A hydrological prediction method based on a variable-order generalized nash confluence model in karst areas
By combining the variable-order generalized Nash confluence model with Caputo fractional derivatives and computational calculus, the problem of describing the flow patterns in karst areas was solved, and high-precision hydrological and flood forecasts were achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- HUAZHONG UNIV OF SCI & TECH
- Filing Date
- 2023-09-15
- Publication Date
- 2026-06-26
Smart Images
Figure CN117494381B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of hydrological forecasting technology, and more specifically, relates to a hydrological forecasting method for karst areas based on a variable-order generalized Nash confluence model. Background Technology
[0002] In karst-developed areas, the karst aquifer system, composed of multiple media such as fissures, caves, and conduits, is a complex and constantly evolving dynamic system. Its hydrological cycle with atmospheric precipitation, surface water, and soil water is quite intricate. Furthermore, the water flow in karst aquifer systems exhibits characteristics such as the coexistence of fissure flow and conduit flow, laminar flow and turbulent flow, linear flow and nonlinear flow, and continuous flow and isolated water bodies. Accurately constructing a mathematical model for the karst aquifer system is a complex problem. To date, no complete and systematic research model has been established that can accurately describe the movement patterns of water flow in karst aquifer media.
[0003] Most existing conceptual hydrological models generalize different structures or hydrological processes within karst aquifers as different tanks, using several tanks connected in series and parallel to represent the spatial layer structure of the karst aquifer medium, and simulating the hydraulic connection between fissures and conduits in the karst aquifer medium by water exchange between the series tanks. These models are generally complex in structure and have many parameters, leading to increased uncertainty, or they have a certain regional limitation and are difficult to generalize.
[0004] Currently, fractional-order instantaneous unit hydrographs exhibit a significant tailing phenomenon, making them suitable for calculating runoff in karst basins where runoff has long memory. However, existing runoff calculation methods based on fractional-order instantaneous unit hydrographs suffer from incomplete calculated runoff processes and mismatches between the runoff generation description and the actual physical processes.
[0005] Therefore, constructing a universally applicable hydrological model for karst areas that reflects the characteristics of water flow in karst regions is of great practical significance. Summary of the Invention
[0006] To address the shortcomings and improvement needs of existing technologies, this invention provides a hydrological forecasting method for karst areas based on a variable-order generalized Nash confluence model. The aim is to more realistically reflect the characteristics of water flow in karst areas and improve the accuracy of hydrological forecasting in karst areas.
[0007] To achieve the above objectives, according to one aspect of the present invention, a hydrological forecasting method for karst areas based on a variable-order generalized Nash confluence model is provided, comprising:
[0008] Based on the potential evapotranspiration and precipitation of a karst region to be predicted for a certain period, the runoff forecast is calculated using the variable-order generalized Nash confluence model corresponding to the karst region to be predicted, which has been calibrated with parameters.
[0009] The variable-order generalized Nash confluence model is expressed as follows:
[0010]
[0011]
[0012] In the formula, O(t) represents the runoff at time t; n represents the number of Nash linear reservoirs; α∈(0,1) represents the order of the Caputo fractional order; O (j) (0) represents the j-th derivative of the outflow at the initial time; The formula for calculating the number of combinations is given; K represents the regulation and storage parameters of the Nash linear reservoir. and Let γ = n, β = (ni)α + j + 1 or β = nα, respectively. u(·) represents the fractional-order instantaneous unit line corresponding to the instantaneous inflow of 1 unit; R(·) represents the watershed runoff; let α be the time-varying order, and let it be linearly related to R(·), i.e., α(t)=α[1+λR(t)]; the parameters calibrated by this model include the runoff parameters and the runoff parameters. The runoff parameters are n, α, K and λ, and the runoff parameters are determined according to the specific formula of R(·).
[0013] Furthermore, the structure of the variable-order generalized Nash confluence model is constructed as follows:
[0014] (1) Based on the generalized Nash confluence theory, a higher-order differential equation is constructed to describe the water flow motion in the confluence system: (1+KD) n O(t) = I(t), where, K represents the regulation parameter of the Nash linear reservoir, I(t) represents the inflow at time t, and n represents the number of Nash linear reservoirs;
[0015] (2) The integer derivatives in the higher-order differential equations are expressed using Caputo fractional derivatives, and the higher-order differential equations are expanded using binomial expansion to obtain the fractional differential equations of the merging system: In the formula, This represents the Caputo fractional derivative;
[0016] (3) The fractional differential equation is solved by using the operational calculus method to obtain the variable-order generalized Nash confluence model.
[0017] Furthermore, step (3) includes:
[0018] (31) The relationship between the Caputo fractional derivative and the Riemann-Liouville fractional derivative Substituting into the fractional differential equation and simplifying, we obtain a new fractional differential equation:
[0019]
[0020] In the formula, U α =I / h α I = h α / h α , indicating the unit yuan; The kernel function representing the Riemann-Liouville fractional derivative;
[0021] (32)(32) According to U α The properties of, will Substituting the first term on the right-hand side of the new fractional differential equation Let R(t) be the unit instantaneous inflow I, then we get in
[0022] Additionally, according to U α Definition and nature get Substituting this relation into the second term on the right-hand side of the new fractional differential equation And using U α Properties and Simplifying the formula, we get: Further according to U α Properties right Simplify to obtain
[0023] Therefore, the new fractional differential equation described in step (31) is transformed into in The variable-order generalized Nash confluence model is obtained.
[0024] Furthermore, the SCE-UA optimization algorithm was used for parameter calibration.
[0025] Furthermore, during parameter calibration, Nash efficiency (NSE) is used as the objective function.
[0026] Furthermore, the watershed runoff R(t) was calculated using the Xin'anjiang model:
[0027]
[0028]
[0029] In the formula, P(t) is the precipitation at time t, E0(t) is the potential evapotranspiration at time t, and W0 is the initial water storage of the basin.
[0030] The generated flow parameters are k, B, and WMM.
[0031] The present invention also provides a computer-readable storage medium comprising a stored computer program, wherein, when the computer program is executed by a processor, it controls the device where the storage medium is located to execute a hydrological forecasting method for karst areas based on a variable-order generalized Nash confluence model as described above.
[0032] In summary, the above-described technical solutions conceived in this invention can achieve the following beneficial effects:
[0033] Fractional calculus, a nonlocal operator expressed in differential-integral form, can accurately characterize the motion of particles in complex media structures or flow fields, making it particularly suitable for describing the complex flow processes in karst regions. The variable-order generalized Nash confluence model proposed in this invention incorporates the receding process of initial water storage in karst regions and employs a variable-order structure to reflect the time-varying response of runoff intensity to the confluence process, thus providing a more realistic representation of the confluence process in karst areas. The aquifers in karst regions are highly complex, with alternating light and dark water systems, frequent transitions between surface and underground rivers, and intertwined and difficult-to-distinguish confluence processes of surface and underground runoff, posing challenges to runoff simulation. Since the variable-order generalized Nash confluence model can directly describe the flow patterns in the complex media of karst regions without the need for source division, it simplifies the model structure, reduces model parameters, and lowers model uncertainty, providing an effective theoretical tool for high-precision flood forecasting and hydrological simulation in karst regions. Attached Figure Description
[0034] Figure 1 This is a comparison chart of the calculation results of the model of this invention with the measured values and the Xin'anjiang model. Detailed Implementation
[0035] To make the objectives, technical solutions, and advantages of this invention clearer, the 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 merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.
[0036] Example 1
[0037] A hydrological forecasting method for karst areas based on a variable-order generalized Nash confluence model includes:
[0038] Based on the potential evapotranspiration and precipitation of a karst region to be predicted for a certain period, the runoff forecast is calculated using the variable-order generalized Nash confluence model corresponding to the karst region to be predicted, which has been calibrated with parameters.
[0039] The variable-order generalized Nash confluence model is expressed as follows:
[0040]
[0041]
[0042] In the formula, O(t) represents the runoff at time t; n represents the number of Nash linear reservoirs; α∈(0,1) represents the order of the Caputo fractional order; O (j) (0) represents the j-th derivative of the flow rate at the initial time; The formula for calculating the number of combinations is given; K represents the regulation and storage parameters of the Nash linear reservoir. and Let γ = n, β = (ni)α + j + 1 or β = nα, respectively. u(·) represents the fractional-order instantaneous unit flow formed by instantaneous unit inflow; R(·) represents the watershed runoff; let α be the time-varying order, and let it be linearly related to R(·), i.e., α(t)=α[1+λR(t)]; the parameters calibrated by this model include the runoff parameters and the runoff parameters. The runoff parameters are n, α, K and λ, and the runoff parameters are determined according to the specific formula of R(·).
[0043] Existing runoff calculation methods based on fractional-order instantaneous unit hydrographs suffer from two main drawbacks: firstly, they fail to consider the influence of initial water storage in the basin, resulting in incomplete calculations of the runoff process; secondly, they neglect the time-varying nature of the order in the fractional-order runoff differential equations, causing different runoff intensities to exhibit the same response process, which is clearly inconsistent with actual physical processes. The variable-order generalized Nash runoff model proposed in this paper incorporates the receding process of initial water storage in karst regions and employs a variable-order structure to reflect the time-varying response of runoff intensity to the runoff process, thus providing a more realistic representation of the runoff process in karst regions. The aquifer in karst regions is highly complex, with alternating surface and underground river systems, frequent transitions between surface and underground rivers, and intertwined and difficult-to-distinguish runoff processes, posing significant challenges to runoff simulation. Since the variable-order generalized Nash confluence model can directly describe the water flow patterns in complex media in karst areas without the need for water source division, it can simplify the model structure, reduce model parameters, and lower model uncertainty. It can provide an effective theoretical tool for high-precision flood forecasting and hydrological simulation in karst areas.
[0044] It should be noted that, to calibrate the parameters of the variable-order generalized Nash confluence model corresponding to each karst region, it is necessary to first collect and preprocess historical hydrological and meteorological data of the karst region to be forecasted, obtain the potential evapotranspiration, precipitation and runoff data of the karst region to be forecasted, and construct a hydrological and meteorological data sample set; then, using this hydrological and meteorological data sample set, the parameters of the constructed variable-order generalized Nash confluence model structure are calibrated.
[0045] Fractional calculus is a natural generalization of integer calculus, and it is a non-standard operator theory and application for studying differentiation and integration of arbitrary orders. Fractional calculus was proposed almost simultaneously with classical calculus, boasting a history of over 300 years. However, its development has been slow due to a lack of clear physical meaning and uncertain application prospects. Fractional calculus possesses time memory and long-range spatial correlation, enabling it to more accurately describe physical phenomena and biochemical processes with memory, heredity, and path dependence properties. In recent decades, due to these unique advantages in describing complex physical processes, fractional calculus has been widely applied in high-energy physics, anomalous diffusion, system control, bioengineering, and fluid mechanics, gradually becoming one of the international research hotspots.
[0046] In the development of fractional calculus theory, many definitions of fractional calculus have emerged. Currently, the four most commonly used definitions are the Grünwald-Letnikov fractional derivative, the Riemann-Liouville fractional derivative, the Caputo fractional derivative, and the Weyl fractional derivative. Among these, the initial conditions of the Caputo definition are given in the form of integer calculus, which are relatively easy to obtain in practical engineering problems; therefore, the Caputo definition is often used in engineering applications.
[0047] Suppose the function f(t) is defined on the interval [a,t], and The α-th order Caputo derivative of the function f(t) is defined as follows:
[0048]
[0049] In the formula, f is the notation for the fractional derivative of Caputo, representing a fractional differential operator; in the symbol, C represents the Caputo operator; α∈[n-1,n), representing the order of the differential; f (n) (s) is the nth derivative of the function f(s). As α→n, the Caputo derivative degenerates into the ordinary nth derivative.
[0050] As a preferred implementation, the structure of the above-mentioned variable-order generalized Nash confluence model is constructed in the following manner:
[0051] (1) Based on the generalized Nash confluence theory, a higher-order differential equation is constructed to describe the water flow motion in the confluence system: (1+KD) n O(t) = R(t), where, K represents the regulation parameter of the Nash linear reservoir, R(t) represents the runoff at time t, and n represents the number of Nash linear reservoirs;
[0052] (2) The integer derivatives in the higher-order differential equations are expressed using Caputo fractional derivatives, and the higher-order differential equations are expanded using binomial expansion to obtain the fractional differential equations of the merging system: In the formula, This represents the Caputo fractional derivative;
[0053] (3) The fractional differential equation is solved by using the operational calculus method to obtain the variable-order generalized Nash confluence model.
[0054] Specifically, according to the generalized Nash confluence theory, the higher-order differential equations describing the water flow motion in a confluence system are:
[0055]
[0056] In the formula: R(t) and O(t) represent the inflow and outflow processes, respectively, and n and K are parameters. If we use... Let the differential operator be denoted, then the above expression can be written as:
[0057] (1+KD) n O(t)=I(t) (2)
[0058] Expressing the integer derivatives in terms of Caputo fractional derivatives, the above equation becomes the following fractional differential equation:
[0059]
[0060] Expanding the above expression as a binomial, we get:
[0061]
[0062] In the formula, This is the formula for calculating combinations.
[0063] Equation (4) is the fractional differential equation of the bus system. Generally, fractional differential equations are difficult to solve analytically. Only some special equations can be solved analytically using the Mellin transform method, Laplace transform method, operational calculus method, and Green's function method. This application attempts to solve the above fractional differential equation using operational calculus method.
[0064] According to the research of Luchko and Gorenflo, the Caputo fractional derivative and the Riemann-Liouville fractional derivative have the following relationship:
[0065]
[0066] In the formula, U α =I / h α , is the inverse operation of the Riemann-Liouville fractional derivative. Where, I = h α / h α , is the unit yuan; h α (t)=t α-1 / Γ(α) is the kernel function of the Riemann-Liouville fractional derivative. U α It has the following properties:
[0067]
[0068]
[0069]
[0070]
[0071] In the formula, λ and ρ are arbitrary constants. The generalized Mittag-Leffler function is defined as follows:
[0072]
[0073] In the formula, α, β, and γ are parameters greater than 0.
[0074] Substituting equation (5) into equation (4), we get:
[0075]
[0076] Further simplification and organization yield:
[0077]
[0078] It can be seen that O(t) consists of two parts on the right side of equation (7). According to the generalized Nash confluence theory, these two parts are the inflow R(t) and the outflow process generated after the initial water storage is regulated, respectively represented by O. I (t) and O s (t) represents, then
[0079]
[0080]
[0081] If we assume the inflow R(t) is an instantaneous inflow I of 1 unit, then the corresponding outflow O I (t) is the fractional instantaneous unit line u(t). Substituting equation (8) into equation (13), we get:
[0082]
[0083] According to U α From the definition and properties of equation (6), we can know that:
[0084]
[0085] Substitute equation (16) into equation (14), and use U α By simplifying equation (14) with the properties of equations (6) and (7), we can obtain:
[0086]
[0087] Substituting equation (9) into the above equation and rearranging, we get:
[0088]
[0089] If remember
[0090]
[0091] The analytical solution to the fractional differential equation (4) describing the merging system is:
[0092]
[0093] Equation (20) is the calculation formula for the fractional-order generalized Nash confluence model. It has a similar model structure to the generalized Nash confluence model, both consisting of a receding process of the initial storage and a response process of the inflow. The main difference between the two lies in the S in the fractional-order generalized Nash confluence model. i,j The long tails of (t) and u(·) better simulate the long-range dependence of runoff processes. The aquifers in karst regions are highly complex, with alternating light and dark water systems, frequent transitions between surface and underground rivers, and intertwined and difficult-to-distinguish confluence processes of surface and underground runoff, posing challenges to runoff simulation. The fractional-order generalized Nash confluence model can directly describe the flow patterns in the complex media of karst regions without the need for source identification, thus simplifying the model structure, reducing model parameters, and lowering model uncertainty. This provides an effective theoretical tool for high-precision flood forecasting and hydrological simulation in karst regions.
[0094] As a preferred implementation method, the SCE-UA optimization algorithm can be used for parameter calibration.
[0095] As a preferred implementation, when performing parameter calibration, the Nash efficiency (NSE) can be used as the objective function, expressed as:
[0096]
[0097] In the formula, m represents the length of the runoff sequence; Q sim,i and Q obs,i Let i represent the simulated flow rate and the observed flow rate at time i, respectively. This represents the average observed flow rate.
[0098] As a preferred implementation method, the watershed runoff R(t) can be calculated using the Xin'anjiang model:
[0099]
[0100]
[0101] In the formula, P(t) is the precipitation at time t, E0(t) is the potential evapotranspiration at time t, and W0 is the soil moisture content at the beginning of the time period;
[0102] The generated flow parameters are k, B, and WMM.
[0103] The above-mentioned flow model parameters include the flow generation parameters k, B, WMM and the flow parameters n, K, α and λ. The range of model parameter values is shown in Table 1.
[0104] Table 1 Model parameter range
[0105] parameter k B WMM n K α λ Range of values 0.5-1.5 0.2-0.8 60-200 1-10 10-30 0-1 0-1
[0106] The variable-order generalized Nash confluence model proposed in this embodiment is derived from the fractional-order differential equations of the watershed confluence system, possessing a strong theoretical foundation. The model has a simple structure with only seven parameters, and the physical meaning of these parameters is clearly defined, reducing the model's uncertainty. This invention's model exhibits high simulation accuracy, has stronger applicability in karst regions, and is easy to promote and apply.
[0107] To better illustrate the feasibility and reliability of the present invention, the following example is provided:
[0108] For the upper reaches of the Qingjiang River, a hydrological forecasting model for karst areas based on a variable-order generalized Nash confluence model was constructed. The construction method includes the following steps:
[0109] S1: Data collection: Collect hydrological and meteorological data for the basin from 2010 to 2018, including daily precipitation data, daily pan evaporation observations and daily runoff data, with daily pan evaporation observations used as daily potential evapotranspiration data.
[0110] S2: Data partitioning: Based on a 7:3 ratio, the model rate-up period and the model validation period are divided. In this example, 2010-2016 is the model rate-up period, and 2017-2018 is the model validation period.
[0111] S3: Model Construction: Construct a hydrological forecasting model for karst areas based on the variable-order generalized Nash confluence model.
[0112] S4: Parameter Calibration: Model parameters were calibrated using hydrological and meteorological data from the watershed calibration period. The Nash efficiency coefficient (NSE) was used as the objective function, and the SCE-UA optimization algorithm was employed for parameter calibration. S5: Model Validation and Accuracy Evaluation: Model validation was performed using validation period data. The calibrated model parameters were used for calculations, and the simulated and measured flow process curves were compared to evaluate accuracy. Simulation results are shown in the figure. Figure 1 The corresponding comparison results (NSE values) with the Xin'anjiang model are shown in Table 2:
[0113] Table 2 Comparison results of NSE simulations by the models
[0114] Model Rate Regular Verification period This invention model 0.77 0.75 Xinanjiang model 0.79 0.72
[0115] The NSE of this example model was 0.77 during the calibration period and 0.75 during the validation period. The accuracy during the calibration period was slightly lower than that of the Xin'anjiang model, while the accuracy during the validation period was higher. However, since this example model has only 7 parameters, far fewer than the 15 parameters of the Xin'anjiang model, it reduces the complexity and uncertainty of the model to some extent, and also demonstrates that the model of this invention has strong applicability to this karst region.
[0116] Example 2
[0117] A computer-readable storage medium includes a stored computer program, wherein when the computer program is run by a processor, it controls the device where the storage medium is located to execute a hydrological forecasting method for karst areas based on a variable-order generalized Nash confluence model as described in Embodiment 1 above.
[0118] The relevant technical solutions are the same as in Embodiment 1, and will not be repeated here.
[0119] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.
Claims
1. A hydrological forecasting method for karst areas based on a variable-order generalized Nash confluence model, characterized in that, include: Based on the potential evapotranspiration and precipitation of a karst region to be predicted for a certain period, the runoff forecast is calculated using the variable-order generalized Nash confluence model corresponding to the karst region to be predicted, which has been calibrated with parameters. The variable-order generalized Nash confluence model is expressed as follows: ； ； ； In the formula, O ( t )express The flow at any given moment; This indicates the number of Nash linear reservoirs. Indicates the order of the Caputo fraction; Represents the flow rate at the initial time. First derivative; The formula for calculating combinations; This represents the regulation and storage parameters of the Nash linear reservoir; and Each of these represents a generalized Mittag-Leffler function, let , or , ; Represents the fractional-order instantaneous unit line formed by an instantaneous unit inflow; Indicates the watershed runoff; let Let be the time-varying order, and its order is equal to... It is a linear relationship, that is The parameters calibrated for this model include flow convergence parameters and flow generation parameters. The flow convergence parameters include... , , and The production flow parameters are based on The specific formula is determined; The structure of the variable-order generalized Nash confluence model is constructed as follows: (1) Based on the generalized Nash confluence theory, construct a higher-order differential equation to describe the water flow motion in the confluence system: In the formula, , This represents the storage and regulation parameters of the Nash linear reservoir. Indicates the current generation at time t. This indicates the number of Nash linear reservoirs; (2) The integer derivatives in the higher-order differential equations are expressed using Caputo fractional derivatives, and the higher-order differential equations are expanded using binomial expansion to obtain the fractional differential equations of the merging system: In the formula, This represents the Caputo fractional derivative; (3) The fractional differential equation is solved by computational calculus to obtain the variable-order generalized Nash confluence model; Step (3) includes: (31) The relationship between the Caputo fractional derivative and the Riemann-Liouville fractional derivative Substituting into the fractional differential equation and simplifying, we obtain a new fractional differential equation: ； In the formula, ; , indicating the unit yuan; , which represents the kernel function of the Riemann-Liouville fractional derivative; (32) According to The properties will Substituting the first term on the right-hand side of the new fractional differential equation At this time, Instantaneous inflow per unit I ,get ,in ; In addition, according to Definition and nature ,get Substituting this relation into the second term on the right-hand side of the new fractional differential equation and utilize Properties and Simplifying the formula, we get: Further based on Properties ,right After simplification, we get ; Therefore, the new fractional differential equation described in step (31) is transformed into ,in Thus, the variable-order generalized Nash confluence model is obtained; The watershed runoff was calculated using the Xin'anjiang model. R ( t ): ； ， ， ； In the formula, P ( t )for Rainfall at any given time E 0( t )for Potential evapotranspiration at any given moment W 0 represents the soil moisture content at the beginning of the time period; The production flow parameters are: k , B and WMM .
2. The hydrological forecasting method for karst areas according to claim 1, characterized in that, The SCE-UA optimization algorithm was used for parameter calibration.
3. The hydrological forecasting method for karst areas according to claim 1, characterized in that, When performing parameter calibration, Nash efficiency (NSE) is used as the objective function.
4. A computer-readable storage medium, characterized in that, The computer-readable storage medium includes a stored computer program, wherein, when the computer program is run by a processor, it controls the device where the storage medium is located to execute a hydrological forecasting method for karst areas based on a variable-order generalized Nash confluence model as described in any one of claims 1 to 3.