A hydrological model parameter calibration method and system based on system differential response relationship
By using a hydrological model parameter calibration method based on the system differential response relationship, and optimizing the parameters using the elastic network solution method and the iterative shrinkage threshold algorithm, the ill-conditioned problem in hydrological model parameter calibration is solved, and efficient and accurate parameter optimization is achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- HOHAI UNIV
- Filing Date
- 2026-02-12
- Publication Date
- 2026-05-29
AI Technical Summary
Existing hydrological model parameter calibration algorithms suffer from ill-conditioned problems, resulting in poor parameter optimization effects. Furthermore, existing intelligent algorithms lack convergence and result determinism, making it difficult to guarantee accuracy and efficiency.
A hydrological model parameter calibration method based on the system differential response relationship is adopted. By constructing an elastic network solution method for initial parameter error and model output simulation error, combined with iterative shrinkage threshold algorithm and cross-validation, the parameters are optimized to overcome ill-conditioned problems and ensure parameter convergence and accuracy.
It effectively solves the ill-conditioned problem in the calibration of hydrological model parameters, improves the accuracy and efficiency of parameter calibration, and ensures the stability and accuracy of parameter optimization.
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Abstract
Description
Technical Field
[0001] This invention relates to a method and system for calibrating hydrological model parameters, belonging to the field of hydrological forecasting and hydrological simulation technology. Background Technology
[0002] Hydrological models, as the primary tool for simulating and analyzing hydrological processes, represent an abstract and conceptual understanding of highly complex and nonlinear hydrological processes. They reflect the physical mechanisms of hydrological phenomena' evolution, exhibiting "deterministic dynamic laws." However, hydrological processes are extremely complex, and due to limitations in human cognition, model construction inevitably involves some degree of distortion. Furthermore, in practical applications, the model's input data, structure, and parameter calibration all affect simulation accuracy and prediction results, leading to "uncertain statistical laws" in hydrological simulations. Constrained by our understanding of hydrological physical processes and the current level of hydrological modeling technology, hydrological model parameters typically cannot be directly derived from watershed characteristics but must be determined indirectly, such as through historical rainfall-runoff data calibration. With numerous existing parameter calibration algorithms, improving efficiency while maintaining accuracy is the primary consideration for parameter calibration algorithms.
[0003] Automatic parameter calibration can be categorized into local search methods, statistical methods, and intelligent algorithms based on their algorithm structure and search mechanism. Local search methods have been widely used in various fields, but they are prone to getting trapped in local optima. Statistical methods, primarily based on random sampling (e.g., the commonly used Markov Monte Carlo method), can estimate the posterior probability distribution of parameters. While this can mitigate the decision-making risks caused by distorted results to some extent, the enormous computational cost makes it difficult to promote in practical applications. Intelligent algorithms, due to their comprehensive and efficient search capabilities within the parameter space, are widely used and have become the main means of solving single-objective model parameter optimization. However, in recent years, scholars have pointed out convergence issues with intelligent algorithms, making it impossible to verify whether the obtained parameter calibration results are globally optimal. Furthermore, due to the complementarity and correlation between model parameters, different parameter combinations often achieve the same simulation effect, i.e., "different parameters with the same effect." Therefore, when using intelligent algorithms based on sample information for parameter calibration, the results are highly uncertain due to this "different parameters with the same effect." To address this issue, some scholars have proposed a linearization calibration method suitable for nonlinear model parameters—the system differential response method. This method is effective in finding unique solutions for parameters and has achieved good results in parameter optimization for the Xin'anjiang model, SAC model, and VIC model. However, it should be noted that although the system differential response algorithm has shown superiority in optimizing hydrological model parameters, ill-conditioned coefficient matrices in the normal equations can affect the parameter calibration results. Summary of the Invention
[0004] Purpose of the invention: To address the shortcomings of existing technologies, the purpose of this invention is to provide a method and system for calibrating hydrological model parameters based on the differential response relationship of the system, which can solve the ill-conditioned problem of the original system differential response and ensure the efficiency of parameter calibration.
[0005] Technical Solution: To achieve the above-mentioned objectives, the present invention proposes a method for calibrating hydrological model parameters based on the system's differential response relationship, comprising the following steps:
[0006] Obtain the necessary input data for running the hydrological model, including hydrological observation time series data; construct the hydrological model based on the input data and parameter matrix;
[0007] Randomly sample within the given range of model parameter values to generate an initial parameter set;
[0008] The hydrological model is driven by the initial parameter set to obtain the initial simulation model output; the simulation error of the model output is obtained based on the hydrological observation time series data and the initial simulation model output; and the system differential response relationship between the initial parameter error and the model output simulation error is constructed based on the elastic network solution method.
[0009] Based on the system differential response relationship, the iterative solution formula for the initial parameter error is obtained based on the iterative shrinkage threshold algorithm. The cross-validation method is used to determine the optimal penalty coefficient and weight coefficient in the solution formula based on the minimum sum of squared residuals. The optimal coefficients are then substituted into the solution formula to solve for the initial parameter error.
[0010] The initial parameters are optimized based on the initial parameter error. It is then determined whether the parameters have converged. If they have not converged, the optimized parameters are used as the initial values to drive the hydrological model again until the parameters converge, and the final parameter calibration results are obtained.
[0011] Furthermore, the necessary input data for the hydrological model to run is obtained, including hydrological observation time-series data; the hydrological model is expressed based on the input data and parameter matrix, including:
[0012] The necessary input data for running a hydrological model is represented as a matrix x = (x1, x2, …, x…). k ) T Let k be the number of input variables, and let θ represent the parameter matrix θ=(θ1, θ2, …, θ n ) T n is the number of parameters, the superscript T indicates transpose, the hydrological model is represented as F=f(θ, x), F is the output variable matrix, and f(θ, x) represents the hydrological model function relationship;
[0013] Hydrological observation time series data are represented as F obs .
[0014] Furthermore, based on the hydrological observation time series data and the initial simulation model output, the model output simulation error is obtained, including:
[0015] Let the initial parameter set be θ (0) The initial simulation model output is F (0) =f(θ (0) According to hydrological observation time series data F obs Using the model parameters as independent variables, for f(θ) (0) Perform a first-order Taylor series expansion on x):
[0016] ,
[0017] in, Let the true value of the parameter be the one to be found. This is the random error term;
[0018] Assume the lengths of the input data and the observed time series data are m, i.e., (x1, F1), (x2, F2), ..., (x m , F m The hydrological model function relationship vector form is:
[0019] ,
[0020] Among them, F obs =(F1, F2, …, F m ) T ;F (0) =(F1 (0) F2 (0) , …, F m (0) ) T E (0) =(e1 (0) e2 (0) ,…, e m (0) ) T S is the Jacobian matrix, represented as follows:
[0021] ,
[0022] The partial derivatives of each term in matrix S are calculated using the following formula:
[0023] ,
[0024] Output simulation error for the model. This represents the initial parameter error.
[0025] Furthermore, the system differential response relationship based on the elastic net solution method, relating the initial parameter error to the model output simulation error, is expressed as:
[0026] ,
[0027] ,
[0028] in, Let λ be the parameter error to be determined, λ be the penalty coefficient, and α be the weight coefficient of the L1 penalty function.
[0029] Furthermore, based on the iterative shrinkage threshold algorithm, the iterative solution formula for the initial parameter error is obtained, including:
[0030] Based on gradient descent method, Iterative:
[0031] ,
[0032] in, and The calibration results are for the k-th and (k-1)-th calibrations, respectively; t k For fixed step size; for exist gradient at;
[0033] but In the current step By iteratively approximating the solution and neglecting the constant term, we have:
[0034]
[0035] The iterative threshold shrinkage algorithm solves the quadratic iterative equation separately and adds a soft threshold function to the iterative equation. Solve the problem. Representing the threshold of the soft thresholding function, we obtain:
[0036] ,
[0037] Where I is the identity matrix;
[0038] For the iterative process, a tolerance value tol is set. The algorithm stops iterating when the difference between the solutions between two iterations is less than tol.
[0039] Furthermore, using cross-validation, the optimal penalty coefficient and weight coefficient in the solution are determined based on minimizing the sum of squared residuals. These optimal coefficients are then substituted into the solution to calculate the initial parameter error, including:
[0040] The optimal λ and α coefficients are determined by minimizing the sum of squared residuals using ten-fold cross-validation. The initial range of the penalty coefficient λ is based on sampling using a logarithmic equipartition function, and the initial range of the weight coefficient α is [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0].
[0041] By substituting the optimal coefficients into the iterative equation introducing the soft threshold function, the final solution is obtained. .
[0042] The hydrological model parameter calibration system based on system differential response relationship provided by this invention includes:
[0043] The data acquisition module is used to acquire the necessary input data for the operation of the hydrological model, including hydrological observation time series data; and to construct the hydrological model based on the input data and parameter matrix.
[0044] The initial parameter set construction module is used to randomly sample within a given range of model parameter values to generate an initial parameter set;
[0045] The system differential response relation construction module is used to drive the hydrological model with the initial parameter set to obtain the initial simulation model output; based on the hydrological observation time series data and the initial simulation model output, the model output simulation error is obtained; and the system differential response relation between the initial parameter error and the model output simulation error is constructed based on the elastic network solution method.
[0046] The initial parameter error solving module is used to obtain the iterative solution formula for the initial parameter error based on the system differential response relationship and the iterative shrinking threshold algorithm. The cross-validation method is used to determine the optimal penalty coefficient and weight coefficient in the solution formula based on the minimum sum of squared residuals. The optimal coefficients are then substituted into the solution formula to solve for the initial parameter error.
[0047] The parameter optimization module is used to optimize the initial parameters based on the initial parameter error, determine whether the parameters have converged, and if not, use the optimized parameters as the initial values to drive the hydrological model to continue processing until the parameters converge and the final parameter calibration results are obtained.
[0048] The present invention also provides an electronic device, comprising: one or more processors; a memory; and one or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors, wherein when the programs are executed by the processors, they implement the steps of the hydrological model parameter calibration method based on the system differential response relationship as described above.
[0049] The present invention also provides a computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements the steps of the hydrological model parameter calibration method based on the system differential response relationship as described above.
[0050] The present invention also provides a computer program product, including a computer program, characterized in that, when the computer program is executed by a processor, it implements the steps of the hydrological model parameter calibration method based on the system differential response relationship as described above.
[0051] Beneficial effects: This invention uses the system differential response description formula based on the elastic network solution method to overcome the problem that current parameter calibration methods based on system differential response algorithms fail due to ill-conditioned coefficient matrices; it adopts the iterative shrinkage threshold method with high numerical stability to solve the parameter error iteratively, thus ensuring the efficiency of parameter calibration. Attached Figure Description
[0052] Figure 1 This is a flowchart of the method of the present invention;
[0053] Figure 2 In the example of the present invention and The comparison results;
[0054] Figure 3 This describes the iterative optimization process of parameter θ in this invention example. Detailed Implementation
[0055] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.
[0056] Reference Figure 1 This invention provides a method for calibrating hydrological model parameters based on the differential response relationship of a system, comprising the following steps:
[0057] Step (1): Obtain the necessary input time series data for the operation of the hydrological model, including hydrology, meteorology, topography, vegetation, etc., obtain hydrological observation time series data, and construct the hydrological model.
[0058] The necessary input data for model operation, such as hydrological, meteorological, topographic, and vegetation data, can be represented as a matrix x = (x1, x2, …, x…). k ) T Let k be the number of input variables. The hydrological model can be represented as F = f(θ, x), where F is the output variable matrix and θ is the parameter matrix θ = (θ1, θ2, …, θ). n ) T Here, n represents the number of parameters, and f(θ, x) represents the functional relationship of the hydrological model. Hydrological observation time-series data can be represented as F... obs .
[0059] In this invention example, the hydrological model adopts a vertical mixing runoff model, taking the Ili River Basin in Xinjiang as the study area. Daily gridded precipitation and evaporation data from 2011 to 2013 in the study area are used as model inputs to drive the vertical mixing runoff model. The output variable matrix is represented by the flow rate Q, and the hydrological observation time series data is represented by Q. obs The vertical mixing runoff model has 19 parameters, and their symbols, physical meanings, and value ranges are shown in Table 1. In the field of parameter calibration, conducting simulation experiments to demonstrate the ideal calibration effect is an effective means of verifying parameter calibration methods. In the simulation experiment, a set of true parameter values is preset. The hydrological model is driven by measured input data and the given true parameter values to obtain the true values of the model output. White noise error is applied to the model output to obtain a set of artificially generated observation outputs. Then, parameter calibration is performed using the given artificially generated measured outputs and measured inputs, assuming that the true parameter values are unknown. The goodness of fit between the calibrated parameters and the preset true parameter values is then tested to evaluate the effectiveness of the parameter calibration method. Therefore, in this invention example, by preset a set of true parameter values (see Table 2), combined with precipitation and evaporation input data, the vertical mixing runoff model is driven to simulate the runoff at the outlet section. Gaussian white noise with a mean of 0 and a standard deviation of 5% of the simulated runoff value is added to the simulated runoff as the observed runoff.
[0060] Table 1 Parameters of the Vertical Mixing Flow Generation Model
[0061]
[0062] Table 2. Preset True Values of Vertical Mixing Flow Generation Model Parameters in Simulation Experiments
[0063]
[0064] Step (2): Randomly sample within the given range of model parameter values to generate an initial parameter set. Use the initial parameter set to drive the hydrological model and construct the system differential response relationship between the initial parameter error and the model output simulation error based on the elastic network solution method.
[0065] In this invention, a set of initial parameters was generated by random sampling within the given range of parameters for the vertical mixing flow generation model, as shown in Table 3.
[0066] Table 3 Initial values of vertical mixing flow generation model parameters in simulation experiment
[0067]
[0068] The model is driven by this set of output parameters to obtain the simulated output. Combined with observation data Q obs Using model parameters as independent variables, the function model f(θ) (0)Linearization of x), i.e., first-order Taylor series expansion:
[0069]
[0070] in, Let the true value of the parameter be the one to be found. This is the random error term.
[0071] The lengths of the input and observed time series are 1096, namely (x1, Q1), (x2, Q2), ..., (x... 1096 Q 1096 The function model vector form is
[0072]
[0073] Among them, Q obs =(Q1, Q2, …, Q 1096 ) T Q (0) =(Q1 (0) Q2 (0) , …, Q 1096 (0) ) T E (0) =(e1 (0) e2 (0) , …, e 1096 (0) ) T S is the Jacobian matrix, represented as follows:
[0074]
[0075] Since the model is not a single differentiable mathematical expression, forward difference is used to calculate the partial derivatives of each term in matrix S. Set to 0.001, the partial derivative can be calculated using the following formula, taking parameter K as an example. The calculation formula is as follows:
[0076]
[0077] Furthermore, a system differential response relationship based on the elastic net solution method is constructed between the initial parameter error and the model output simulation error:
[0078]
[0079]
[0080] in, Let λ be the parameter error to be determined, λ be the penalty coefficient, and α be the weight coefficient of the L1 penalty function. λ and α are usually determined through cross-validation.
[0081] Step (3): Based on the iterative shrinkage threshold algorithm, the initial parameter error iterative solution formula is obtained. Ten-fold cross-validation is used to determine the optimal penalty coefficient and L1 weight coefficient in the solution formula based on the minimum residual square sum. The optimal coefficient is substituted into the solution formula to solve the initial parameter error. The initial parameters are optimized based on the initial parameter error.
[0082] Due to the inability to obtain The analytical solution needs to be obtained iteratively. First, based on the gradient descent method, we obtain... Iterative:
[0083]
[0084] in, and The calibration results are for the k-th and (k-1)-th calibrations, respectively; t k For fixed step size; for exist The gradient at that point.
[0085] therefore, In the current step The solution can be approximated through iteration and the constant term can be ignored:
[0086]
[0087] The iterative threshold shrinkage algorithm solves the quadratic iterative equation separately and adds a soft threshold function to the iterative equation. Solve the problem. Representing the threshold of the soft thresholding function, we obtain:
[0088]
[0089] In the formula Soft threshold function The threshold in, when Greater than Time to take , Less than Take 0 at that time.
[0090] For the iterative process, the tolerance tol=10 is set. -8 The algorithm stops iterating when the difference between the solutions of two iterations is less than tol.
[0091] The penalty coefficient λ and the L1 penalty function weight coefficient α are both used as hyperparameters. Ten-fold cross-validation (dividing the dataset into 10 parts, 9 for training and 1 for validation) is employed. The optimal λ and α parameters are determined based on minimizing the sum of squared residuals. The initial range of parameter λ is based on sampling using a logarithmic equipartition function, such as logspace(-4, 4, 100), which falls within 10... -3 Up to 10 4 Divide the given information into 100 logarithmic elements, with the initial value of α ranging from [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0]. Substitute the optimal coefficients into the iterative equation above to perform an iterative solution, yielding the final result. .
[0092] In this invention example, during the first parameter iteration optimization step, the hyperparameters λ and α, determined using ten-fold cross-validation, are 3125.7 and 1, respectively. These optimal coefficients are then substituted into... The solution is obtained by iterative process to obtain the final result. The results are shown in Table 4.
[0093] Table 4 Solution results
[0094]
[0095] Based on parameter error Optimize the initial parameters, i.e. Optimized parameters The results are shown in Table 5.
[0096] Table 5 Optimized parameters result
[0097]
[0098] Step (4): Determine whether the parameters have converged. If they have not converged, let the optimized parameters be the initial values and continue processing until the parameters converge and the final parameter calibration result is obtained.
[0099] With the new parameter θ (1) Run the hydrological model and calculate the difference between observed runoff and corresponding calculated runoff. ; Determine whether it converges, i.e. Is it equal to If convergence occurs, output θ. (1) Set the final calibration parameters; if it does not converge, let θ (0) =θ (1) Then return to step (2) to perform a new round of parameter optimization.
[0100] In the examples of this invention, and The comparison results are shown in Figure 2 It can be seen that after parameter optimization The magnitude is significantly smaller than ,but Not equal to If the convergence condition is not met, let θ (0) =θ (1) Then return to step (2) to perform a new round of parameter optimization.
[0101] In this invention example, before reaching the convergence condition, the parameter θ underwent multiple iterative optimization processes, the iterative process of which is described below. Figure 3 As can be seen, after 40 iterations, all parameter values converged to fixed values, indicating that the calibration algorithm has convergence. In fact, in the early stages of iteration, when the number of iterations reached 5, the optimized parameters were already very close to the preset true values. Furthermore, in the later stages of iteration, the parameters steadily converged to their true values with a smaller optimization step size, indicating that the algorithm has an effective step-size decay mechanism. The result θ of the 40th iteration is shown below. (40) The results are shown in Table 6. It can be seen that the final calibrated θ (40 Only WUM differs from the preset true value by 0.02, while the calibration results of the other parameters are the same as the preset true values. This indicates that the hydrological model parameter calibration method based on the system differential response iterative shrinkage threshold algorithm of the present invention can effectively calibrate the parameters.
[0102] Table 6 Optimized parameters θ (40) result
[0103]
[0104] Based on the same technical concept as the method embodiments, the present invention also provides a hydrological model parameter calibration system based on the system differential response relationship, comprising:
[0105] The data acquisition module is used to acquire the necessary input data for the operation of the hydrological model, including hydrological observation time series data; and to construct the hydrological model based on the input data and parameter matrix.
[0106] The initial parameter set construction module is used to randomly sample within a given range of model parameter values to generate an initial parameter set;
[0107] The system differential response relation construction module is used to drive the hydrological model with the initial parameter set to obtain the initial simulation model output; based on the hydrological observation time series data and the initial simulation model output, the model output simulation error is obtained; and the system differential response relation between the initial parameter error and the model output simulation error is constructed based on the elastic network solution method.
[0108] The initial parameter error solving module is used to obtain the iterative solution formula for the initial parameter error based on the system differential response relationship and the iterative shrinking threshold algorithm. The cross-validation method is used to determine the optimal penalty coefficient and weight coefficient in the solution formula based on the minimum sum of squared residuals. The optimal coefficients are then substituted into the solution formula to solve for the initial parameter error.
[0109] The parameter optimization module is used to optimize the initial parameters based on the initial parameter error, determine whether the parameters have converged, and if not, use the optimized parameters as the initial values to drive the hydrological model to continue processing until the parameters converge and the final parameter calibration results are obtained.
[0110] It should be understood that the hydrological model parameter calibration system based on the system differential response relationship in the embodiments of the present invention can realize all the technical solutions in the above method embodiments. The functions of each functional module can be specifically implemented according to the methods in the above method embodiments. The specific implementation process can be referred to the relevant descriptions in the above embodiments, which will not be repeated here.
[0111] The present invention also provides an electronic device, comprising: one or more processors; a memory; and one or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors, wherein when the programs are executed by the processors, they implement the steps of the hydrological model parameter calibration method based on the system differential response relationship as described above.
[0112] The present invention also provides a computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements the steps of the hydrological model parameter calibration method based on the system differential response relationship as described above.
[0113] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, apparatus (systems), electronic devices, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.
[0114] This invention is described with reference to a flowchart of a method according to embodiments of the invention. It should be understood that each step in the flowchart and combinations thereof can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing device to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing device, generate instructions for implementing the process. Figure 1A device for a function specified in one or more processes.
[0115] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 The function specified in one or more processes.
[0116] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 Steps of a specified function in one or more processes.
Claims
1. A method for calibrating hydrological model parameters based on the differential response relationship of a system, characterized in that, The method includes the following steps: Obtain the necessary input data for running the hydrological model, including hydrological observation time series data; construct the hydrological model based on the input data and parameter matrix; Randomly sample within the given range of model parameter values to generate an initial parameter set; The hydrological model is driven by the initial parameter set to obtain the initial simulation model output; the simulation error of the model output is obtained based on the hydrological observation time series data and the initial simulation model output; and the system differential response relationship between the initial parameter error and the model output simulation error is constructed based on the elastic network solution method. Based on the system differential response relationship, the iterative solution formula for the initial parameter error is obtained based on the iterative shrinkage threshold algorithm. The cross-validation method is used to determine the optimal penalty coefficient and weight coefficient in the solution formula based on the minimum sum of squared residuals. The optimal coefficients are then substituted into the solution formula to solve for the initial parameter error. The initial parameters are optimized based on the initial parameter error. It is then determined whether the parameters have converged. If they have not converged, the optimized parameters are used as the initial values to drive the hydrological model again until the parameters converge, and the final parameter calibration results are obtained.
2. The method according to claim 1, characterized in that, Obtain the necessary input data for running the hydrological model, including hydrological observation time series data; express the hydrological model based on the input data and parameter matrix, including: The necessary input data for running a hydrological model is represented as a matrix x = (x1, x2, …, x…). k ) T Let k be the number of input variables, and let θ represent the parameter matrix θ=(θ1, θ2, …, θ n ) T n is the number of parameters, the superscript T indicates transpose, the hydrological model is represented as F=f(θ,x), F is the output variable matrix, and f(θ, x) represents the hydrological model function relationship; Hydrological observation time series data are represented as F obs .
3. The method according to claim 2, characterized in that, Based on the hydrological observation time series data and the initial simulation model output, the simulation error of the model output is obtained, including: Let the initial parameter set be θ (0) The initial simulation model output is F (0) =f(θ (0) According to hydrological observation time series data F obs Using the model parameters as independent variables, for f(θ) (0) Perform a first-order Taylor series expansion on x): ， in, Let the true value of the parameter be the one to be found. This is the random error term; Assume the lengths of the input data and the observed time series data are m, i.e., (x1, F1), (x2, F2), ..., (x m , F m The hydrological model function relationship vector form is: ， Among them, F obs =(F1, F2, …, F m ) T ;F (0) =(F1 (0) F2 (0) , …, F m (0) ) T E (0) =(e1 (0) e2 (0) , …, e m (0) ) T S is the Jacobian matrix, represented as follows: ， The partial derivatives of each term in matrix S are calculated using the following formula: ， Output simulation error for the model. This represents the initial parameter error.
4. The method according to claim 3, characterized in that, The system differential response relationship based on the elastic net solution method, relating the initial parameter error to the model output simulation error, is expressed as: ， ， in, Let λ be the parameter error to be determined, λ be the penalty coefficient, and α be the weight coefficient of the L1 penalty function.
5. The method according to claim 4, characterized in that, The initial parameter error iterative solution formula is obtained based on the iterative shrinkage threshold algorithm, including: Based on gradient descent method, Iterative: ， in, and The calibration results are for the k-th and (k-1)-th calibrations, respectively; t k For fixed step size; for exist gradient at; but In the current step By iteratively approximating the solution and neglecting the constant term, we have: ， The iterative threshold shrinkage algorithm solves the quadratic iterative equation separately and adds a soft threshold function to the iterative equation. Solve the problem. Representing the threshold of the soft thresholding function, we obtain: ， Where I is the identity matrix; For the iterative process, a tolerance value tol is set. The algorithm stops iterating when the difference between the solutions between two iterations is less than tol.
6. The method according to claim 5, characterized in that, The cross-validation method is used to determine the optimal penalty coefficient and weight coefficient in the solution formula based on minimizing the sum of squared residuals. The optimal coefficients are then substituted into the solution formula to solve for the initial parameter error, including: The optimal λ and α coefficients are determined by minimizing the sum of squared residuals using ten-fold cross-validation. The initial range of the penalty coefficient λ is based on sampling using a logarithmic equipartition function, and the initial range of the weight coefficient α is [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0]. By substituting the optimal coefficients into the iterative equation introducing the soft threshold function, the final solution is obtained. .
7. A hydrological model parameter calibration system based on the differential response relationship of the system, characterized in that, include: The data acquisition module is used to acquire the necessary input data for the operation of the hydrological model, including hydrological observation time series data; and to construct the hydrological model based on the input data and parameter matrix. The initial parameter set construction module is used to randomly sample within a given range of model parameter values to generate an initial parameter set; The system differential response relation construction module is used to drive the hydrological model with the initial parameter set to obtain the initial simulation model output; based on the hydrological observation time series data and the initial simulation model output, the model output simulation error is obtained; and the system differential response relation between the initial parameter error and the model output simulation error is constructed based on the elastic network solution method. The initial parameter error solving module is used to obtain the iterative solution formula for the initial parameter error based on the system differential response relationship and the iterative shrinking threshold algorithm. The cross-validation method is used to determine the optimal penalty coefficient and weight coefficient in the solution formula based on the minimum sum of squared residuals. The optimal coefficients are then substituted into the solution formula to solve for the initial parameter error. The parameter optimization module is used to optimize the initial parameters based on the initial parameter error, determine whether the parameters have converged, and if not, use the optimized parameters as the initial values to drive the hydrological model to continue processing until the parameters converge and the final parameter calibration results are obtained.
8. An electronic device, characterized in that, include: One or more processors; Memory; And one or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors, wherein when the programs are executed by the processors, they implement the steps of the hydrological model parameter calibration method based on the system differential response relationship as described in any one of claims 1-7.
9. A computer-readable storage medium, characterized in that, It stores a computer program, which, when executed by a processor, implements the steps of the hydrological model parameter calibration method based on the system differential response relationship as described in any one of claims 1-7.
10. A computer program product, comprising a computer program, characterized in that, When the computer program is executed by the processor, it implements the steps of the hydrological model parameter calibration method based on the system differential response relationship as described in any one of claims 1-7.