Method for monthly runoff probability prediction based on diffusion model of fusion teleconnection factor

Abstract

This invention discloses a method for monthly runoff probability forecasting using a diffusion model that integrates teleconnection factors. The method includes the following steps: collecting and preprocessing monthly baseline data of the study watershed to obtain a training set and a validation set; the monthly baseline data includes hydrological and meteorological factor data, teleconnection factor data, and basic watershed characteristic data; constructing a diffusion model that integrates teleconnection factors; iteratively updating and obtaining the optimal parameter combination of the diffusion model based on the particle swarm optimization algorithm and the training set; and using the diffusion model with the optimal parameter combination and the validation set, obtaining the monthly runoff probability distribution and the monthly runoff probability forecast interval at multiple confidence levels through Monte Carlo simulation. This invention solves the factor time-delay matching problem through time sliding windows, and the diffusion model that integrates teleconnection factors enhances the interpretability of its physical mechanisms and long-term forecast performance while retaining the advantages of the diffusion model's probability output, thus overcoming the key technical bottlenecks in monthly runoff probability forecasting.

Description

Technical Field

[0001] This invention relates to the field of hydrological forecasting technology, and more specifically, to a method for predicting monthly runoff probability using a diffusion model that integrates teleconnection factors. Background Technology

[0002] Monthly runoff forecasting is a core technical support for long-term water resource management, and its accuracy directly affects the efficiency of watershed water resource allocation and flood control safety. Current monthly forecasting methods can be mainly divided into two categories: one is statistical models based on measured hydrological and meteorological data, represented by ARIMA and random forests. While these models are relatively mature, they heavily rely on measured watershed data, have limited applicability in areas with no or scarce data, and struggle to effectively quantify forecast uncertainties. The other category is deep learning models that integrate teleconnection factors, such as the BO-BiLSTM model. These models overcome the limitations of data dependence to some extent, but due to their prominent black-box characteristics and unclear physical meaning of parameters, they struggle to clearly reveal the inherent mechanism of the coupling of determinism and randomness in runoff evolution.

[0003] Teleconnection factors, encompassing El Niño index and subtropical high ridge position, influence regional precipitation patterns by modulating atmospheric circulation, thus dominating the long-term evolution trend of monthly runoff and becoming key drivers for improving monthly forecasting capabilities. Diffusion models, as important tools in stochastic process theory, can describe the deterministic evolution trend and stochastic fluctuation characteristics of state variables through drift coefficients and diffusion coefficients, respectively. Their deterministic and stochastic modeling framework is inherently consistent with the runoff generation process. However, existing runoff forecasting research based on diffusion models has not effectively incorporated teleconnection factors, resulting in insufficient long-term forecasting capabilities. Furthermore, the lack of targeted design for time lag effects in monthly forecasts makes it difficult to fully utilize the lead time advantage provided by teleconnection factors.

[0004] Therefore, constructing a diffusion model that integrates teleconnection factors and adapts to monthly time lag characteristics, while retaining the advantages of the diffusion model's probabilistic output, and enhancing its physical mechanism interpretability and long-term forecast performance, has important theoretical value and practical significance for breaking through the key technical bottleneck of monthly runoff probability forecasting. Summary of the Invention

[0005] This invention provides a method for monthly runoff probability forecasting using a diffusion model that integrates teleconnection factors. It solves the factor time-delay matching problem by using a time sliding window, constructs a diffusion model that integrates teleconnection factors, uses the PSO algorithm to achieve global parameter optimization, and finally outputs multi-confidence level probability forecast results through Monte Carlo simulation.

[0006] The technical solution adopted in this invention is: A diffusion model method for predicting monthly runoff probability by incorporating teleconnection factors includes the following steps: S1. Collect hydrological and meteorological factor data, teleconnection factor data, and basic characteristic data of the watershed for research, and preprocess them to obtain training set and validation set; wherein, the hydrological and meteorological factor data includes monthly precipitation data and monthly runoff data; the teleconnection factor data includes atmospheric circulation factor and sea surface temperature factor; the basic characteristic data of the watershed includes watershed area data; S2. Set the boundary conditions of the model and construct the diffusion model that integrates teleconnection factors; S3. Based on the particle swarm optimization algorithm and the training set, the optimal parameter combination and target diffusion model are obtained through iterative updates. S4. Based on the target diffusion model and validation set, the monthly runoff probability distribution and the monthly runoff probability forecast interval at multiple confidence levels are obtained through Monte Carlo simulation output. S5. Based on multi-dimensional accuracy indicators, the NGBoost model is used as a control model to evaluate the model simulation accuracy. The multi-dimensional accuracy indicators include Nash efficiency coefficient (NSE), root mean square error (RMSE), and continuous sorting probability score (CRPS).

[0007] Further, the preprocessing in step S1 includes: S11. Based on the time sliding window mechanism, time-delay matching is performed on the teleconnection factor data and monthly runoff data to obtain the teleconnection factor data and monthly runoff data under the same time-delay gradient, which are used as the original dataset. S12. Perform missing value imputation, outlier removal and standardization on the original dataset in sequence to obtain the target dataset; S13. Divide the target dataset into training and validation sets according to a set ratio.

[0008] Furthermore, step S11 is as follows: 1) Monthly runoff depth data are calculated based on monthly runoff data and watershed area data; 2) Establish the relationship between monthly runoff and monthly runoff depth, obtain the forecasted monthly runoff, and then obtain the forecasted monthly runoff depth; 3) In the forecast period scenario, select teleconnection factor data and monthly runoff data corresponding to the forecast month in advance; 4) Using the predicted monthly runoff depth as the sliding window reference point, set the first time-delay gradient and the second time-delay gradient based on the selected teleconnection factor data and monthly runoff data, respectively. Slide the teleconnection factor data forward according to the first time-delay gradient and the monthly runoff data forward according to the second time-delay gradient to obtain the teleconnection factor data and monthly runoff data under the same time-delay gradient, which are used as the original dataset.

[0009] Furthermore, step S12 is as follows: 1) By using linear interpolation, a small amount of missing data is filled in to obtain the first dataset; 2) By using the multiple standard deviation method, outliers are removed to obtain the second dataset; 3) By standardizing the data, all second datasets are converted to the same size to obtain the third dataset, which is the target dataset.

[0010] Further, step S2 specifically includes: S21. Define the monthly runoff depth as a one-dimensional diffusion process, and establish the Itō stochastic differential equation representing the runoff diffusion process; the expression is as follows:

[0011] In the formula, for The differential increment of monthly runoff depth reflects The minute change in monthly runoff depth relative to the previous moment corresponds to the continuous evolution of runoff. This aligns with the core logic of Itoh's diffusion process, which describes the dynamic changes of state variables. for The moon's path flows deep at every moment; The drift coefficient reflects the deterministic trend of runoff change. The diffusion coefficient quantifies the intensity of runoff fluctuations caused by random factors. For the increment of the Wiener process, satisfying E[ ]=0 and Var[ ]= It is used to characterize random disturbances in runoff evolution; S22. The stochastic differential equation is transformed into an evolution equation of the probability density function using the Fock-Planck equation, which includes drift and diffusion terms; the expression is as follows:

[0012] In the formula, for time During runoff, time The conditional probability density of runoff has a solution that directly corresponds to the predicted monthly runoff probability. The predicted monthly runoff probability distribution for any interval can be obtained by integration. S23. Based on hydrophysical constraints, set the model boundary conditions: 1) When the monthly runoff depth approaches the low water limit, that is, when there is no effective runoff generation in the watershed, both the drift coefficient and the diffusion coefficient approach zero. 2) When the monthly runoff depth approaches the flood limit, the drift coefficient is related to the watershed storage coefficient, and the diffusion coefficient tends to a stable constant; among them, the watershed storage coefficient is used to reflect the inhibitory effect of the river channel's flood discharge capacity on runoff growth; S24. Parametric fusion of teleconnection factors: Transforming abstract teleconnection factors into parameter-driven terms that the model can recognize.

[0013] Further, step S24 specifically includes: 1) Calculation of the driving force of the teleconnection comprehensive factor: Based on principal component analysis, the multi-dimensional teleconnection factor data is reduced and fused to obtain the teleconnection comprehensive influence factor; based on the principal component coefficients and standardized values ​​of each teleconnection influence factor, the weighted sum of multiple teleconnection comprehensive influence factors is obtained to obtain the driving force of the teleconnection comprehensive factor; the calculation formula is as follows:

[0014] In the formula, This is a driving factor of the teleconnection comprehensive factor; For the first Principal component coefficients of the teleconnection factor; For the first Standardized values ​​of teleconnection factors; This represents the number of teleconnection factor terms; 2) Parametric design of drift coefficient: Based on the driving factors of teleconnection, monthly precipitation, and runoff autoregressive characteristics, the drift coefficient is calculated; the formula is as follows:

[0015] In the formula, This is the drift coefficient; This is a driving factor of the teleconnection comprehensive factor; This refers to the rainfall in the previous month; The depth of the lunar runoff; This represents the influence coefficient of the teleconnection factor, corresponding to the driving force of the comprehensive teleconnection factor in the drift coefficient. The weights reflect the intensity of the teleconnection factors' regulation of the deterministic trend of runoff; This is the precipitation influence coefficient, corresponding to the precipitation in the previous month in the drift coefficient. The weight of the precipitation reflects the intensity of the replenishment effect of short-term precipitation on the monthly runoff; This is the runoff autoregressive coefficient, corresponding to the monthly runoff depth in the drift coefficient. The weight of the watershed reflects the attenuation effect of watershed regulation on runoff; 3) Parametric design of diffusion coefficient: Based on the coefficient of variation of the driving factors of teleconnection, the coefficient of variation of monthly precipitation, and the characteristics of runoff itself, the diffusion coefficient is calculated; the formula is as follows:

[0016] In the formula, The diffusion coefficient is a quantification of the intensity of runoff fluctuations caused by random factors. The coefficient of variation of the teleconnection factor is the driving force of the comprehensive teleconnection factor. The ratio of the standard deviation to the mean; The coefficient of variation is the monthly precipitation data, which is the monthly precipitation series. The ratio of the standard deviation to the mean; The depth of the lunar runoff; This represents the influence coefficient of teleconnection factor fluctuations, corresponding to the coefficient of variation of teleconnection factor in the diffusion coefficient. The weights reflect the impact of the stability of teleconnection factors on random fluctuations in runoff. This is the influence coefficient of precipitation fluctuation, corresponding to the precipitation variation coefficient in the diffusion coefficient. The weight of the metric reflects the regulatory effect of short-term precipitation fluctuations on random changes in runoff; This is the runoff fluctuation coefficient, corresponding to the square of the monthly runoff depth in the diffusion coefficient. The weights reflect the impact of the runoff's own magnitude on random fluctuations; The constant term coefficient corresponds to the basic fluctuation intensity of the diffusion coefficient, quantifying runoff fluctuations caused by inherent random factors.

[0017] Further, step S3 specifically includes: S31. Constructing the fitness function: With the objectives of minimizing the Continuous Ranking Probability Score (CRPS) and ensuring the Predicted Interval Coverage Rate (PICP) approaches a pre-set confidence level, a weighted comprehensive fitness function is constructed; the expression is as follows:

[0018] in,

[0019]

[0020] In the formula, The weighted comprehensive fitness function is used; CRPS is the continuous ranking probability score, which is used to quantify the overall deviation between the monthly runoff probability distribution and the measured value. Forecast coverage rate; The cumulative monthly runoff distribution function output by the diffusion model is obtained through... The result is obtained by integration; For actual measured monthly runoff; As an indicator function, when the predicted monthly runoff depth Measured monthly runoff depth hour =1, otherwise =0; The total number of samples; For the first Real-time measurement of monthly runoff; For the first The lower limit of the confidence interval for time forecast (corresponding to the 5th percentile); For the first The upper limit of the confidence interval for time forecast (corresponding to the 95th percentile); This is an indicator function, used when the measured monthly runoff falls within the corresponding forecast interval. ,otherwise ; S33. Update and iterate based on the POS algorithm: 2) Algorithm parameter initialization: particle population size, maximum number of iterations, inertia weight, learning factor; initial position of particles, initial velocity of particles; where each particle corresponds to a set of 7-dimensional parameter combinations. The dimension of each particle is consistent with the number of parameters to be calibrated; the inertial weight adopts an adaptive strategy, i.e., in the initial stage of iteration. =0.9 enhances global search and avoids getting trapped in local optima; in the later stages of iteration =0.4 enhances local convergence and improves parameter accuracy; the learning factor is set with... and and = =2, used to balance the influence of individual experience and group experience; the initial position and initial velocity of the particles are randomly generated; 3) For each initialized particle, substitute it into the diffusion model to calculate the fitness value: 4) Update the individual particle optimal pbest and the group optimal gbest; 5) Update particle velocity and position; 6) Update and iterate, and determine whether the number of iterations has reached the preset maximum number of iterations: if it has, stop the iteration and proceed to the next step; if it has not, return to step 3) and repeat the operation until the maximum number of iterations is reached; 7) After the iteration terminates, the 7-dimensional parameter combination corresponding to the global optimum gbest of the population is used as the optimal calibration parameters of the model. The specific process for calculating the fitness value of each particle is as follows: ① Combine the parameters in the 7-dimensional parameter combination and parameters Substitute these values ​​into the drift coefficient formula and the diffusion coefficient formula respectively; ②The runoff probability density function is obtained by solving the Fock-Planck equation, which includes drift and diffusion terms; ③ Based on the runoff probability density function, the CRPS and PICP indices are calculated; ④ The fitness value is calculated based on the weighted comprehensive fitness function; S34. After determining the optimal parameter combination, the corresponding target drift coefficient and target diffusion coefficient are obtained; S35, the target drift coefficient and the target diffusion coefficient, are used to train the target diffusion model through the training set.

[0021] Furthermore, in step S4, Monte Carlo simulation is used to generate the monthly runoff probability distribution, specifically as follows: 1) Spatiotemporal matching of parameters: Based on the validation set, obtain the driving forces of the teleconnection comprehensive factors for each forecast month. and the precipitation in the month preceding the forecast month Calculate the drift coefficient and diffusion coefficient at the corresponding time. This ensures that the parameters are dynamically updated over time. 2) Random sample generation: Multiple sets of independent monthly runoff evolution samples are generated by solving stochastic differential equations using the Euler-Markov method; 3) Probability result extraction: Sort multiple groups of samples and calculate the 5%, 25%, 50%, 75%, and 95% quantiles; among them, the 50% quantile is the deterministic forecast value; the 5%-95% quantiles constitute the monthly runoff probability forecast interval at a 90% confidence level; at the same time, the monthly runoff probability distribution, i.e., the monthly runoff probability density curve, is obtained through kernel density estimation, realizing full-dimensional output of point forecast, interval forecast, and distribution forecast.

[0022] Furthermore, the Nash efficiency coefficient (NSE) is used to evaluate the overall accuracy of monthly runoff deterministic forecasts, primarily reflecting the degree of fit between simulated and measured runoff; the calculation formula is as follows:

[0023] In the formula, NSE is the point forecast accuracy; For the first Real-time measurement of monthly runoff; For the first Real-time forecast of monthly runoff; This represents the mean of the measured monthly runoff series; denoted as the total number of samples; where NSE ranges from (-∞, 1], and the model performance is rated as excellent when NSE ≥ 0.75; The root mean square error (RMSE) is used to quantify the average deviation between simulated and measured monthly runoff, and its unit is the same as that of runoff volume; the calculation formula is as follows:

[0024] In the formula, This is the root mean square error; For the first Real-time measurement of monthly runoff; For the first Real-time forecast of monthly runoff; The total number of samples; where, The smaller the value, the smaller the deviation between the simulated and measured values, and the higher the simulation accuracy of the overall runoff fluctuation; conversely, the larger the value, the greater the deviation. The Continuous Ranking Probability Score (CRPS) is used to evaluate the accuracy of monthly runoff probability forecasts. It measures the overall deviation between the runoff probability distribution output by the core quantification model and the measured values, with units consistent with runoff volume. The calculation formula is as follows:

[0025] In the formula, For continuous sorting probability scoring; The cumulative monthly runoff distribution function output by the diffusion model is obtained through... The result is obtained by integration; For actual measured monthly runoff; As an indicator function, when the predicted monthly runoff depth Measured monthly runoff depth hour =1, otherwise =0; where, The smaller the value, the higher the fit between the probability distribution and the measured value, and the more accurate the uncertainty quantification.

[0026] Compared with the prior art, the present invention has the following advantages: 1) This invention integrates teleconnection factors into the parameters of the diffusion model through a parameterization scheme, thereby achieving a quantitative correlation between climate factors, hydrological processes, and model parameters, which solves the problem of weak long-term forecasting capability of traditional diffusion models; the physical meaning of the model parameters is clear, overcoming the "black box" defect of deep learning models and improving the interpretability and credibility of forecast results. 2) This invention designs a targeted time sliding window mechanism to achieve accurate time-lag matching between teleconnection factors and hydro-meteorological factors, ensuring that the lag relationship between factors and target monthly runoff is fully reflected, while meeting the high efficiency requirements of operational forecasting. 3) Particle swarm optimization algorithm is used for dual-objective parameter calibration. The comprehensive fitness function is used to balance the prediction accuracy and reliability, avoid the prediction bias caused by a single objective, and improve the overall performance of the model. 4) By outputting multi-dimensional probability results such as point forecasts, interval forecasts, and distribution forecasts through Monte Carlo simulation, it is possible to directly provide a basis for risk quantification for water resource allocation, help decision-makers fully grasp the uncertainty of runoff forecasts, and improve the scientificity and rationality of decision-making.

[0027] 5) The teleconnection factor is publicly available data globally. In areas where there is no measured data, basic data can be supplemented through remote sensing inversion and other methods, so that the applicable scope of the method of this invention covers watersheds in different climate zones such as humid and semi-arid regions. It is especially suitable for monthly runoff probability forecasting in areas with no data or scarce data, thus expanding the application scenarios of monthly runoff forecasting methods. Attached Figure Description

[0028] The present invention will be further described below with reference to the accompanying drawings and specific embodiments: Figure 1 This is a flowchart illustrating a method for predicting the probability of monthly runoff using a diffusion model that integrates teleconnection factors, as described in this invention. Figure 2 This is a schematic diagram of the monthly runoff probability forecast results in this invention. Detailed Implementation

[0029] Example

[0030] like Figure 1 As shown, a diffusion model method for predicting monthly runoff probability by incorporating teleconnection factors includes the following steps: S1. Collect hydrological and meteorological factor data, teleconnection factor data, and basic characteristic data of the watershed for research, and preprocess them to obtain training set and validation set; wherein, the hydrological and meteorological factor data includes monthly precipitation data and monthly runoff data; the teleconnection factor data includes atmospheric circulation factor and sea surface temperature factor; the basic characteristic data of the watershed includes watershed area data; Preprocessing includes: S11. Based on the time sliding window mechanism, time-lag matching is performed on the teleconnection factor data and monthly runoff data to obtain teleconnection factor data and monthly runoff data under the same time-lag gradient, which serves as the original dataset; the details are as follows: 1) Monthly runoff depth data are calculated based on monthly runoff data and watershed area data; 2) Establish the relationship between monthly runoff and monthly runoff depth, obtain the forecasted monthly runoff, and then obtain the forecasted monthly runoff depth; 3) In the forecast period scenario, select teleconnection factor data and monthly runoff data corresponding to the forecast month in advance; 4) Using the predicted monthly runoff depth as the sliding window reference point, set the first time-delay gradient and the second time-delay gradient based on the selected teleconnection factor data and monthly runoff data, respectively. Slide the teleconnection factor data forward according to the first time-delay gradient and the monthly runoff data forward according to the second time-delay gradient to obtain the teleconnection factor data and monthly runoff data under the same time-delay gradient, which are used as the original dataset. S12. Perform missing value imputation, outlier removal, and standardization on the original dataset sequentially to obtain the target dataset; the details are as follows: 1) By using linear interpolation, a small amount of missing data is filled in to obtain the first dataset; 2) By using the multiple standard deviation method, outliers are removed to obtain the second dataset; 3) By standardizing all the second datasets, convert them to the same size to obtain the third dataset, which is the target dataset; S13. Divide the target dataset into a 7:3 ratio to obtain the training set and the validation set.

[0031] Note: By establishing a quantitative relationship between monthly runoff and monthly runoff depth, runoff data from different watershed areas can be converted into a regionally comparable runoff depth index, eliminating the impact of watershed area differences on runoff analysis and providing standardized basic data for subsequent time-lag matching of multi-source factors and model construction. Simultaneously, this conversion process follows basic hydrological principles; the formula for calculating monthly runoff depth is the ratio of monthly runoff to watershed area, i.e., monthly runoff depth = monthly runoff / watershed area, with the unit uniformly set to millimeters, ensuring data consistency in dimensions and laying a reliable data foundation for the fusion analysis of model input factors.

[0032] S2. Set the boundary conditions for the model and construct the diffusion model that integrates teleconnection factors; specifically: S21. Define the monthly runoff depth as a one-dimensional diffusion process, and establish the Itō stochastic differential equation representing the runoff diffusion process; the expression is as follows:

[0033] In the formula, for The differential increment of monthly runoff depth reflects The minute change in monthly runoff depth relative to the previous moment corresponds to the continuous evolution of runoff. This aligns with the core logic of Itoh's diffusion process, which describes the dynamic changes of state variables. for The moon's path flows deep at every moment; The drift coefficient reflects the deterministic trend of runoff change. The diffusion coefficient quantifies the intensity of runoff fluctuations caused by random factors. For the increment of the Wiener process, satisfying E[ ]=0 and Var[ ]= This means that the expected value of the Wiener process increment is 0, indicating that the random disturbances in runoff evolution have no systematic bias; variance and time increment The correlation is proportional, indicating that the intensity of random fluctuations in runoff has a cumulative effect as the forecast duration increases. This is in line with the evolution law of uncertainty in hydrological processes and can accurately characterize the random disturbances in runoff evolution caused by uncontrollable factors such as fluctuations in watershed meteorological input and heterogeneity of underlying surface. S22. The stochastic differential equation is transformed into an evolution equation of the probability density function using the Fock-Planck equation, which includes drift and diffusion terms; the expression is as follows:

[0034] In the formula, for time During runoff, time The conditional probability density of runoff has a solution that directly corresponds to the predicted monthly runoff probability. The predicted monthly runoff probability distribution for any interval can be obtained by integration. S23. Based on hydrophysical constraints, set the model boundary conditions: 1) When the monthly runoff depth approaches the low water limit, that is, when there is no effective runoff generation in the watershed, both the drift coefficient and the diffusion coefficient approach zero. 2) When the monthly runoff depth approaches the flood limit, the drift coefficient is related to the watershed storage coefficient, and the diffusion coefficient tends to a stable constant; among them, the watershed storage coefficient is used to reflect the inhibitory effect of the river channel's flood discharge capacity on runoff growth; S24. Parametric Fusion of Teleconnection Factors: Transforming abstract teleconnection factors into parameter-driven terms that the model can recognize; specifically: 1) Calculation of the driving force of the teleconnection comprehensive factor: Based on principal component analysis, the multi-dimensional teleconnection factor data is reduced and fused to obtain the teleconnection comprehensive influence factor; based on the principal component coefficients and standardized values ​​of each teleconnection influence factor, the weighted sum of multiple teleconnection comprehensive influence factors is obtained to obtain the driving force of the teleconnection comprehensive factor; the calculation formula is as follows:

[0035] In the formula, This is a driving factor of the teleconnection comprehensive factor; For the first Principal component coefficients of the teleconnection factor; For the first Standardized values ​​of teleconnection factors; This represents the number of teleconnection factor terms; 2) Parametric design of drift coefficient: Based on the driving factors of teleconnection, monthly precipitation, and runoff autoregressive characteristics, the drift coefficient is calculated; the formula is as follows:

[0036] In the formula, This is the drift coefficient; This is a driving factor of the teleconnection comprehensive factor; This refers to the rainfall in the previous month; The depth of the lunar runoff; This represents the influence coefficient of the teleconnection factor, corresponding to the driving force of the comprehensive teleconnection factor in the drift coefficient. The weights reflect the intensity of the teleconnection factors' regulation of the deterministic trend of runoff; This is the precipitation influence coefficient, corresponding to the precipitation in the previous month in the drift coefficient. The weight of the precipitation reflects the intensity of the replenishment effect of short-term precipitation on the monthly runoff; This is the runoff autoregressive coefficient, corresponding to the monthly runoff depth in the drift coefficient. The weight of the watershed reflects the attenuation effect of watershed regulation on runoff; 3) Parametric design of diffusion coefficient: Based on the coefficient of variation of the driving factors of teleconnection, the coefficient of variation of monthly precipitation, and the characteristics of runoff itself, the diffusion coefficient is calculated; the formula is as follows:

[0037] In the formula, The diffusion coefficient is a quantification of the intensity of runoff fluctuations caused by random factors. The coefficient of variation of the teleconnection factor is the driving force of the comprehensive teleconnection factor. The ratio of the standard deviation to the mean; The coefficient of variation is the monthly precipitation data, which is the monthly precipitation series. The ratio of the standard deviation to the mean; The depth of the lunar runoff; This represents the influence coefficient of teleconnection factor fluctuations, corresponding to the coefficient of variation of teleconnection factor in the diffusion coefficient. The weights reflect the impact of the stability of teleconnection factors on random fluctuations in runoff. This is the influence coefficient of precipitation fluctuation, corresponding to the precipitation variation coefficient in the diffusion coefficient. The weight of the metric reflects the regulatory effect of short-term precipitation fluctuations on random changes in runoff; This is the runoff fluctuation coefficient, corresponding to the square of the monthly runoff depth in the diffusion coefficient. The weights reflect the impact of the runoff's own magnitude on random fluctuations; The constant term coefficient corresponds to the basic fluctuation intensity of the diffusion coefficient, quantifying runoff fluctuations caused by inherent random factors.

[0038] Note: In this embodiment, the monthly runoff depth is used as the state variable, and the Fokker-Planck equation containing drift and diffusion terms is derived. The teleconnection factor is quantified as the driving variable of the core parameters of the diffusion model.

[0039] S3. Based on the particle swarm optimization algorithm and the training set, the optimal parameter combination and target diffusion model are obtained through iterative updates; specifically: S31. Constructing the fitness function: With the objectives of minimizing the Continuous Ranking Probability Score (CRPS) and ensuring the Predicted Interval Coverage Rate (PICP) approaches a pre-set confidence level, a weighted comprehensive fitness function is constructed; the expression is as follows:

[0040] in,

[0041]

[0042] In the formula, The weighted comprehensive fitness function is used; CRPS is the continuous ranking probability score, which is used to quantify the overall deviation between the monthly runoff probability distribution and the measured value. Forecast coverage rate; The cumulative monthly runoff distribution function output by the diffusion model is obtained through... The result is obtained by integration; For actual measured monthly runoff; As an indicator function, when the predicted monthly runoff depth Measured monthly runoff depth hour =1, otherwise =0; The total number of samples; For the first Real-time measurement of monthly runoff; For the first The lower limit of the confidence interval for time forecast (corresponding to the 5th percentile); For the first The upper limit of the confidence interval for time forecast (corresponding to the 95th percentile); This is an indicator function, used when the measured monthly runoff falls within the corresponding forecast interval. ,otherwise ; S33. Update and iterate based on the POS algorithm: 2) Algorithm parameter initialization: particle population size, maximum number of iterations, inertia weight, learning factor; initial position of particles, initial velocity of particles; where each particle corresponds to a set of 7-dimensional parameter combinations. The dimension of each particle is consistent with the number of parameters to be calibrated; the inertial weight adopts an adaptive strategy, i.e., in the initial stage of iteration. =0.9 enhances global search and avoids getting trapped in local optima; in the later stages of iteration =0.4 enhances local convergence and improves parameter accuracy; the learning factor is set with... and and = =2, used to balance the influence of individual experience and group experience; the initial position and initial velocity of the particles are randomly generated; 3) For each initialized particle, substitute it into the diffusion model to calculate the fitness value: 4) Update the individual particle optimal pbest and the group optimal gbest; 5) Update particle velocity and position; 6) Update and iterate, and determine whether the number of iterations has reached the preset maximum number of iterations: if it has, stop the iteration and proceed to the next step; if it has not, return to step 3) and repeat the operation until the maximum number of iterations is reached; 7) After the iteration terminates, the 7-dimensional parameter combination corresponding to the global optimum gbest of the population is used as the optimal calibration parameters of the model. The particle position boundaries are set as follows: The location boundary range is [0.2, 2.0], and the teleconnection factor has a positive regulatory effect on runoff. To facilitate abortion, (negative benefit to flow reduction), therefore Non-negative; combined with actual measurements in the watershed, the teleconnection effect can be ignored when the coefficient is less than 0.2, while it will over-amplify the teleconnection effect when it is greater than 2.0, leading to simulation distortion. The location boundary range is [0.3, 1.8]. The effect of precipitation on runoff replenishment is positive, therefore... The coefficient is always positive; when it is less than 0.3, it cannot reflect the core replenishment role of precipitation, and when it is greater than 1.8, it will overemphasize the impact of precipitation, resulting in higher simulated values ​​during the flood season. The location boundary range is [-0.5, -0.05]. The watershed's water storage capacity has a attenuating effect on runoff, therefore... The coefficient is always negative; when the absolute value of the coefficient is less than 0.05, the regulation and storage effect can be ignored, but when it is greater than 0.5, it will excessively attenuate the runoff, resulting in lower simulated values ​​during the dry season. The location boundary range is [0.05, 0.30], and the variation of teleconnection factors has a positive impact on runoff fluctuations. The larger the value, the stronger the fluctuation. Non-negative; the coefficient values ​​are combined with the weights of the teleconnection factors to ensure consistency with... Matching the intensity of regulation; The location boundary range is [0.08, 0.35, and the impact of precipitation fluctuations on runoff fluctuations is positive ( ). The larger the value, the stronger the fluctuation. Always positive; coefficient slightly greater than This reflects the dominant role of short-term precipitation fluctuations in random changes in runoff; The location boundary range is [0.02, 0.20]. The magnitude of the runoff itself has a positive impact on the fluctuation (the larger the runoff, the stronger the fluctuation). Non-negative; the coefficient values ​​are moderate to avoid over-amplifying the fluctuation intensity during periods of high runoff; The location boundary range is [0.01, 0.15], and the diffusion coefficient reflects the intensity of random fluctuations and is always positive. As a basic fluctuation term, if the value is too small, the inherent random factors will be ignored; if the value is too large, the fluctuation simulation will be distorted. The particle velocity boundary is set as follows: The velocity boundaries correspond to the position boundaries of each parameter; the following settings are used: Velocity boundary = Position boundary range × 0.15; The specific process for calculating the fitness value of each particle is as follows: ① Combine the parameters in the 7-dimensional parameter combination and parameters Substitute these values ​​into the drift coefficient formula and the diffusion coefficient formula respectively; ②The runoff probability density function is obtained by solving the Fock-Planck equation, which includes drift and diffusion terms; ③ Based on the runoff probability density function, the CRPS and PICP indices are calculated; ④ The fitness value is calculated based on the weighted comprehensive fitness function; S34. After determining the optimal parameter combination, the corresponding target drift coefficient and target diffusion coefficient are obtained; S35, the target drift coefficient and the target diffusion coefficient, are used to train the target diffusion model through the training set.

[0043] Explanation: By using the particle swarm optimization algorithm to globally optimize the model parameters, the shortcomings of traditional parameter calibration methods that are prone to getting trapped in local optima can be effectively overcome, ensuring the global optimality of the model parameter combination. At the same time, the weighted integrated fitness function constructed by minimizing CRPS and approximating PICP to the preset confidence level not only focuses on the overall deviation between the probability forecast distribution and the measured values, but also takes into account the coverage of the forecast interval. This results in a significant improvement in the accuracy and reliability of the optimized target diffusion model in probability forecasting, providing a solid model foundation for subsequent monthly runoff probability forecasting.

[0044] S4. Based on the target diffusion model and validation set, the monthly runoff probability distribution and the monthly runoff probability forecast interval at multiple confidence levels are obtained through Monte Carlo simulation output; specifically, the monthly runoff probability distribution is generated using Monte Carlo simulation as follows: 1) Spatiotemporal matching of parameters: Based on the validation set, obtain the driving forces of the teleconnection comprehensive factors for each forecast month. and the precipitation in the month preceding the forecast month Calculate the drift coefficient and diffusion coefficient at the corresponding time. This ensures that the parameters are dynamically updated over time. 2) Random sample generation: Multiple sets of independent monthly runoff evolution samples are generated by solving stochastic differential equations using the Euler-Markov method; 3) Probability Result Extraction: Multiple groups of samples are sorted, and the 5%, 25%, 50%, 75%, and 95% quantiles are calculated. Among them, the 50% quantile is the deterministic forecast value; the 5%-95% quantiles constitute the monthly runoff probability forecast interval at a 90% confidence level; at the same time, the monthly runoff probability distribution, i.e., the monthly runoff probability density curve, is obtained through kernel density estimation, realizing full-dimensional output of point forecast, interval forecast, and distribution forecast. Note: In this embodiment, 1000 sets of independent monthly runoff evolution samples are generated to balance forecast accuracy and computational efficiency, as determined by sensitivity analysis.

[0045] S5. Based on multi-dimensional accuracy indicators, the NGBoost model is used as a control model to evaluate the model simulation accuracy; the multi-dimensional accuracy indicators include Nash efficiency coefficient (NSE), root mean square error (RMSE), and continuous ranking probability score (CRPS). The Nash efficiency coefficient (NSE) is used to evaluate the overall accuracy of deterministic monthly runoff forecasts, and it primarily reflects the degree of agreement between simulated and measured runoff. The calculation formula is as follows:

[0046] In the formula, NSE is the point forecast accuracy; For the first Real-time measurement of monthly runoff; For the first Real-time forecast of monthly runoff; This represents the mean of the measured monthly runoff series; denoted as the total number of samples; where NSE ranges from (-∞, 1], and the model performance is rated as excellent when NSE ≥ 0.75; In this embodiment, NSE=0.87 during training on the model training set and NSE=0.85 during validation on the validation set, both of which meet the excellent standard, indicating that the simulation accuracy is high and the stability is strong. The root mean square error (RMSE) is used to quantify the average deviation between simulated and measured monthly runoff, with units consistent with runoff volume; the calculation formula is as follows:

[0047] In the formula, This is the root mean square error; For the first Real-time measurement of monthly runoff; For the first Real-time forecast of monthly runoff; The total number of samples; where, The smaller the value, the smaller the deviation between the simulated and measured values, and the higher the simulation accuracy of the overall runoff fluctuation; conversely, the larger the value, the greater the deviation. The Continuous Ranking Probability Score (CRPS) is used to evaluate the accuracy of monthly runoff probability forecasts. It represents the overall deviation between the runoff probability distribution output by the core quantification model and the measured values, with units consistent with runoff volume. The calculation formula is as follows:

[0048] In the formula, For continuous sorting probability scoring; The cumulative monthly runoff distribution function output by the diffusion model is obtained through... The result is obtained by integration; For actual measured monthly runoff; As an indicator function, when the predicted monthly runoff depth Measured monthly runoff depth hour =1, otherwise =0; where, The smaller the value, the higher the fit between the probability distribution and the measured value, and the more accurate the uncertainty quantification. The accuracy assessment of the model simulation results was obtained through the calculation of the above multi-dimensional accuracy indicators, as shown in Table 1: Table 1 Accuracy Evaluation of Model Simulation Results

[0049] As shown in Table 1, the model of this invention exhibits excellent comprehensive performance in monthly runoff probability forecasting: the NSE in the rate period and validation set reaches 0.87 and 0.85 respectively, with a difference of only 0.02. This not only meets the superior standard of NSE ≥ 0.75 in my country's "Hydrological Information Forecasting Specification", but also shows significantly better stability than the NGBoost model (whose validation set NSE decreased by 0.06 to 0.82 compared to the rate period). In terms of the core probabilistic forecasting indicators, the CRPS of the model in this invention are 1509.05 m³ / s in the rate period and 1593.93 m³ / s in the validation set, which are 40.0% and 30.2% lower than the corresponding periods of the NGBoost model, respectively, and the uncertainty quantification accuracy is greatly improved. At the same time, the RMSE of the model validation set is 3527.29 m³ / s, which is 10.2% lower than NGBoost's 3926.19 m³ / s, making the simulation of overall runoff fluctuations and extreme value characteristics more robust. This result fully demonstrates that the technical framework integrating teleconnection factor time-delay matching and diffusion model physical parameter design can accurately capture the coupling law between long-term climate trends and short-term hydrological driving forces of runoff, and reduce probabilistic forecast errors through native stochastic processes. Its overall performance is superior to traditional statistical models and has reliable engineering application value.

[0050] The method of the present invention is used to obtain, as follows Figure 2 The monthly runoff probability forecast results shown intuitively demonstrate the probabilistic forecasting performance advantages of the model of this invention: 1) The 90% confidence interval (light blue shaded area) output by the model almost completely covers the measured flow curve (black solid line) throughout the entire time period, echoing the quantification results of nearly 90% in the validation set PICP mentioned above, proving that the reliability of the probability interval fully meets the risk coverage requirements; 2) The predicted median (red dashed line) closely matches the fluctuation trend of the measured flow. Whether it is the low flow fluctuation during the dry season or the high flow peak during the flood season (such as the time series at about 100 locations), the median curve can accurately follow the changes in the measured sequence, corresponding to the high deterministic forecast accuracy of NSE≥0.85 mentioned above; at the same time, the interval performance in extreme scenarios is more practical. The confidence interval in the high flow segment is appropriately widened to cover the peak fluctuation, and the interval in the dry season is narrowed to improve the forecast resolution. This avoids the risk of underreporting extreme floods and does not reduce the decision reference value due to the interval being too wide, intuitively demonstrating the comprehensive performance of the model of this invention, which has accurate deterministic trends, stable probability intervals, and strong adaptability to extreme scenarios.

[0051] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made by those skilled in the art to the technical solutions of the present invention without departing from the principles and essence of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A method for predicting the probability of monthly runoff using a diffusion model that integrates teleconnection factors, characterized in that, Includes the following steps: S1. Collect hydrological and meteorological factor data, teleconnection factor data, and basic characteristic data of the watershed for research, and preprocess them to obtain training set and validation set; wherein, the hydrological and meteorological factor data includes monthly precipitation data and monthly runoff data; the teleconnection factor data includes atmospheric circulation factor and sea surface temperature factor; the basic characteristic data of the watershed includes watershed area data; S2. Set the boundary conditions of the model and construct the diffusion model that integrates teleconnection factors; S3. Based on the particle swarm optimization algorithm and the training set, the optimal parameter combination and target diffusion model are obtained through iterative updates. S4. Based on the target diffusion model and validation set, the monthly runoff probability distribution and the monthly runoff probability forecast interval at multiple confidence levels are obtained through Monte Carlo simulation output. S5. Based on multi-dimensional accuracy indicators, the NGBoost model is used as a control model to evaluate the model simulation accuracy. The multi-dimensional accuracy indicators include Nash efficiency coefficient (NSE), root mean square error (RMSE), and continuous sorting probability score (CRPS).

2. The method for predicting monthly runoff probability using a diffusion model that integrates teleconnection factors according to claim 1, characterized in that, The preprocessing described in step S1 includes: S11. Based on the time sliding window mechanism, time-delay matching is performed on the teleconnection factor data and monthly runoff data to obtain the teleconnection factor data and monthly runoff data under the same time-delay gradient, which are used as the original dataset. S12. Perform missing value imputation, outlier removal and standardization on the original dataset in sequence to obtain the target dataset; S13. Divide the target dataset into training and validation sets according to a set ratio.

3. The method for predicting monthly runoff probability using a diffusion model that integrates teleconnection factors according to claim 2, characterized in that, Step S11 is as follows: 1) Monthly runoff depth data are calculated based on monthly runoff data and watershed area data; 2) Establish the relationship between monthly runoff and monthly runoff depth, obtain the forecasted monthly runoff, and then obtain the forecasted monthly runoff depth; 3) In the forecast period scenario, select teleconnection factor data and monthly runoff data corresponding to the forecast month in advance; 4) Using the predicted monthly runoff depth as the sliding window reference point, set the first time-delay gradient and the second time-delay gradient based on the selected teleconnection factor data and monthly runoff data, respectively. Slide the teleconnection factor data forward according to the first time-delay gradient and the monthly runoff data forward according to the second time-delay gradient to obtain the teleconnection factor data and monthly runoff data under the same time-delay gradient, which are used as the original dataset.

4. The method for predicting monthly runoff probability using a diffusion model that integrates teleconnection factors according to claim 2, characterized in that, Step S12 is as follows: 1) By using linear interpolation, a small amount of missing data is filled in to obtain the first dataset; 2) By using the multiple standard deviation method, outliers are removed to obtain the second dataset; 3) By standardizing the data, all second datasets are converted to the same size to obtain the third dataset, which is the target dataset.

5. The method for predicting monthly runoff probability using a diffusion model incorporating teleconnection factors according to claim 1, characterized in that, Step S2 specifically involves: S21. Define the monthly runoff depth as a one-dimensional diffusion process, and establish the Itō stochastic differential equation representing the runoff diffusion process; the expression is as follows: In the formula, for The differential increment of monthly runoff depth reflects The minute change in monthly runoff depth relative to the previous moment corresponds to the continuous evolution of runoff. This aligns with the core logic of Itoh's diffusion process, which describes the dynamic changes of state variables. for The moon's path flows deep at every moment; The drift coefficient reflects the deterministic trend of runoff change. The diffusion coefficient quantifies the intensity of runoff fluctuations caused by random factors. For the increment of the Wiener process, satisfying E[ ]=0 and Var[ ]= It is used to characterize random disturbances in runoff evolution; S22. The stochastic differential equation is transformed into an evolution equation of the probability density function using the Fock-Planck equation, which includes drift and diffusion terms; the expression is as follows: In the formula, for time During runoff, time The conditional probability density of runoff has a solution that directly corresponds to the predicted monthly runoff probability. The predicted monthly runoff probability distribution for any interval can be obtained by integration. S23. Based on hydrophysical constraints, set the model boundary conditions: 1) When the monthly runoff depth approaches the low water limit, that is, when there is no effective runoff generation in the watershed, both the drift coefficient and the diffusion coefficient approach zero. 2) When the monthly runoff depth approaches the flood limit, the drift coefficient is related to the watershed storage coefficient, and the diffusion coefficient tends to a stable constant; among them, the watershed storage coefficient is used to reflect the inhibitory effect of the river channel's flood discharge capacity on runoff growth; S24. Parametric fusion of teleconnection factors: Transforming abstract teleconnection factors into parameter-driven terms that the model can recognize.

6. The method for predicting monthly runoff probability using a diffusion model incorporating teleconnection factors according to claim 1, characterized in that, Step S24 specifically involves: 1) Calculation of the driving force of the teleconnection comprehensive factor: Based on principal component analysis, the multi-dimensional teleconnection factor data is reduced and fused to obtain the teleconnection comprehensive influence factor; based on the principal component coefficients and standardized values ​​of each teleconnection influence factor, the weighted sum of multiple teleconnection comprehensive influence factors is obtained to obtain the driving force of the teleconnection comprehensive factor; the calculation formula is as follows: In the formula, This is a driving factor of the teleconnection comprehensive factor; For the first Principal component coefficients of the teleconnection factor; For the first Standardized values ​​of teleconnection factors; This represents the number of teleconnection factor terms; 2) Parametric design of drift coefficient: Based on the driving factors of teleconnection, monthly precipitation, and runoff autoregressive characteristics, the drift coefficient is calculated; the formula is as follows: In the formula, This is the drift coefficient; This is a driving factor of the teleconnection comprehensive factor; This refers to the rainfall in the previous month; The depth of the lunar runoff; This represents the influence coefficient of the teleconnection factor, corresponding to the driving force of the comprehensive teleconnection factor in the drift coefficient. The weights reflect the intensity of the teleconnection factors' regulation of the deterministic trend of runoff; This is the precipitation influence coefficient, corresponding to the precipitation in the previous month in the drift coefficient. The weight of the precipitation reflects the intensity of the replenishment effect of short-term precipitation on the monthly runoff; This is the runoff autoregressive coefficient, corresponding to the monthly runoff depth in the drift coefficient. The weight of the watershed reflects the attenuation effect of watershed regulation on runoff; 3) Parametric design of diffusion coefficient: Based on the coefficient of variation of the driving factors of teleconnection, the coefficient of variation of monthly precipitation, and the characteristics of runoff itself, the diffusion coefficient is calculated; the formula is as follows: In the formula, The diffusion coefficient is a quantification of the intensity of runoff fluctuations caused by random factors. The coefficient of variation of the teleconnection factor is the driving force of the comprehensive teleconnection factor. The ratio of the standard deviation to the mean; The coefficient of variation is the monthly precipitation data, which is the monthly precipitation series. The ratio of the standard deviation to the mean; The depth of the lunar runoff; This represents the influence coefficient of teleconnection factor fluctuations, corresponding to the coefficient of variation of teleconnection factor in the diffusion coefficient. The weights reflect the impact of the stability of teleconnection factors on random fluctuations in runoff. This is the influence coefficient of precipitation fluctuation, corresponding to the precipitation variation coefficient in the diffusion coefficient. The weight of the metric reflects the regulatory effect of short-term precipitation fluctuations on random changes in runoff; This is the runoff fluctuation coefficient, corresponding to the square of the monthly runoff depth in the diffusion coefficient. The weights reflect the impact of the runoff's own magnitude on random fluctuations; The constant term coefficient corresponds to the basic fluctuation intensity of the diffusion coefficient, quantifying runoff fluctuations caused by inherent random factors.

7. The method for predicting monthly runoff probability using a diffusion model incorporating teleconnection factors according to claim 6, characterized in that, Step S3 specifically involves: S31. Constructing the fitness function: With the objectives of minimizing the Continuous Ranking Probability Score (CRPS) and ensuring the Predicted Interval Coverage Rate (PICP) approaches a pre-set confidence level, a weighted comprehensive fitness function is constructed; the expression is as follows: in, In the formula, The weighted comprehensive fitness function is used; CRPS is the continuous ranking probability score, which is used to quantify the overall deviation between the monthly runoff probability distribution and the measured value. Forecast coverage rate; The cumulative monthly runoff distribution function output by the diffusion model is obtained through... The result is obtained by integration; For actual measured monthly runoff; As an indicator function, when the predicted monthly runoff depth Measured monthly runoff depth hour =1, otherwise =0; The total number of samples; For the first Real-time measurement of monthly runoff; For the first The lower limit of the confidence interval for time forecast (corresponding to the 5th percentile); For the first The upper limit of the confidence interval for time forecast (corresponding to the 95th percentile); This is an indicator function, used when the measured monthly runoff falls within the corresponding forecast interval. ,otherwise ; S33. Update and iterate based on the POS algorithm: 2) Algorithm parameter initialization: particle population size, maximum number of iterations, inertia weight, learning factor; initial position of particles, initial velocity of particles; where each particle corresponds to a set of 7-dimensional parameter combinations. The dimension of each particle is consistent with the number of parameters to be calibrated; the inertial weight adopts an adaptive strategy, i.e., in the initial stage of iteration. =0.9 enhances global search and avoids getting trapped in local optima; in the later stages of iteration =0.4 enhances local convergence and improves parameter accuracy; the learning factor is set with... and and = =2, used to balance the influence of individual experience and group experience; the initial position and initial velocity of the particles are randomly generated; 3) For each initialized particle, substitute it into the diffusion model to calculate the fitness value: 4) Update the individual particle optimal pbest and the group optimal gbest; 5) Update particle velocity and position; 6) Update and iterate, and determine whether the number of iterations has reached the preset maximum number of iterations: if it has, stop the iteration and proceed to the next step; if it has not, return to step 3) and repeat the operation until the maximum number of iterations is reached; 7) After the iteration terminates, the 7-dimensional parameter combination corresponding to the global optimum gbest of the population is used as the optimal calibration parameters of the model. The specific process for calculating the fitness value of each particle is as follows: ① Combine the parameters in the 7-dimensional parameter combination and parameters Substitute these values ​​into the drift coefficient formula and the diffusion coefficient formula respectively; ②The runoff probability density function is obtained by solving the Fock-Planck equation, which includes drift and diffusion terms; ③ Based on the runoff probability density function, the CRPS and PICP indices are calculated; ④ The fitness value is calculated based on the weighted comprehensive fitness function; S34. After determining the optimal parameter combination, the corresponding target drift coefficient and target diffusion coefficient are obtained; S35, the target drift coefficient and the target diffusion coefficient, are used to train the target diffusion model through the training set.

8. The method for predicting monthly runoff probability using a diffusion model that integrates teleconnection factors according to claim 1, characterized in that, In step S4, Monte Carlo simulation is used to generate the monthly runoff probability distribution, specifically as follows: 1) Spatiotemporal matching of parameters: Based on the validation set, obtain the driving forces of the teleconnection comprehensive factors for each forecast month. and the precipitation in the month preceding the forecast month Calculate the drift coefficient and diffusion coefficient at the corresponding time. This ensures that the parameters are dynamically updated over time. 2) Random sample generation: Multiple sets of independent monthly runoff evolution samples are generated by solving stochastic differential equations using the Euler-Markov method; 3) Probability result extraction: Sort multiple groups of samples and calculate the 5%, 25%, 50%, 75%, and 95% quantiles; among them, the 50% quantile is the deterministic forecast value; the 5%-95% quantiles constitute the monthly runoff probability forecast interval at a 90% confidence level; at the same time, the monthly runoff probability distribution, i.e., the monthly runoff probability density curve, is obtained through kernel density estimation, realizing full-dimensional output of point forecast, interval forecast, and distribution forecast.

9. The method for predicting monthly runoff probability using a diffusion model incorporating teleconnection factors according to claim 1, characterized in that, The Nash efficiency coefficient (NSE) is used to evaluate the overall accuracy of deterministic monthly runoff forecasts, and it primarily reflects the degree of agreement between simulated and measured runoff. The calculation formula is as follows: In the formula, NSE is the point forecast accuracy; For the first Real-time measurement of monthly runoff; For the first Real-time forecast of monthly runoff; This represents the mean of the measured monthly runoff series; denoted as the total number of samples; where NSE ranges from (-∞, 1], and the model performance is rated as excellent when NSE ≥ 0.75; The root mean square error (RMSE) is used to quantify the average deviation between simulated and measured monthly runoff, and its unit is the same as that of runoff volume; the calculation formula is as follows: In the formula, This is the root mean square error; For the first Real-time measurement of monthly runoff; For the first Real-time forecast of monthly runoff; The total number of samples; where, The smaller the value, the smaller the deviation between the simulated and measured values, and the higher the simulation accuracy of the overall runoff fluctuation; conversely, the larger the value, the greater the deviation. The Continuous Ranking Probability Score (CRPS) is used to evaluate the accuracy of monthly runoff probability forecasts. It measures the overall deviation between the runoff probability distribution output by the core quantification model and the measured values, with units consistent with runoff volume. The calculation formula is as follows: In the formula, For continuous sorting probability scoring; The cumulative monthly runoff distribution function output by the diffusion model is obtained through... The result is obtained by integration; For actual measured monthly runoff; As an indicator function, when the predicted monthly runoff depth Measured monthly runoff depth hour =1, otherwise =0; where, The smaller the value, the higher the fit between the probability distribution and the measured value, and the more accurate the uncertainty quantification.

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