CN106897526B

CN106897526B - Hydrological model efficiency coefficient calculation method based on weighted minimum-multiplication

Info

Publication number: CN106897526B
Application number: CN201710115825.1A
Authority: CN
Inventors: 占车生; 韩建
Original assignee: Institute of Geographic Sciences and Natural Resources of CAS
Current assignee: Institute of Geographic Sciences and Natural Resources of CAS
Priority date: 2017-03-01
Filing date: 2017-03-01
Publication date: 2020-04-10
Anticipated expiration: 2037-03-01
Also published as: CN106897526A

Abstract

The invention discloses a hydrological model efficiency coefficient calculation method based on weighted minimum-multiplication, which comprises the following steps of: step 1, acquiring an observation and simulation runoff sequence; step 2, calculating the arithmetic mean of the observation and simulation runoff sequence to obtain average runoff; step 3, taking the reciprocal of the average flow as the weight of the weighted minimum multiplication; and 4, dividing the weighted sum of the observation analog value residual absolute value sequence subtracted by 1 by the sequence length by the improved efficiency coefficient. The method corrects the problem that the Nash efficiency coefficient is excessively sensitive to the peak value by quantitatively considering heteroscedasticity widely existing in hydrological simulation and combining with weighted minimum multiplication. The method also has the advantages of boundedness, time domain independence, robustness and the like, and is easy to popularize in other model evaluation researches.

Description

Hydrological model efficiency coefficient calculation method based on weighted minimum-multiplication

Technical Field

The invention relates to the technical field of model efficiency coefficient calculation of a hydrological model, in particular to a hydrological model efficiency coefficient calculation method based on weighted minimum multiplication.

Background

The hydrological model is the main tool for researching the water circulation process of the basin at present. In model applications, it is generally necessary to evaluate the simulation effect by calculating certain efficiency coefficient(s) so as to obtain the optimal parameter combination by means of an optimization algorithm. The most widely used efficiency coefficient at present is the Nash-Sutcliffe efficiency coefficient^[1](hereinafter referred to as Nash efficiency coefficient) is calculated as follows:

wherein NSE represents the Nash efficiency coefficient; n is the sequence length; qbs is the observed runoff sequence; qsim is a simulated runoff sequence; RMSE is the root mean square error; sigma_obsIs the variance of the observed runoff sequence.

Practical application and intensive research show that the Nash efficiency coefficient has some defects: (1) hydrologic simulations are generally heteroscedastic^[2]The Nash efficiency coefficient is essentially a standard least squares method, which assumes that the simulated residuals are homogeneous in variance; (2) efficiency of modelThe coefficients are essentially a measure operator^[3]The error part must satisfy the nonnegativity, symmetry and triangle inequality. But the error part of the Nash coefficient does not meet the latter two requirements; (3) the Nash efficiency coefficient is very sensitive to the peak value, so that the peak value in the calibration result is better in simulation and the low-flow area is poorer in simulation; (4) the Nash efficiency coefficient is sensitive to the variance of the observed runoff^[4]In some cases, the phenomenon that the variance of the observed runoff is large and the simulation effect is common, but the Nash efficiency coefficient is also large can occur.

Aiming at the defects of the Nash efficiency coefficient, a plurality of improved methods are provided. For example, the second order index in the Nash efficiency coefficient is changed into the first order index^[5]To reduce its sensitivity to peaks; grouping calculation residual square sum and observation runoff variance^[6]It may also reduce its sensitivity to peaks; the weighted least square method is used for replacing the standard least square method to solve the problem of heteroscedastic difference, and the important step of the method is weight calculation; gupta and other decomposition based on Nash efficiency coefficient designs new efficiency coefficient KGE^[7]. In addition, there are many indexes different from Nash efficiency coefficients, such as deterministic coefficient and water balance coefficient, which have more obvious defects and are not described herein again.

The references referred to herein are as follows:

[1]NASH J E, SUTCLIFFE J V. River flow forecasting through conceptualmodels part I — A discussion of principles [J]. J Hydrol, 1970, 10(3): 282-90.

[2]SOROOSHIAN S, DRACUP J A. Stochastic Parameter-EstimationProcedures for Hydrologic Rainfall-Runoff Models - Correlated andHeteroscedastic Error Cases [J]. Water Resour Res, 1980, 16(2): 430-42.

[3]GUINOT V, CAPPELAERE B, DELENNE C, et al. Towards improvedcriteria for hydrological model calibration: theoretical analysis ofdistance- and weak form-based functions [J]. J Hydrol, 2011, 401(1-2): 1-13.

[4]SCHAEFLI B, GUPTA H V. Do Nash values have value. [J]. HydrolProcess, 2007, 21(15): 2075-80.

[5]LEGATES D R, MCCABE G J. Evaluating the use of "goodness-of-fit"measures in hydrologic and hydroclimatic model validation [J]. Water ResourRes, 1999, 35(1): 233-41.

[6] liujiajiajia, Zhouzhuhao, Jia Shang Wen, a calculation method of improved Nash efficiency coefficient, CN104143025A [ P/OL ]. 2014-11-12.

[7]GUPTA H V, KLING H, YILMAZ K K, et al. Decomposition of the meansquared error and NSE performance criteria: Implications for improvinghydrological modelling [J]. J Hydrol, 2009, 377(1-2): 80-91.

Disclosure of Invention

Aiming at the defects of the prior art, the invention provides a hydrological model efficiency coefficient calculation method based on weighted minimum multiplication, aiming at the difficult problems that the Nash efficiency coefficient does not consider heteroscedasticity and does not meet the second two axioms of a measurement operator.

In order to solve the technical problems, the invention adopts the following technical scheme:

a hydrological model efficiency coefficient calculation method based on weighted minimum-multiplication comprises the following steps:

step 1, acquiring an observation and simulation runoff sequence;

step 2, calculating the arithmetic mean of the observation and simulation runoff sequence to obtain the average runoff q_mean；

And 3, taking the reciprocal of the average runoff as the weight of the weighted minimum multiplication. It is assumed here that the residual magnitude is proportional to the magnitude of the average runoff, and to improve the robustness of the efficiency coefficient, the least squares method is changed to a minimum-one multiplication, where the weights do not need to be normalized. (ii) a

Step 4, the improved efficiency coefficient is equal to 1 minus the weighted sum of the observation analog value residual absolute value sequence divided by the sequence length;

step 5, the specific calculation method according to the method is as follows:

wherein, HSE represents an improved hydrological model efficiency coefficient considering heteroscedasticity and symmetry; n is the sequence length; q. q.s_obsTo observe runoff sequence; q. q.s_simSimulating a runoff sequence; q. q.s_meanIs the average runoff sequence.

The invention is characterized in that: assuming that the size of the residual error is proportional to the size of the average runoff, the inverse of the average runoff is used as the weight of the weighted least-one multiplication, and finally the model efficiency coefficient HSE is calculated.

Compared with the prior art, the invention has the following advantages and beneficial effects:

firstly, the method quantitatively considers the residual heteroscedasticity widely existing in hydrological simulation, and takes the reciprocal of the average runoff as the weight of the weighted minimum multiplication. The larger the average runoff is, the smaller the weight occupied by the corresponding error is, i.e. the greater the tolerance to the corresponding error is, which is consistent with the experience of hydrological practice. The problem that the Nash efficiency coefficient is excessively sensitive to the peak value is corrected by changing the standard least square method into the weighted least square method. The simulation effect obtained by the method is closer in the high, medium and low flow sections.

Secondly, the error part (namely, the number of the subtractions of 1) of the method meets three axioms (nonnegativity, symmetry and triangle inequality) of a measurement operator, and combines the problems in the specific hydrological field with a strict mathematical theory (measurement space part), thereby being beneficial to deepening the understanding and the cognition of the hydrological model evaluation and calibration research.

Thirdly, the efficiency coefficient obtained by the method has the characteristics of boundedness (the minimum is-1 and the maximum is 1), time domain independence, robustness and the like which are superior to the existing indexes such as Nash efficiency coefficient and the like.

The method is not limited in the field of runoff simulation, and can be popularized to the simulation evaluation and calibration research of other hydrological processes, ecological processes and environmental processes as long as the method basically conforms to the assumption that the simulation error is proportional to the average value sequence.

Drawings

FIG. 1 is a flow chart of the process of the present invention;

FIG. 2 is a sequence of observed and simulated runoff in an embodiment of the method of the invention.

Detailed Description

The technical solution of the present invention is further specifically described below by way of examples with reference to the accompanying drawings.

As shown in fig. 1, a method for calculating efficiency coefficients of a hydrological model based on weighted minimum-one multiplication includes the following steps:

step 1, data preparation. Observing runoff sequence and sorting the data from the hydrological station; after the research area, the used model and the parameter value are determined, the model is operated to obtain a simulated runoff sequence. Wherein the parameter value is a sampling point obtained by using a random sampling algorithm such as Monte Carlo in a parameter space;

step 2, calculating an average runoff sequence, namely an arithmetic average value of the observation runoff sequence and the simulation runoff sequence;

and 3, taking the reciprocal of the average runoff as the weight of the weighted minimum multiplication. The situation that the residual error is in direct proportion to the average runoff is assumed, meanwhile, in order to improve the robustness of the efficiency coefficient, the least square method is changed into the minimum multiplication, and the weight does not need to be normalized;

step 4, calculating an improved efficiency coefficient which is equal to 1 minus the weighted sum of the observation analog value residual absolute value sequence divided by the sequence length; the step is equivalent to calculating the efficiency coefficient of each time point and then calculating the average value of the efficiency coefficients;

step 5, the specific calculation method according to the method is as follows:

In the actual model calibration work, the method is a calculation method of an objective function and should be used together with a hydrological model parameter random sampling method, a hydrological model parameter optimization algorithm and the like to obtain the optimal parameter combination.

Claims

1. A hydrological model efficiency coefficient calculation method based on weighted minimum-multiplication is characterized by comprising the following steps:

step 1, acquiring an observation runoff sequence, and operating a model by determining a research area, the model and a parameter value to obtain a simulation runoff sequence; wherein the parameter value is a sampling point obtained by adopting a Monte Carlo random sampling algorithm;

Step 3, if the residual error is in direct proportion to the average runoff, taking the reciprocal of the average runoff as the weight of weighted minimum multiplication, wherein the weight is not normalized;

step 5, the specific calculation method according to the method is as follows:

wherein, HSE represents an improved hydrological model efficiency coefficient considering heteroscedasticity and symmetry; n is the sequence length；q_obs(t) is the observation runoff sequence; q. q.s_sim(t) is a simulated runoff sequence; q. q.s_mean(t) is the mean runoff sequence.

2. The method of claim 1, wherein the inverse of the average runoff is used as a weight for the weighted least one multiplication in step 3, where the average runoff is used instead of the observed runoff or the simulated runoff to ensure that it meets the symmetry axiom of the metric operator.

3. The method of claim 1, wherein the inverse of the average runoff is used as a weight for the weighted least squares multiplication in step 3, and the method includes the assumption that the residual error of the hydrological model simulation results is proportional to the average runoff.

4. The method of claim 1, wherein the inverse of the average runoff is used as the weight for the weighted least-squares multiplication in step 3, where the weight is not normalized, because the weight, i.e., the product dimension of the inverse of the average runoff and the residual error, is exactly 1, and no normalization is necessary.

5. The method of claim 1, wherein in step 3, the inverse of the average runoff is used as a weight for weighting the least squares method instead of the least squares method to reduce the sensitivity of the rating result to extreme values.

6. A method as claimed in claim 1, wherein the efficiency coefficients are calculated in steps 4 and 5 in a manner which differs from the order of the Nash efficiency coefficients, the former summing first and then the latter summing and then the quotient.