Evaluation method for multi-period non-stationary hydrological variables

Abstract

The invention discloses a multi-period non-stationary hydrological variable valuation method, and relates to the technical field of hydrological prediction. Comprising the steps of obtaining hydrological variables of a plurality of observation stations in a drainage basin at different time points as hydrological time sequences of the observation stations; constructing an experimental variation function according to the hydrological time sequence of each observation station; selecting a drift model through a trend surface analysis method and model inspection; constructing a multi-period non-stationary variable equation set comprising a Lagrange multiplier and a hydrological variable weight based on the experimental variation function and the drift model; expressing the multi-period non-stationary variable equation set as a matrix form, and calculating a Lagrange multiplier and a hydrological variable weight through a matrix inversion method; and constructing an estimation formula according to the calculation result of the Lagrange multiplier and the hydrological variable weight, and performing multi-period spatial estimation on the hydrological variable of the to-be-measured drainage basin through the estimation formula. Dependence on the number of observation stations can be reduced, and accurate estimation can be achieved only through a small amount of observation station data.

Description

Technical Field

[0001] The present invention relates to the technical field of hydrological prediction, and in particular to a method for estimating multi-period non-stationary hydrological variables. Background Art

[0002] Kriging, a widely used tool for processing both stationary and nonstationary geological information, has been gradually applied to hydrology since 1976. Its application process includes: collecting hydrological data (including spatially distributed data such as groundwater levels, water quality parameters, rainfall, and soil moisture); preprocessing the hydrological data by checking for normality, performing logarithmic transformation on nonnormally distributed data, and removing outliers to ensure data representativeness; spatial normalization (e.g., unifying the coordinate system and elevation datum); and analyzing spatial structure by calculating experimental variograms and fitting theoretical models. Regarding kriging model selection and optimization, ordinary kriging is suitable for regional groundwater level prediction, assuming second-order stationarity; universal kriging introduces a drift term for nonstationary rainfall interpolation; and collaborative kriging integrates auxiliary variables to improve the accuracy of the primary variable estimates; and model reliability is assessed through leave-one-out cross-validation, calculating the root mean square error and mean absolute error. Spatial interpolation and uncertainty quantification steps include calculating weights and outputting the results.

[0003] However, Kriging still has limitations when dealing with the spatiotemporal randomness of hydrological variables, particularly in terms of information utilization. Traditional Kriging relies primarily on spatial variability for interpolation, focusing on characterizing random components while paying less attention to the temporal continuity of hydrological variables. This results in a large amount of valuable information in the temporal dimension not being fully utilized, thus affecting interpolation accuracy. This is especially true in small watersheds with sparsely distributed observation stations. This waste of temporal information further exacerbates the uncertainty of spatial inferences. Summary of the Invention

[0004] Based on this, it is necessary to provide a valuation method for multi-period non-stationary hydrological variables to address the above technical issues.

[0005] An embodiment of the present invention provides a method for estimating non-stationary hydrological variables over multiple time periods, including: The hydrological variables at different time points of multiple stations in the basin are obtained as the basin hydrological time series; the first-order stationarity assumption, the stationarity assumption of the mutual variance function, and the non-stationarity assumption of the increment are determined based on the neighborhood correlation between any two stations in the basin; Based on the assumptions of first-order stationarity, stationarity of the cross-variance function, and non-stationarity of the increments, an experimental variogram was constructed based on the basin hydrological time series to describe the variation of hydrological variables between stations with distance. Trend surface analysis and model testing were used to select a drift model to simulate the non-random variation trend of hydrological variables in the temporal dimension. Based on the experimental variogram and drift model, a multi-period non-stationary variable equation system including Lagrange multipliers and hydrological variable weights is constructed; the multi-period non-stationary variable equation system is expressed in matrix form, and the Lagrange multipliers and hydrological variable weights are calculated using the matrix inversion method; An estimation formula is constructed based on the calculation results of Lagrange multipliers and hydrological variable weights, and the hydrological variables of the measured basin are estimated in multiple time periods using the estimation formula.

[0006] Optionally, obtain hydrological variables at different time points at multiple stations within the basin as basin hydrological time series, specifically including: The geographical location of the measuring station in the basin is expressed as spatial coordinates. The stations are marked as ; If a specific moment or period in the basin is expressed as a time dimension, then The moments are marked as ; The first Station at time The true value of the hydrological variable is marked as , then the observation data of all stations in the basin at different time points form the hydrological time series of each station.

[0007] Optionally, the neighborhood correlation between any two stations in the watershed is negatively correlated with the distance between the two stations.

[0008] Optionally, a first-order stationarity assumption is made based on the following equation: ; in, is the mathematical expectation of the true value of the hydrological variable, is the long-term trend or drift of the hydrological variable; The stationarity assumption of the cross-variance function is determined based on the following formula: ; in, The true value of the hydrological variable at the first measuring station and the true value of the hydrological variable at the second measuring station The mutual variance function between is the distance between the first and second measuring stations, is the true value of the hydrological variable at the first measuring station, is the true value of the hydrological variable at the second measuring station, For the first measuring station, It is the second measuring station; The non-stationary assumption of the increments is determined based on the following formula: ; in, is the long-period drift, Remaining for a short period.

[0009] Alternatively, an experimental variogram describing the variation of hydrological variables between stations with distance can be constructed based on the basin hydrological time series using the following formula based on the assumptions of first-order stationarity, stationarity of the mutual variance function, and non-stationarity of the increments: ; in, is the experimental variogram, The distance is The number of station pairs, The first station at time The true value of the hydrological variable, The second station at time The true value of the hydrological variables; When the variability changes only with distance and has nothing to do with direction, the experimental variogram is isotropic and can be expressed as: ; When the variability changes with direction When changes, the experimental variation function is anisotropic, and the formula is expressed as: ; in, For distance, For direction.

[0010] Optionally, a drift model for simulating the non-random trend of hydrological variables in the temporal dimension is selected through trend surface analysis and model testing, including: The drift model includes a linear drift model and a quadratic drift model. The linear drift model is determined based on the following formula: ; The quadratic drift model is determined based on the following formula: ; The trend surface was fitted by the least square method, and the parameters of the linear drift model and the quadratic drift model were estimated. The mean square error and determination coefficient of the linear drift model and the quadratic drift model were calculated respectively. When the mean square error of the quadratic drift model is smaller than that of the linear drift model and the coefficient of determination of the quadratic drift model is larger than that of the linear drift model, the quadratic drift model is selected; otherwise, the linear drift model is selected.

[0011] Optionally, based on the experimental variogram and drift model, a multi-period non-stationary variable equation system including Lagrange multipliers and hydrological variable weights is constructed, specifically including: The unbiasedness condition is determined based on the following formula: ; The condition for minimizing the estimated variance is determined based on the following formula: ; Based on the experimental variogram and drift model, a multi-period non-stationary variable equation system including Lagrange multipliers and hydrological variable weights is constructed according to the following formula through the unbiasedness condition and the minimum estimated variance condition: ; in, is the hydrological variable weight of the second station, is the variogram value between the first and second stations, is the variogram value between the target station and the first station, is the Lagrange multiplier, is the estimate of the hydrological variables, is the true value of the hydrological variable.

[0012] Optionally, the multi-period non-stationary variable equations are expressed in matrix form, and the Lagrange multipliers and hydrological variable weights are calculated by matrix inversion method, specifically including: The multi-period non-stationary variable equation system is expressed in matrix form based on the following formula: ; Solve the matrix using the matrix inversion method based on the following formula: ; in, is composed of the variogram value between the first and second stations matrix, for the reason The vector formed, To include hydrological variable weights vector, is the Lagrange multiplier.

[0013] Optionally, an estimation formula is constructed based on the calculation results of the Lagrange multiplier and the hydrological variable weights, specifically including: The estimated value of the target station is determined by calculating the Lagrange multiplier and the hydrological variable weight based on the following formula: ; in, The target station is at time The estimated value of For the station The weights of hydrological variables; The formula for determining the estimated variance of the target station is based on the following formula using the calculation results of the Lagrange multiplier and the hydrological variable weight: in, is the estimated variance; The smaller the estimated variance, the more reliable the valuation result; the larger the estimated variance, the less reliable the valuation result.

[0014] Optionally, a multi-period spatial valuation of the hydrological variables of the basin to be measured is performed using valuation formulas, specifically including: The hydrological variables of the target station during the missing period are calculated using the estimation formula based on the following formula to interpolate the missing data: ; Based on the following formula, any target station in the basin to be measured is estimated at time To estimate the point value in the watershed to be measured: ; The time of the watershed to be measured is determined based on the following formula The average value of , in order to calculate the average value of the watershed to be measured: ; in, is the watershed area to be measured, The watershed to be tested.

[0015] Compared with the prior art, the above-mentioned multi-period non-stationary hydrological variable estimation method provided by the embodiment of the present invention has the following beneficial effects: The present invention uses a small amount of station data to efficiently construct a multi-period non-stationary variable equation group, and obtains Lagrange multipliers and hydrological variable weights through the multi-period non-stationary variable equation group to construct an estimation formula. This can reduce dependence on the number of stations and achieve accurate estimation with only a small amount of station data. It can also realize reliable interpolation and analysis of missing data in small watersheds, and provide a convenient method for completing missing data. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 The present invention is a flowchart of a method for estimating multi-period non-stationary hydrological variables provided in one embodiment. DETAILED DESCRIPTION

[0017] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0018] In one embodiment, a method for estimating a multi-period non-stationary hydrological variable is provided, the method comprising: Step 1: Define variables (1) Define the true value of hydrological variables : : represents the coordinates of a spatial point In the time dimension Hydrological variables, usually the observation values ​​of the observation station (such as precipitation, evaporation, etc.).

[0019] (2) Define the coordinates of a spatial point: : Target station, indicating the geographical location of the station; For the first measuring station, It is the second measuring station; There are a total of stations, marked as ; and : Indicates the coordinates of the first and second survey stations.

[0020] (3) Define the time dimension : : Time dimension, used to represent a specific moment or period; Assume that the time series is moments, marked as ; :Indicates the a point in time.

[0021] (4) Define random time series clusters : : Indicates the first measuring station at time point Observed values ​​of These observations form a cluster of hydrological time series , describes the observation data of all stations at different time points.

[0022] (5) Neighborhood correlation: For any two stations in space, the true value of the hydrological variable at the first station is and the true value of the hydrological variables at the second station is negatively correlated and increases with the distance between the two stations. increases, and its correlation gradually weakens.

[0023] Clarifying the spatial and temporal dimensions of variables and the correlation between stations provides a clear basis for subsequent modeling.

[0024] Step 2: Make basic assumptions (1) Assumption 1: First-order stationarity (assuming that hydrological variables do not drift significantly in the spatial dimension, so that the drift model focuses more on temporal trend characteristics) (1) : True value of hydrological variable The mathematical expectation of : Only over time Variation, which represents the long-term trend or drift of a hydrological variable, independent of spatial location .

[0025] (2) Assumption 2: Stationarity of the cross-variance function (assuming that the correlation of hydrological variables is only related to the distance between stations and not to the specific location, which facilitates the description of the spatial variability of hydrological variables) (2) : The true value of the hydrological variable at the first measuring station and the true value of the hydrological variable at the second measuring station The mutual variance function between : the distance between the first and second measuring stations; : the true value of the hydrological variable at the first measuring station; : The true value of the hydrological variable at the second measuring station.

[0026] (3) Assumption 3: Non-stationarity of increments (decomposing hydrological variables into temporal drift and spatial random components can handle long-term trends and local fluctuation characteristics separately) (3) : Long-term drift, used to represent the long-term trend in the time dimension; : Short-period residual, used to capture random fluctuations in the spatial dimension.

[0027] Step 3: Calculate the experimental variogram (1) Using the basic assumptions in step 2, based on the basin hydrological time series Compute the experimental variogram.

[0028] (4) : experimental variogram, describing the variation of hydrological variables between stations with distance; : The distance is The number of station pairs; The first station at time The true value of the hydrological variable, The second station at time The true value of the hydrological variables.

[0029] When calculating the variogram, only the distance between observations is considered, not their absolute positions. The assumption of stationarity of the cross-variance function ensures that the variogram remains consistent when calculated at different spatial locations. This allows for the establishment of a reasonable spatial variation model by fitting the experimental variogram.

[0030] (2) Isotropic case (5) When variability varies only with distance When the variogram depends only on .

[0031] (3) Anisotropic situation (6) in, For distance, For direction.

[0032] When the variability changes with direction When the value changes, the variation function in different directions needs to be calculated separately.

[0033] Analyze the spatial variability of hydrological variables and construct a variogram model to provide a basis for subsequent valuation.

[0034] Step 4: Drift model selection and optimization (1) Preliminary selection of drift model Linear Drift Model: (7) Quadratic drift model: (8) The drift function is used to simulate the non-random change trend of variables in the time dimension, that is, to simulate the long-term trend of the time dimension.

[0035] (2) Optimization model Trend surface analysis and model testing were used to select the drift model with the smallest error.

[0036] ① Use the least squares method to fit the trend surface and estimate the parameters of the selected drift model (such as linear or quadratic drift model).

[0037] ② Calculate fitting error (optional) The fitting error measures the accuracy of the model and usually uses the following indicators: Mean square error MSE : in, is the observed value, is the drift model calculated value, is the sample size.

[0038] Coefficient of determination R 2 : in, is the mean of the observed data, The closer it is to 1, the better the model fitting effect is.

[0039] ③ Conduct model testing Use statistical test methods to evaluate the fitting quality of different drift models: F-test (ANOVA): Compare the variances of different drift models to determine if there is a significant improvement.

[0040] Residual analysis: Observe whether the distribution of residuals satisfies normality and randomness to ensure the rationality of the drift model.

[0041] ④ Select the optimal drift model Comparison of different models MSE and R 2 value, and select the drift model with the smallest error and the best fitting effect.

[0042] If the mean square error of the quadratic drift model is significantly lower than that of the linear drift model and the coefficient of determination is R 2 If the value of the drift coefficient is significantly improved, the quadratic drift model is selected; otherwise, the linear drift model is selected.

[0043] Step 5: Establish a multi-period non-stationary variable equation system (1) Unbiasedness condition Target point valuation The mathematical expectation is equal to the true value : (9) It can be expressed as: (2) Estimation variance minimization condition Target point valuation With the smallest mean square error: (10) (3) Constructing a system of equations with non-stationary variables On the basis of assuming the drift model, the system of equations is constructed by combining the unbiasedness condition and the minimum estimation variance condition: (11) : weight of hydrological variables at the second station; : the variogram value between the first and second stations; : the variogram value between the target station and the first station; : Lagrange multiplier, used to meet the unbiased condition; : Hydrological variable estimation; : The true value of the hydrological variable.

[0044] Step 6: Solve for hydrological variable weights and Lagrange multipliers (1) In order to facilitate the solution, the system of equations is expressed in matrix form: (12) : The variogram value between the first and second stations matrix; :Depend on The vector formed; : Contains hydrological variable weights vector of : Lagrange multiplier.

[0045] (2) Solution: By matrix inversion or other numerical methods, solve: (13) (3) Verify weights: Ensure that the weights meet the unbiasedness condition.

[0046] Step 7: Determine the valuation formula and estimate the variance The estimated value and variance of the target point are calculated using the solved weights and Lagrange multipliers.

[0047] (1) Valuation formula (14) : Target station In time estimated value of; : Weight of hydrological variables at the first measuring station.

[0048] (2) Estimated variance (15) : Estimated variance, used to evaluate the accuracy of the valuation; The smaller the estimated variance, the more reliable the valuation result; the larger the estimated variance, the less reliable the valuation result.

[0049] Step 8: Multi-period spatial valuation (1) Interpolation of missing data at measuring stations Use the estimation formula to calculate the value of the station during the period of missing measurement: (16) (2) Estimation of point values ​​within the region For any station in the target area, estimate its Value: (17) (3) Calculation of regional average values Calculate the time of the watershed to be measured The average value of: (18) in, is the watershed area to be measured, The watershed to be tested.

[0050] Complete multi-period spatial interpolation and regional average analysis to provide support for practical applications (such as watershed hydrological analysis).

[0051] The present invention has broad application prospects and can provide theoretical support and practical tools for the fields of hydrology and water conservancy. It is particularly suitable for the analysis of hydrological variables in small watersheds or data-scarce areas.

[0052] The above-described embodiments merely illustrate several implementations of the present invention. While the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the present invention. It should be noted that a person skilled in the art would be able to make numerous variations and improvements without departing from the spirit of the present invention, and all such variations and improvements fall within the scope of protection of the present invention.

Claims

1. A method for estimating non-stationary hydrological variables over multiple time periods, characterized in that: include: Obtain hydrological variables at different time points at multiple stations within the basin as basin hydrological time series; According to the neighborhood correlation between any two stations in the basin, the first-order stationarity assumption, the stationarity assumption of the mutual variance function and the non-stationarity assumption of the increment are determined; Based on the assumptions of first-order stationarity, stationarity of the cross-variance function, and non-stationarity of the increments, an experimental variogram was constructed based on the basin hydrological time series to describe the variation of hydrological variables between stations with distance. Trend surface analysis and model testing were used to select a drift model to simulate the non-random variation trend of hydrological variables in the temporal dimension. Based on the experimental variogram and drift model, a multi-period non-stationary variable equation system including Lagrange multipliers and hydrological variable weights is constructed; the multi-period non-stationary variable equation system is expressed in matrix form, and the Lagrange multipliers and hydrological variable weights are calculated using the matrix inversion method; An estimation formula is constructed based on the calculation results of Lagrange multipliers and hydrological variable weights, and the hydrological variables of the measured basin are estimated in multiple time periods using the estimation formula.

2. The method for estimating a multi-period non-stationary hydrological variable according to claim 1, wherein: The step of obtaining hydrological variables at different time points at multiple stations within the basin as a basin hydrological time series specifically includes: The geographical location of the measuring station in the basin is expressed as spatial coordinates. The stations are marked as ; If a specific moment or period in the basin is expressed as a time dimension, then The moments are marked as ; The first Station at time The true value of the hydrological variable is marked as , then the observation data of all stations in the basin at different time points form the hydrological time series of each station.

3. The method for estimating a multi-period non-stationary hydrological variable according to claim 1, wherein: The neighborhood correlation between any two stations in a watershed is negatively correlated with the distance between the two stations.

4. The method for estimating a multi-period non-stationary hydrological variable according to claim 1, wherein: The first-order stationarity assumption is determined based on the following formula: ； in, is the mathematical expectation of the true value of the hydrological variable, is the long-term trend or drift of the hydrological variable; The stationarity assumption of the cross-variance function is determined based on the following formula: ； in, The true value of the hydrological variable at the first measuring station and the true value of the hydrological variable at the second measuring station The mutual variance function between is the distance between the first and second measuring stations, is the true value of the hydrological variable at the first measuring station, is the true value of the hydrological variable at the second measuring station, For the first measuring station, It is the second measuring station; The non-stationary assumption of the increments is determined based on the following formula: ； in, is the long-period drift, Remaining for a short period.

5. The method for estimating a multi-period non-stationary hydrological variable according to claim 1, wherein: Based on the assumptions of first-order stationarity, stationarity of the mutual variance function, and non-stationarity of the increments, the experimental variogram is constructed according to the basin hydrological time series to describe the variation of the differences in hydrological variables between stations with distance: ； in, is the experimental variogram, The distance is The number of station pairs, The first station at time The true value of the hydrological variable, The second station at time The true value of the hydrological variables; When the variability changes only with distance and has nothing to do with direction, the experimental variogram is isotropic and can be expressed as: ； When the variability changes with direction When changes, the experimental variation function is anisotropic, and the formula is expressed as: ； in, For distance, For direction.

6. The method for estimating a multi-period non-stationary hydrological variable according to claim 1, wherein: The drift model selected by trend surface analysis and model testing for simulating the non-random change trend of hydrological variables in the time dimension specifically includes: The drift model includes a linear drift model and a quadratic drift model. The linear drift model is determined based on the following formula: ； The quadratic drift model is determined based on the following formula: ； The trend surface was fitted by the least square method, and the parameters of the linear drift model and the quadratic drift model were estimated. The mean square error and determination coefficient of the linear drift model and the quadratic drift model were calculated respectively. When the mean square error of the quadratic drift model is smaller than that of the linear drift model and the coefficient of determination of the quadratic drift model is larger than that of the linear drift model, the quadratic drift model is selected; otherwise, the linear drift model is selected.

7. The method for estimating a multi-period non-stationary hydrological variable according to claim 1, wherein: The multi-period non-stationary variable equations including Lagrange multipliers and hydrological variable weights are constructed based on the experimental variogram and drift model, specifically including: The unbiasedness condition is determined based on the following formula: ； The condition for minimizing the estimated variance is determined based on the following formula: ； Based on the experimental variogram and drift model, a multi-period non-stationary variable equation system including Lagrange multipliers and hydrological variable weights is constructed according to the following formula through the unbiasedness condition and the minimum estimated variance condition: ； in, is the hydrological variable weight of the second station, is the variogram value between the first and second stations, is the variogram value between the target station and the first station, is the Lagrange multiplier, is the estimate of the hydrological variables, is the true value of the hydrological variable.

8. The method for estimating a multi-period non-stationary hydrological variable according to claim 1, wherein: The multi-period non-stationary variable equations are expressed in matrix form, and the Lagrange multipliers and hydrological variable weights are calculated by matrix inversion method, specifically including: The multi-period non-stationary variable equation system is expressed in matrix form based on the following formula: ； Solve the matrix using the matrix inversion method based on the following formula: ； in, is composed of the variogram value between the first and second stations matrix, for the reason The vector formed, To include hydrological variable weights vector, is the Lagrange multiplier.

9. The method for estimating a multi-period non-stationary hydrological variable according to claim 1, wherein: The valuation formula is constructed based on the calculation results of the Lagrange multiplier and the hydrological variable weight, specifically including: The estimated value of the target station is determined by calculating the Lagrange multiplier and the hydrological variable weight based on the following formula: ； in, The target station is at time The estimated value of is the weight of the hydrological variable of the first measuring station; The formula for determining the estimated variance of the target station is based on the following formula using the calculation results of the Lagrange multiplier and the hydrological variable weight: in, is the estimated variance; The smaller the estimated variance, the more reliable the valuation result; the larger the estimated variance, the less reliable the valuation result.

10. The method for estimating multi-period non-stationary hydrological variables according to claim 1, wherein: The multi-period spatial valuation of the hydrological variables of the measured basin by the valuation formula specifically includes: The hydrological variables of the target station during the missing period are calculated using the estimation formula based on the following formula to interpolate the missing data: ； Based on the following formula, any target station in the basin to be measured is estimated at time To estimate the point value in the watershed to be measured: ； The time of the watershed to be measured is determined based on the following formula The average value of , in order to calculate the average value of the watershed to be measured: ； in, is the watershed area to be measured, The watershed to be tested.

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