Multi-period non-stationary multi-hydrological variable space valuation method

Abstract

The invention discloses a multi-period non-stationary multi-hydrological variable spatial valuation method, and relates to the technical field of hydrogeology. Comprising the steps of obtaining a univariate time sequence; combining the univariate time sequences of all observation stations in the drainage basin to obtain a multivariate random time sequence; calculating a long-term average level of each observation station through a space-time random function to obtain a long-term trend; subtracting the multivariable random time sequence of each observation station from the corresponding long-term trend to obtain a random residual error; according to the random residual error, determining a covariance function and a variation function between any two observation stations in the drainage basin; and on the basis, a multi-period non-stationary multivariable space valuation model is constructed by combining a multivariable Kriging model theory, and multi-hydrological variable valuation is performed on the drainage basin to be measured. According to the invention, internal correlation between variables can be revealed, and the prediction precision and reliability can be effectively improved.

Description

Technical Field

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

[0002] In the field of hydrogeology, geostatistical methods, such as cokriging, are widely used to estimate the spatial distribution of hydrological variables. For example, Hoeksema used cokriging to estimate water surface height in mountainous areas, and Neuman used residual kriging to study the spatial variability of groundwater levels.

[0003] However, traditional methods are mainly based on the assumption of stationarity. The implementation process includes: first, setting up monitoring points in the study area to measure hydrological variables such as groundwater level and precipitation, and recording the locations and values; assuming that the mean value of hydrological variables is constant within the region, and that the relationship between variables is only determined by distance and does not change with geographical location; comparing and analyzing the measured data to explore the variation pattern of hydrological variables with spatial distance, and thus establish a spatial correlation model; selecting the location to be estimated, and using the data of its surrounding observation points and the spatial variation model to calculate the predicted value of the location, so as to achieve a reasonable estimation of the hydrological variables of the unmonitored basin; calculating the prediction error, and intuitively presenting the spatial distribution of hydrological variables through visualization methods such as contour maps and three-dimensional maps.

[0004] Traditional methods based on the stationarity assumption rely on the spatial stationarity of the data. However, since hydrological variables often exhibit non-stationarity in complex environments, traditional models cannot effectively describe the changing patterns of variables with spatial positions in complex environments, and it is difficult to capture the dynamic changes of variables over time, resulting in insufficient prediction accuracy. Summary of the Invention

[0005] Based on this, it is necessary to provide a spatial valuation method for multi-period non-stationary multi-hydrological variables to address the above technical issues.

[0006] The embodiment of the present invention provides a spatial estimation method for multi-period non-stationary multi-hydrological variables, including: Obtain the observation value of a single hydrological variable at each time point at a single observation station in the basin as a univariate time series; combine the univariate time series of all observation stations in the basin to obtain a multivariate random time series; According to the spatial correlation characteristics and temporal correlation characteristics between the hydrological variables at each station at each time point, the first-order stationarity hypothesis, the covariance function stationarity hypothesis, the random variable decomposition hypothesis and the second-order stationarity hypothesis are determined; The long-term average level of each station is calculated through the spatiotemporal random function to obtain the long-term trend corresponding to each station; the random residual corresponding to each station is obtained by subtracting the multivariate random time series of each station from the corresponding long-term trend; Based on the assumptions of first-order stationarity, covariance function stationarity, random variable decomposition, and second-order stationarity, the covariance function and variogram between any two stations in the basin are determined according to the random residuals corresponding to each station. According to the covariance function and variogram between any two measuring stations in the basin, combined with the multivariate Kriging model theory, a multi-period non-stationary multivariate spatial valuation model is constructed, and the multi-hydrological variables of the measured basin are estimated through the multi-period non-stationary multivariate spatial valuation model.

[0007] Optionally, based on the spatial correlation characteristics and temporal correlation characteristics between the hydrological variables at each observation station at each time point, the first-order stationarity assumption, the covariance function stationarity assumption, the random variable decomposition assumption, and the second-order stationarity assumption are determined, specifically including: Based on the hydrological variables at different stations in the basin at different times, and the fact that the mean of the variables varies with spatial location but is independent of time, a first-order stationary hypothesis is determined. The first-order stationary hypothesis is used to reflect the long-term spatial trend characteristics of the hydrological variables. Determining a covariance function stationary hypothesis based on the spatial correlation characteristics of hydrological variables at different stations within the basin; the covariance function stationary hypothesis indicates that the covariance function is only related to the relative positions between the stations; Determine a random variable decomposition hypothesis based on the structural characteristics of hydrological variables at different stations within the basin; the random variable decomposition hypothesis is used to split the hydrological variables into long-term trend terms and short-term fluctuation terms. The long-term trend term is used to capture the spatial structure of the hydrological variables, and the short-term fluctuation term is used to characterize local anomalies of the hydrological variables. Based on the correlation characteristics of short-term fluctuation terms of different measuring stations in the basin in spatial position and time dimensions, the second-order stationarity hypothesis is determined; the second-order stationarity hypothesis is used to characterize the correlation between the covariance function and the time lag and spatial distance of the measuring stations.

[0008] Alternatively, the observation values ​​of a single hydrological variable at each time point at a single observation station in the basin can be obtained based on the following formula to form a univariate time series: ; The univariate time series of all stations in the basin are combined based on the following formula to obtain a multidimensional random time series: ; in, For the measuring station, For time point, is the first hydrological variable, M is the total number of hydrological variables, N is the total number of measuring stations, T is the total number of time points, i and j For formal parameters.

[0009] Optionally, the long-term average level of each station is calculated using a spatiotemporal random function based on the following formula: ; The multidimensional random time series of each station is subtracted from the corresponding long-term trend based on the following formula: in, Indicates the first i A measuring station, Indicates the first i The long-term trend corresponding to each station.

[0010] Optionally, the covariance function and variogram between any two stations in the watershed are determined based on the following formula, specifically including: For the first and second stations in the watershed, the cross-covariance function is determined based on the following formula: ; when When , it is the direct covariance function; For the first and second stations in the watershed, the cross-variogram is determined based on the following formula: ; when When , it is the direct variogram; in, is the first hydrological variable, represents the second hydrological variable, For the first measuring station, The second measuring station.

[0011] Alternatively, based on the covariance function and variogram between any two stations in the basin, combined with the multivariate Kriging model theory, a multi-period non-stationary multivariate spatial valuation model is constructed using the following formula: ; in, For the station The first hydrological variable at any time point The estimated value of are the multivariate Kriging weights, is the station number of the first hydrological variable, , is the total number of stations measuring the first hydrological variable; The multivariate Kriging weights satisfy the first set of equations under the conditions of unbiasedness and minimum variance: in, is the cross covariance of the first hydrological variable and the second hydrological variable between the third and fourth stations, is the cross covariance between the initial hydrological variable and the first hydrological variable at the target station and the third station, is the Lagrange multiplier; Multivariate Kriging Weights Under the eigenvalue assumption, the second set of equations is satisfied: in, is the cross-variogram of the first hydrological variable and the second hydrological variable between the third and fourth stations, is the variation function of the initial hydrological variable and the first hydrological variable between the target station and the third station, For the third measuring station, The fourth measuring station, is the first hydrological variable, Represents the second hydrological variable.

[0012] The above-mentioned spatial estimation method of multi-period non-stationary multi-hydrological variables provided by the embodiment of the present invention has the following beneficial effects compared with the prior art: By integrating multi-source spatiotemporal observation data, the present invention reveals the complex evolution of hydrological variables in the spatiotemporal dimensions, and establishes a robust covariance structure based on the stationarity assumption, effectively quantifying the spatial dependence and temporal non-stationary characteristics between measuring stations, thereby separating long-term trends from random residuals, and significantly improving the ability to characterize the dynamic changes of hydrological processes; on this basis, combined with the multivariate Kriging theory, it breaks through the limitations of traditional single-variable modeling and realizes the collaborative valuation of multiple factors. It can solve the problem that traditional models cannot effectively describe the changing laws of variables with spatial positions in complex environments, capture the dynamic changes of hydrological variables over time, and significantly improve the accuracy of basin hydrological predictions. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] Figure 1 The present invention is a flowchart of a spatial estimation method for multi-period non-stationary multi-hydrological variables provided in one embodiment. DETAILED DESCRIPTION

[0014] 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.

[0015] Traditional methods are mainly based on the stationary assumption, and their limitations are reflected in: (1) Ignoring spatial non-stationarity: It is impossible to effectively describe the variation of variables with spatial position in complex environments; (2) Underestimation of temporal local stationarity: Traditional models have difficulty capturing the dynamic changes of variables over time, resulting in insufficient prediction accuracy; (3) Insufficient multivariate joint modeling: Current technologies have limited performance in dealing with the intercorrelations between multiple variables and it is difficult to reveal their potential internal connections.

[0016] In one embodiment, a spatial estimation method for multi-period non-stationary multi-hydrological variables is provided, the method comprising: The observation values ​​of a single hydrological variable at each time point at a single observation station in the basin are obtained as a univariate time series; the univariate time series of all observation stations in the basin are combined to obtain a multivariate random time series.

[0017] According to the spatial correlation characteristics and temporal correlation characteristics between the hydrological variables at each measuring station at each time point, the first-order stationarity hypothesis, the covariance function stationarity hypothesis, the random variable decomposition hypothesis and the second-order stationarity hypothesis are determined.

[0018] The long-term average level of each station is calculated through the spatiotemporal random function to obtain the long-term trend corresponding to each station. The random residual corresponding to each station is obtained by subtracting the multivariate random time series of each station from the corresponding long-term trend.

[0019] Based on the assumptions of first-order stationarity, covariance function stationarity, random variable decomposition, and second-order stationarity, the covariance function and variogram between any two stations in the basin are determined based on the random residuals corresponding to each station. Based on the covariance function and variogram between any two stations in the basin, combined with multivariate Kriging model theory, a multi-period non-stationary multivariate spatial valuation model is constructed. This model is then used to estimate multiple hydrological variables in the basin under test.

[0020] Specific implementation: Step 1: Define spatiotemporal random functions and variables (1) Space-time random function set up represents a spatiotemporal random function, where: : spatial location (such as geographic coordinates); For the first measuring station, For the second measuring station, For the third measuring station, It is the fourth measuring station; : time dimension; : variable number (such as precipitation, temperature, evaporation, etc.), k 0 is the initial hydrological variable, is the first hydrological variable, Represents the second hydrological variable.

[0021] (2) Univariate random time series Obtain the observed value of the first hydrological variable at the first station to form a univariate time series: .

[0022] (3) Multivariate random time series For all first variables and the first measuring station , each time point The observations of constitute a multidimensional random time series: .

[0023] (4) Spatial correlation between hydrological variables In a limited neighborhood, the time series of hydrological variables between any two stations have a negative correlation. In general, the correlation increases with the distance between the stations. increases and gradually weakens.

[0024] Step 2: Formulate a basic hypothesis In order to construct a multi-period non-stationary multi-variable spatial valuation model, the present invention uses random functions Assume the following: Based on the variable values ​​of multiple stations in the basin at different times, and combined with the characteristics that the mean of the variable changes with spatial position but is independent of time, the first-order stationarity hypothesis is determined to reflect the spatial long-term trend characteristics of hydrological variables.

[0025] According to the spatial correlation characteristics of hydrological variables between different stations, the stationary assumption of the covariance function is determined, that is, the covariance is only related to the relative positions between the stations, which makes it easier to construct and simplify the covariance model.

[0026] According to the structural characteristics of the hydrological variables at the measuring station, the random variable decomposition hypothesis is determined, and the hydrological variables are split into long-term trend terms and short-term fluctuation terms. The long-term trend term is used to capture the spatial structure of the hydrological variables, and the short-term fluctuation term is used to characterize the local anomalies of the hydrological variables.

[0027] According to the correlation characteristics between short-term fluctuation items in space and time, the second-order stationarity hypothesis is determined, that is, its covariance function is related to time lag and spatial distance, which facilitates the derivation of covariance function and variogram.

[0028] According to the characteristics that some variables are non-stationary but have limited correlation in the time dimension, the quasi-eigenvalue hypothesis is determined, and it is believed that their short-term fluctuation terms satisfy loose stationarity in time, thereby expanding the applicability of the model to non-stationary processes.

[0029] (1) First-order stationarity assumption (simplifying the analysis of the mean, considering only the changes in the spatial dimension) , the mean of the random process is only related to the spatial position and has nothing to do with time, satisfying: (1) : The first hydrological variable at location The long-term mean reflects the overall trend in space.

[0030] (2) Covariance function stationarity assumption (describing the spatial correlation of random processes and providing a basis for the construction of covariance functions) , random process and The covariance function of is only related to the spatial position and satisfies: (2) (3) Random variable decomposition hypothesis (analyzing the long-term trend and short-term fluctuation of hydrological variables separately to facilitate model construction) , random process It can be decomposed into the superposition of long-term trend (drift part) and random residual (remaining part): (3) : Long-term trend, which indicates the long-term trend of hydrological variables in space, usually determined by external driving factors such as topography and climate conditions; : Random residual, which represents random fluctuations or short-term variability that deviates from the long-term average trend and is used to describe local random characteristics.

[0031] (4) Second-order stationarity assumption (emphasizing the spatial correlation of random processes to facilitate the analysis of covariance and variogram) Cross-covariance function of random residuals It satisfies the stationary property in the time dimension, that is, the covariance is only related to the spatial distance and time: (4) : Hydrological variables in time The mean residual of ; when When , the cross covariance function It degenerates into a direct covariance function, which is used to describe the correlation of the same hydrological variable; when When the covariance function represents the hydrological variable and Correlation at the same point.

[0032] (5) Quasi-eigenvalue hypothesis (provides flexibility and is applicable to non-stationary random processes with certain time correlation) If assumption 4 is too strict, a more relaxed assumption can be adopted, that is, the cross-variogram of random residuals Satisfies stationarity in the time dimension: (5) when When , it represents the direct variogram; when When the variogram represents the hydrological variable and Correlation at the same point.

[0033] Step 3: Calculate the long-term trend and random residuals (1) Calculating long-term trends random variable At each station The long-term average level is calculated as: (6) The long-term mean of each hydrological variable is used to remove its trend effect and provide a basis for the subsequent calculation of the random residual part.

[0034] (2) Calculate random residuals Random residuals after removing the long-term mean Expressed as: (7) Step 4: Calculate the covariance function and variogram Using the basic assumptions from step 2, calculate the covariance function and variogram: (1) Covariance function For any two stations and , the cross-covariance function calculation formula is: (8) when When , it is the direct covariance function.

[0035] (2) Variogram For any two stations and , the cross-variogram calculation formula is: (9) when When , it is the direct variogram.

[0036] Step 5: Establish a multi-period spatial non-stationary multivariate model According to the covariance function and variogram results in step 4, the multivariate Kriging model theory is used to establish A multi-period non-stationary spatial estimation model of hydrological variables is used to estimate the values ​​of hydrological variables at target stations at unobserved locations and time points.

[0037] (1) Valuation formula At any point in time , station Estimates of initial hydrological variables : (10) : The first hydrological variable at the third measuring station at time point The remaining value of : Multivariate Kriging weight, indicating the influence of the third station on the valuation; : The station number of the first hydrological variable, ; : The total number of observation stations for the first hydrological variable.

[0038] (2) Solving multivariate Kriging weights ①Under the conditions of unbiasedness and minimum variance, the expectation of the valuation result should be consistent with the estimated variable, and the valuation variance should be minimum; this condition relies on the assumptions of first-order stationarity and covariance function stationarity proposed in step 2. The system of equations, i.e., the multivariate Kriging weights The following equations must be satisfied:

[0039] (11) : cross covariance of the first hydrological variable and the second hydrological variable between the third and fourth stations; : cross covariance between the initial hydrological variable and the first hydrological variable at the target station and the third station; : Lagrange multiplier, used to satisfy the unbiasedness condition.

[0040] ② Under the quasi-eigenvalue hypothesis, the multivariate Kriging weights The following equations need to be satisfied, that is, the above multivariable Kriging equations should be written as the following equations: (12) : Cross-variogram of the first hydrological variable and the second hydrological variable between the third and fourth measuring stations; : the variogram of the initial hydrological variable and the first hydrological variable between the target station and the third station; (3) Estimated variance The uncertainty of the estimate is measured by the estimated variance Expression, the estimated variance calculation formulas corresponding to the above two equations are: (13) : The autovariance of the target station (distance is 0).

[0041] (14) Step 6: Estimate hydrological variables truth value Because in hypothesis three, the random decomposition variable hypothesis is made, that is, the random process Decomposed into long-term trend (drift part) and random residual (the remainder), that is, .

[0042] In step 5, we get the random residuals (the rest), so step 6 is needed to calculate the final random process True value.

[0043] (1) Valuation formula (15) (2) Determine random residuals Based on step 5, a multi-period spatial non-stationary multivariate model is established to determine the random residuals. .

[0044] (3) Determine the offset The offset was determined using trend surface analysis and multivariate Kriging: ①Trend surface analysis method (17) : regression coefficient of trend basis function; : Trend basis function, usually including linear terms, high-order terms, etc. Determine the coefficients through regression analysis , and test the significance of the trend model.

[0045] ②Multivariate Kriging model method (18) (19) This method uses a multidimensional spatiotemporal random function cluster modeling approach, treating multiple related hydrological variables within the same time period or across different time periods as a whole. Through systematic analysis, it reveals the complex spatiotemporal relationships among these hydrological variables. This not only provides scientific support for the simulation and prediction of hydrological processes, but also, through analysis of different time periods, enables accurate decision-making for long-term water resources planning and management.

[0046] 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 spatial estimation method for multi-period non-stationary multi-hydrological variables, characterized by: include: Obtain the observation value of a single hydrological variable at each time point at a single observation station in the basin as a univariate time series; The univariate time series of all stations in the basin are combined to obtain a multivariate random time series; According to the spatial correlation characteristics and temporal correlation characteristics between the hydrological variables at each station at each time point, the first-order stationarity hypothesis, the covariance function stationarity hypothesis, the random variable decomposition hypothesis and the second-order stationarity hypothesis are determined; The long-term average level of each station is calculated through the spatiotemporal random function to obtain the long-term trend corresponding to each station; Subtract the multivariate random time series of each station from the corresponding long-term trend to obtain the random residual corresponding to each station; Based on the assumptions of first-order stationarity, covariance function stationarity, random variable decomposition, and second-order stationarity, the covariance function and variogram between any two stations in the basin are determined according to the random residuals corresponding to each station. According to the covariance function and variogram between any two measuring stations in the basin, combined with the multivariate Kriging model theory, a multi-period non-stationary multivariate spatial valuation model is constructed, and the multi-hydrological variables of the measured basin are estimated through the multi-period non-stationary multivariate spatial valuation model.

2. The spatial estimation method of multi-period non-stationary multi-hydrological variables according to claim 1, characterized in that: The first-order stationarity hypothesis, the covariance function stationarity hypothesis, the random variable decomposition hypothesis and the second-order stationarity hypothesis are determined based on the spatial correlation characteristics and temporal correlation characteristics between the hydrological variables at each measuring station at each time point, specifically including: Based on the hydrological variables at different stations in the basin at different times, and the fact that the mean of the variables varies with spatial location but is independent of time, a first-order stationary hypothesis is determined. The first-order stationary hypothesis is used to reflect the long-term spatial trend characteristics of the hydrological variables. Determining a covariance function stationary hypothesis based on the spatial correlation characteristics of hydrological variables at different stations within the basin; the covariance function stationary hypothesis indicates that the covariance function is only related to the relative positions between the stations; Determine a random variable decomposition hypothesis based on the structural characteristics of hydrological variables at different stations within the basin; the random variable decomposition hypothesis is used to split the hydrological variables into long-term trend terms and short-term fluctuation terms. The long-term trend term is used to capture the spatial structure of the hydrological variables, and the short-term fluctuation term is used to characterize local anomalies of the hydrological variables. Based on the correlation characteristics of short-term fluctuation terms of different measuring stations in the basin in spatial position and time dimensions, the second-order stationarity hypothesis is determined; the second-order stationarity hypothesis is used to characterize the correlation between the covariance function and the time lag and spatial distance of the measuring stations.

3. The spatial estimation method of multi-period non-stationary multi-hydrological variables according to claim 1, characterized in that: The observation values ​​of a single hydrological variable at each time point at a single observation station in the basin are obtained based on the following formula to form a univariate time series: ； The univariate time series of all stations in the basin are combined based on the following formula to obtain a multidimensional random time series: ； in, For the measuring station, For time point, is the first hydrological variable, M is the total number of hydrological variables, N is the total number of measuring stations, T is the total number of time points, i and j For formal parameters.

4. The spatial estimation method of multi-period non-stationary multi-hydrological variables according to claim 3, characterized in that: The long-term average level of each station is calculated using the spatiotemporal random function based on the following formula: ； The multidimensional random time series of each station is subtracted from the corresponding long-term trend based on the following formula: in, Indicates the first i A measuring station, Indicates the first i The long-term trend corresponding to each station.

5. The spatial estimation method of multi-period non-stationary multi-hydrological variables according to claim 4, characterized in that: The covariance function and variogram between any two stations in the basin are determined based on the following formula: For the first and second stations in the watershed, the cross-covariance function is determined based on the following formula: ； when When , it is the direct covariance function; For the first and second stations in the watershed, the cross-variogram is determined based on the following formula: ； when When , it is the direct variogram; in, is the first hydrological variable, represents the second hydrological variable, For the first measuring station, The second measuring station.

6. The spatial estimation method of multi-period non-stationary multi-hydrological variables according to claim 1, characterized in that: Based on the covariance function and variogram between any two stations in the basin, combined with the multivariate Kriging model theory, a multi-period non-stationary multivariate spatial valuation model is constructed by the following formula: ； in, For the station The first hydrological variable at any time point The estimated value of are the multivariate Kriging weights, is the station number of the first hydrological variable, , is the total number of stations measuring the first hydrological variable; The multivariate Kriging weights satisfy the first set of equations under the conditions of unbiasedness and minimum variance: in, is the cross covariance of the first hydrological variable and the second hydrological variable between the third and fourth stations, is the cross covariance between the initial hydrological variable and the first hydrological variable at the target station and the third station, is the Lagrange multiplier; The multivariate kriging weights Under the eigenvalue assumption, the second set of equations is satisfied: in, is the cross-variogram of the first hydrological variable and the second hydrological variable between the third and fourth stations, is the variation function of the initial hydrological variable and the first hydrological variable between the target station and the third station, For the third measuring station, The fourth measuring station, is the first hydrological variable, Represents the second hydrological variable.

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