CN111709146A - Method for establishing relation between radar reflectivity factor and rainfall intensity based on Copula function - Google Patents
Method for establishing relation between radar reflectivity factor and rainfall intensity based on Copula function Download PDFInfo
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Abstract
The invention discloses a method for establishing a relation between a radar reflectivity factor and rainfall intensity based on a Copula function. The quantitative statistical relationship between the radar reflectivity factor and the precipitation intensity established by the invention can represent the non-normal characteristics of the reflectivity factor and the precipitation intensity, accurately describe the nonlinear correlation structure between the radar reflectivity factor and the precipitation intensity, and is beneficial to improving the radar measurement precipitation precision.
Description
Technical Field
The invention belongs to the field of radar measurement of rainfall, and particularly relates to a method for establishing a relation between a radar reflectivity factor and rainfall intensity based on a Copula function.
Background
Accurate regional rainfall measurement is of great significance to scientifically understanding hydrologic cycle, forecasting and early warning of rainstorm flood and water resource management. Compared with the traditional ground rainfall station, the weather radar has the advantages of wide measurement range, high space-time resolution and capability of timely obtaining large-area quantitative precipitation data, is an effective tool for measuring regional precipitation at present and is widely adopted by meteorological service departments.
The radar measured precipitation mainly utilizes the obtained radar reflectivity factor Z, and R is calculated according to the relation between the radar reflectivity factor Z and precipitation intensity R obtained in advance. At present, common methods for establishing the relation between the radar reflectivity factor Z and the precipitation intensity R mainly comprise a Marshall-Palmer model method, a least square method, a genetic algorithm, an artificial neural network method and the like. However, the methods have some problems and limitations, and the Marshall-Palmer model method needs a large amount of actually measured drop spectrum data, which causes difficulty in actual operation; the least square method and the genetic algorithm both need to presuppose specific function types and do not necessarily accord with the actual situation; the structure of the artificial neural network can only be selected by experience, and a unified theoretical guidance is lacked.
The Copula function can construct the joint distribution of a plurality of random variables with any edge distribution, can well capture the non-normal characteristics of the variables and the nonlinear correlation relationship between the variables, and is widely applied to the field of hydrological weather. At present, no document introduces Copula function into the relation establishment of radar reflectivity factor and precipitation intensity.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides a method for establishing a relation between a radar reflectivity factor and precipitation intensity based on a Copula function.
In order to solve the technical problems, the invention adopts the following technical scheme:
a method for establishing a relation between radar reflectivity factors and rainfall intensity based on a Copula function comprises the following steps:
step 1, collecting radar reflectivity factors and precipitation intensity data;
step 2, selecting a proper marginal probability distribution function line type according to the radar reflectivity factor and precipitation intensity data in the step 1, estimating parameters of the marginal probability distribution function line type, and determining an optimal marginal probability distribution function;
step 3, selecting a proper Copula function to construct a combined probability distribution function of the radar reflectivity factor and the rainfall intensity according to the radar reflectivity factor and the rainfall intensity data in the step 1 and the edge probability distribution function estimated in the step 2, and estimating parameters of the combined probability distribution function;
step 4, solving a conditional probability distribution function of precipitation intensity when the radar reflectivity factor is given according to the edge probability distribution function estimated in the step 2 and the combined probability distribution function constructed in the step 3;
and 5, establishing a relation between the radar reflectivity factor and the precipitation intensity according to the conditional probability distribution function in the step 4.
In the step 2, lognormal distribution, Gamma distribution and pearson type III distribution are used as alternative edge probability distribution function profiles, and parameters of the alternative edge probability distribution function are estimated by using a linear moment method.
In the step 2, the candidate edge probability distribution function with the minimum root mean square error of the one-dimensional theoretical frequency and the empirical frequency is used as the optimal edge probability distribution function.
In the step 3, a Gumbel-Hougaard Copula function is adopted to construct a joint probability distribution function of the radar reflectivity factor and the precipitation intensity, and a maximum likelihood method is adopted to estimate parameters of the Gumbel-Hougaard Copula function.
The method collects radar reflectivity factors and precipitation intensity data, determines an edge probability distribution function, constructs a combined probability distribution function of the radar reflectivity factors and the precipitation intensity based on a Copula function, further solves a conditional probability distribution function of the precipitation intensity when the radar reflectivity factors are given, and establishes a relation between the radar reflectivity factors and the precipitation intensity on the basis.
Compared with the prior art, the invention has the beneficial effects that:
the quantitative statistical relationship between the radar reflectivity factor and the precipitation intensity established by the invention can represent the non-normal characteristics of the reflectivity factor and the precipitation intensity, accurately describe the nonlinear correlation structure between the radar reflectivity factor and the precipitation intensity, and is beneficial to improving the radar measurement precipitation precision.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
FIG. 2 is a schematic diagram of a relationship between radar reflectivity factor and precipitation intensity established based on Copula function.
Detailed Description
The invention is further illustrated by the following examples in conjunction with the accompanying drawings.
As shown in fig. 1-2, a method for establishing a relation between radar reflectivity factors and precipitation intensity based on Copula function collects radar reflectivity factors and precipitation intensity data, determines an edge probability distribution function, constructs a joint probability distribution function of the radar reflectivity factors and the precipitation intensity based on the Copula function, further solves a conditional probability distribution function of the precipitation intensity when the radar reflectivity factors are given, and establishes a relation between the radar reflectivity factors and the precipitation intensity on the basis. Fig. 1 is a calculation flowchart of the present embodiment, which is performed according to the following steps:
1. collecting radar reflectivity factor and precipitation intensity data.
In the embodiment, the radar reflectivity factor data is acquired from a Chinese meteorological data network, the time scale is 6 minutes, and the hourly radar reflectivity factor data is obtained by carrying out arithmetic average on 10 data of 6 minutes in an hour, and the unit is mm6/m3. Precipitation intensity data are obtained from a ground rainfall station, and the time scale is 1 hour and the unit is mm/h.
2. And determining an edge probability distribution function of the radar reflectivity factor and the precipitation intensity.
Selecting a proper marginal probability distribution function line type according to the radar reflectivity factor and precipitation intensity data in the step 1, estimating parameters of the marginal probability distribution function line type, and determining an optimal marginal probability distribution function, wherein the step comprises three substeps:
2.1 alternative edge probability distribution function Linear
Since the overall frequency profile of the radar reflectivity factor and precipitation intensity is unknown, a profile is usually chosen that fits well to most sample data sets.
In the present embodiment, lognormal distribution, Gamma distribution, and pearson type III distribution are used as alternative edge probability distribution function profiles.
2.2 estimating parameters of the edge profile
The conventional methods for estimating edge distribution line parameters mainly include a moment method, a probability weight moment method, a weight function method, a linear moment method, and the like. The linear moment method is mainly characterized in that the method is sensitive to the maximum value and the minimum value of a sequence without the conventional moment, the estimated parameter estimation value is reliable, and the high-precision parameter estimation method is recognized at home and abroad at present.
In this embodiment, the parameters of the candidate edge probability distribution function are estimated by using an L-moment method.
2.3 optimal edge probability distribution function determination
And evaluating the fitting condition of the one-dimensional theoretical frequency and the empirical frequency of the edge distribution by adopting a Root Mean Square Error (RMSE) criterion, wherein the smaller the RMSE value is, the better the fitting effect is.
In the formula: f (x)i) Is an observed value xiThe theoretical frequency of (d); m (i) x is x ≦ x in the measured seriesiN is the sample length.
In this specific implementation, the candidate edge probability distribution function with the minimum RMSE value is used as the optimal edge probability distribution function.
3. And constructing a combined probability distribution function of the radar reflectivity factor and the rainfall intensity based on the Copula function.
Selecting a proper Copula function to construct a combined probability distribution function of the radar reflectivity factor and the rainfall intensity according to the radar reflectivity factor and the rainfall intensity data in the step 1 and the edge probability distribution function estimated in the step 2, and estimating parameters of the combined probability distribution function, wherein the step comprises two substeps:
3.1 selecting Copula function
Let Z, R represent the radar reflectivity factor and the precipitation intensity, respectively, and z and r are the corresponding realizations, respectively. FZ(z)、FR(r) is the edge probability distribution function, corresponding to a probability density function of fZ(z)、fR(r) of (A). The joint probability distribution function of Z, R can be represented by a two-dimensional Copula function C:
FZ,R(z,r)＝Cθ(FZ(z),FR(r))＝Cθ(u,v) (2)
wherein, θ is a parameter of the Copula function; u ═ FZ(z),v＝FR(r) is the edge probability distribution function.
In the specific implementation, a Gumbel-Hougaard Copula function is adopted to construct a combined probability distribution function of the radar reflectivity factor and the rainfall intensity, and the expression is as follows:
3.2 estimating the parameters of the Copula function
In the present embodiment, a maximum likelihood method is used to estimate the parameters of the Gumbel-Hougaard Copula function.
4. And solving the conditional probability distribution function of the precipitation intensity when the radar reflectivity factor is given.
When the value Z of the radar reflectivity factor Z is given, the corresponding value of the rainfall intensity R is not unique, only the probability of different values is different, and a conditional probability distribution function exists
FR|Z(r)＝P(R≤r|Z＝z) (4)
By means of Copula function, conditional probability distribution function FR|Z(r) can be expressed as:
5. and establishing a relation between the radar reflectivity factor and the precipitation intensity.
According to the conditional probability of the precipitation intensity R in the step 4Distribution function FR|Z(R), calculating to obtain the median R of the rainfall intensity RmThe median function of the rainfall intensity R obtained by the method is the relation between the established radar reflectivity factor and the rainfall intensity as an estimated value.
Median R of precipitation intensity RmSolving by:
FR|Z(rm)＝0.5 (6)
in the specific implementation, a numerical solution is obtained by trial calculation of a solution formula (6) by a dichotomy.
By solving the median R of the precipitation intensity R at any given Z-ZmAnd obtaining the relation between the radar reflectivity factor and the rainfall intensity based on the Copula function:
R＝rm(z) (7)
as shown in fig. 2, a schematic diagram of a relationship between a radar reflectivity factor and precipitation intensity established based on a Copula function is given. And (3) solid round points are the radar reflectivity factor and precipitation intensity point data collected in the step (1), and a solid line is a relation curve of the radar reflectivity factor and the precipitation intensity.
In summary, the invention determines the edge probability distribution function by collecting the radar reflectivity factor and the precipitation intensity data, constructs the joint probability distribution function of the radar reflectivity factor and the precipitation intensity based on the Copula function, further solves the conditional probability distribution function of the precipitation intensity when the radar reflectivity factor is given, and establishes the relation between the radar reflectivity factor and the precipitation intensity on the basis. The quantitative statistical relationship between the radar reflectivity factor and the precipitation intensity established by the invention can represent the non-normal characteristics of the reflectivity factor and the precipitation intensity, accurately describe the nonlinear correlation structure between the radar reflectivity factor and the precipitation intensity, and is beneficial to improving the radar measurement precipitation precision.
Claims (4)
1. A method for establishing a relation between radar reflectivity factors and rainfall intensity based on a Copula function is characterized by comprising the following steps:
step 1, collecting radar reflectivity factors and precipitation intensity data;
step 2, selecting a proper marginal probability distribution function line type according to the radar reflectivity factor and precipitation intensity data in the step 1, estimating parameters of the marginal probability distribution function line type, and determining an optimal marginal probability distribution function;
step 3, selecting a proper Copula function to construct a combined probability distribution function of the radar reflectivity factor and the rainfall intensity according to the radar reflectivity factor and the rainfall intensity data in the step 1 and the edge probability distribution function estimated in the step 2, and estimating parameters of the combined probability distribution function;
step 4, solving a conditional probability distribution function of precipitation intensity when the radar reflectivity factor is given according to the edge probability distribution function estimated in the step 2 and the combined probability distribution function constructed in the step 3;
and 5, establishing a relation between the radar reflectivity factor and the precipitation intensity according to the conditional probability distribution function in the step 4.
2. The method for establishing the relation between the radar reflectivity factor and the precipitation intensity based on the Copula function as claimed in claim 1, wherein: in the step 2, lognormal distribution, Gamma distribution and pearson type III distribution are used as alternative edge probability distribution function profiles, and parameters of the alternative edge probability distribution function are estimated by using a linear moment method.
3. The method for establishing the relation between the radar reflectivity factor and the precipitation intensity based on the Copula function as claimed in claim 1, wherein: in the step 2, the candidate edge probability distribution function with the minimum root mean square error of the one-dimensional theoretical frequency and the empirical frequency is used as the optimal edge probability distribution function.
4. The method for establishing the relation between the radar reflectivity factor and the precipitation intensity based on the Copula function as claimed in claim 1, wherein: in the step 3, a Gumbel-Hougaard Copula function is adopted to construct a joint probability distribution function of the radar reflectivity factor and the precipitation intensity, and a maximum likelihood method is adopted to estimate parameters of the Gumbel-Hougaard Copula function.
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|---|---|---|---|---|
| CN104615907A (en) * | 2015-03-11 | 2015-05-13 | 武汉大学 | Method for deriving and designing flood process line based on multi-variable most possible condition combination |
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|---|---|---|---|---|
| CN104615907A (en) * | 2015-03-11 | 2015-05-13 | 武汉大学 | Method for deriving and designing flood process line based on multi-variable most possible condition combination |
| CN108596998A (en) * | 2018-04-24 | 2018-09-28 | 江西省水利科学研究院 | A kind of rainfall runoff correlation drawing drawing method based on Copula functions |
| CN110082842A (en) * | 2019-05-24 | 2019-08-02 | 北京敏视达雷达有限公司 | A kind of precipitation estimation method and device |
| CN110276150A (en) * | 2019-06-27 | 2019-09-24 | 江西省水利科学研究院 | A kind of Mountain Area river basal flow capacity system interpolation extension method based on Copula function |
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Application publication date: 20200925 |