAre all weight coefficients and correspond to different observation error terms, K1An infinite series of approximations for a finite term number of 1, G (r, theta; r ', theta ') being a Green's function, f (r ', theta ') being a function containing muGAnd muRA known term of (a);
(2) the real rainfall u is directly observed by a rain gauge in a circular area0(r, θ) is substituted as a known term on the left side in formula (1) to give formula (2),
carrying out inversion fitting on the formula (2) to obtain a boundary condition g (a, theta'), and then obtaining a corrected value of the rainfall u (r, theta) on each point of the area as a known term to replace the rainfall u (r, theta) equation (1); and Green letterOf the numbers G (r, theta; r ', theta'), KkAn infinite series of finite term approximations,
in the above formula, j is the serial number of the finite term, k is the term number of the finite term, m and n are natural numbers, C is a constant of the Euler equation, C is approximately equal to 0.5772157, and x is the independent variable of the finite term series;
(3) stability condition determination for inversion boundary condition g (a, θ
Setting the rainfall u (r, theta) in the equation (1), and only the boundary condition g (a, theta ') is an unknown condition, the matrix coefficient A of g (a, theta') can be obtained,
in the formula (4), the coordinate number (i, k) belongs to uG，uGJ is the serial number of the jth point of the boundary condition, j is 1,2,3 … n, and n is a matrix dimension; a is the radius of the boundary point, r is the radius of the inner point, and further the equivalent independent equation number MI is obtained,
in formula (5), σjIs the singular value of A (X) coefficient matrix, j is the serial number of the j point of the boundary condition, j is 1,2,3 … n, n is the matrix dimension;
when MI is larger than or equal to L, judging that the equation (2) is stable, and obtaining boundary conditions g (a, theta') through inversion fitting of the equation (2) so as to further obtain the rainfall at each point of the circular area; otherwise equation (2) is unstable.
2. The improved algorithm for radar-rain gauge joint estimation of precipitation based on minimum functional acquisition boundary conditions according to claim 1, characterized by: the number of boundary values of the circular area covered by the weather radar is 360.
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