Hydrological prediction model of hybrid deep neural network

Abstract

The invention discloses a hydrological prediction model of a hybrid deep neural network, which is characterized in that physical equations of different process models are combined, and on the basis of describing a runoff forming process as accurately as possible, the process models and a deep learning network architecture are coupled; and meanwhile, an embedded neural network is used for extracting undefined hydrological knowledge at present. According to the method, the hydrological model based on the process is coupled with the deep learning network model, and the embedded neural network is used for replacing part of the water circulation process which cannot be reasonably explained by a physical formula at present in a differential programming framework, so that the model can extract reasonable hydrological knowledge; the method has process understanding and hydrological consciousness similar to a process-based model, so that the performance of the process-based model is improved.

Description

Technical Field

[0001] The present invention relates to the technical field of hydrological prediction, and in particular to a hydrological prediction model using a hybrid deep neural network. Background Art

[0002] The rainfall-runoff model is widely used in academia and industry to conceptualize the extremely complex natural process of runoff formation. A parameterized scheme for the mutual transfer of water between natural systems in a watershed is established to simulate and predict runoff, providing a scientific basis for agricultural production and early warning of hydrological disasters. The process-based hydrological model is based on established hydrological knowledge, but some empirical formulas involved in the runoff formation process cannot accurately represent the water cycle process due to the lack of comprehensive physics, which further causes some errors in the calculation of runoff, such as the calculation of water evapotranspiration and the allocation of the contribution ratio of soil water at different depths to outlet runoff. The deep learning model that has emerged in recent years has shown superior performance in watershed runoff prediction compared with traditional process-based hydrological models. Although the deep learning hydrological model has excellent performance, compared with the process-based model, the output of the deep learning model cannot be clearly explained and the internal mechanism is not easy to understand. At the same time, the input of the deep learning model is only meteorological data, without considering any prior hydrological knowledge. Therefore, the deep learning model can neither meet nor benefit from basic hydrological knowledge such as water balance. Therefore, people hope to develop a new hybrid model that can maintain high prediction performance similar to deep learning models and achieve physical interpretability similar to process-based modeling. At the same time, the deep learning module extracts implicit hydrological knowledge that is not yet clear by learning a large amount of hydrological data, further improving the accuracy of the process model module's description of the hydrological process.

[0003] The hybrid hydrological model combines the advantages of both process-based and deep learning models, aiming to address the limitations of each model by leveraging their respective strengths. The process-based model provides hydrological knowledge as a water balance constraint, while the deep learning model ensures the predictive performance of the model.

[0004] At present, there are the following types of hybrid hydrological models: (1) Deep learning as a postprocessor: Deep learning models are used as postprocessors of process-based hydrological models to improve their performance by learning simulation errors. Using deep learning as a postprocessor to correct errors in process-based models can improve model performance. (2) Physically informed neural networks: Physically informed neural networks use process-based models to generate synthetic data for training deep learning models, or introduce certain physical constraints in the loss function. This operation can implicitly provide prior hydrological knowledge and reduce the need for large amounts of data. However, the above two hybrid models are still deep learning models in essence, without explicit interpretability and hydrological understanding, and cannot strictly comply with physical constraints. (3) Deep learning for parameter calibration: Deep learning neural networks are used to parameterize the modules of process-based hydrological models to improve the performance of process-based models in unmetered basin areas. Although this type of model retains physical interpretability, due to the errors in its own description of some water cycle links, the model errors will be amplified and the ideal prediction effect cannot be obtained. (4) Deep learning based on process models: The development of differentiable models makes it possible to achieve deep bidirectional integration between process-based models and neural networks. It can connect different prior knowledge to neural networks and train them together. This integration takes advantage of the strengths of both models, resulting in more accurate hydrological relationships and clear process understanding. However, in regional modeling, this strategy makes overall adjustments to the physical parameters of the process model. However, the physical parameters are affected by the properties of the watershed, for example, the maximum water holding capacity is jointly affected by the soil and vegetation conditions of the watershed. When making predictions for unmetered watersheds, the prediction performance of watersheds with different attribute conditions may vary greatly. Using the same physical parameters for different watersheds is also inconsistent with the design principles of the process model.

[0005] In response to the above problems, the present invention provides a hydrological prediction model using a hybrid deep neural network to extract hydrological knowledge from different river basins. Summary of the invention

[0006] The present invention provides a hydrological prediction model of a hybrid deep neural network for extracting hydrological knowledge of different river basins.

[0007] The purpose of the present invention is to provide a hydrological prediction model of a hybrid deep neural network, which uses an embedded neural network to calculate the evapotranspiration in the current evapotranspiration module and the outlet runoff ratio in the runoff response module, and couples the recurrent neural network so that the hybrid hydrological model can also perform more accurate runoff prediction for watersheds with large attribute differences when performing regional modeling.

[0008] Furthermore, the hybrid hydrological model includes a precipitation partitioning module, a snowmelt and snow accumulation module, a soil moisture and effective precipitation module, an evapotranspiration module, and a runoff response module, and complies with the water conservation law, and the relationship is as follows:

[0009]

[0010] Among them, S0 represents the amount of snow, in millimeters; t represents the time, in days; Indicates the daily snow cover status; P snow Indicates snowfall, mm / day; S melt represents snowmelt, mm / day; S1 represents soil moisture, mm; Indicates daily soil moisture status; P eff represents effective precipitation, mm / day; ET represents evapotranspiration, mm / day; Qb represents base flow, mm / day; Qs represents surface runoff, mm / day. snow Obtained from the precipitation partition module, S melt It is obtained from the snowmelt module, ET is obtained from the evapotranspiration module, Qb and Qs are obtained from the runoff response module, and P eff Obtained from the effective precipitation module.

[0011] Furthermore, the precipitation classification module divides precipitation into rain and snow according to temperature. When the temperature is lower than the set threshold, the precipitation exists in the form of snow; when the temperature is higher than the threshold, the precipitation exists in the form of rain. The snowmelt and snow accumulation module is used to simulate the melting process of snow, calculate the amount of snowmelt, and update the change of snow storage. The soil moisture and effective precipitation module is used to calculate the distribution process of precipitation after entering the soil, including infiltration, water storage and surface runoff. This module is responsible for determining the effective precipitation (the amount of water that can participate in the formation of runoff) and simulating soil moisture changes. The evapotranspiration module is used to estimate the actual evapotranspiration, using the potential evapotranspiration calculation method based on meteorological data, and considering the limiting effect of soil moisture on the actual evapotranspiration. The runoff response module is used to simulate the confluence process of surface runoff and base flow, and convert the effective water volume of each source into runoff at the outlet of the basin. The whole prediction process is: the precipitation classification module divides precipitation into rain and snow, in which snowfall is transferred to the snowmelt and snow accumulation module, and rainfall is directly transferred to the soil moisture module. The amount of snowmelt water calculated by the snowmelt and snow accumulation module is passed as input to the soil moisture and effective precipitation module to update the soil moisture state. The output of the soil moisture and effective precipitation module, including effective precipitation and soil moisture state, is passed to the evapotranspiration module to calculate the actual evapotranspiration, and is also passed to the runoff response module to generate runoff. The actual evapotranspiration calculated by the evapotranspiration module is used as feedback to reduce the soil water storage in the soil moisture module. The output of the runoff response module is the total runoff at the outlet of the basin, providing the final result of the basin hydrological response.

[0012] Furthermore, the precipitation partition module is expressed as:

[0013]

[0014] Among them, P snow Indicates snowfall, mm / day; P rain represents rainfall, mm / day; prcp represents precipitation, mm / day; tmean represents temperature, ℃; Tmin is the temperature threshold. When the average temperature is lower than the threshold, the precipitation prcp is in the form of snowfall, otherwise it is in the form of rain.

[0015] Furthermore, the snowmelt and snow accumulation module is expressed as:

[0016]

[0017] Among them, S0 represents the amount of snow, millimeters; Df is the degree-day factor, mm / day / ℃; tmean represents the air temperature, ℃; Tmax is the snowmelt temperature threshold, ℃; when tmean exceeds Tmax and the current S0 is greater than zero, the snow begins to melt. In other cases, snowmelt does not occur and Smelt is zero.

[0018] Furthermore, the soil moisture and effective precipitation modules are expressed as follows:

[0019]

[0020] Among them, P eff represents effective precipitation, mm / day; S1 represents soil moisture, mm; FC represents field capacity, The ratio of represents soil moisture deficit; Prain represents rainfall, mm / day; β is the model parameter; S melt Indicates snowmelt, mm / day.

[0021] Furthermore, the runoff response module is expressed as follows:

[0022]

[0023] Q0=Qs+Qb

[0024] Among them, Qb is base flow, mm / day; Qs is surface runoff, mm / day; Q0 ​​is the total runoff of the basin, which is the sum of base flow Qb and surface runoff Qs, mm / day; Qmax is the maximum underground flow, mm / day; f is the attenuation factor of the runoff attenuation rate; Smax is the maximum soil moisture content, mm; S1 is soil moisture, mm. When the soil moisture S1 is greater than Smax, it means that the soil is fully saturated, and the base flow is the maximum underground flow.

[0025] Furthermore, the recurrent neural network uses the Nash efficiency coefficient as the Loss function, and the formula is expressed as:

[0026]

[0027] Where NSE represents the Nash efficiency coefficient, represents the observed runoff on day t, represents the simulated runoff on day t, Represents the average value of observed runoff over n days.

[0028] Furthermore, the embedded neural network is used to replace the calculation of evapotranspiration. The traditional calculation of potential evapotranspiration is as follows:

[0029]

[0030] Among them, L day Indicates the length of the day (hours), e sat (T) represents saturated vapor pressure (kPa), T represents temperature (Fahrenheit), and potential evapotranspiration PET (mm / day) is affected by day length and temperature, so the embedded neural network is based on T and L day As input, PET as output. On this basis, the evapotranspiration in the evapotranspiration module is calculated as follows:

[0031]

[0032] Where ET stands for evapotranspiration, e sat represents saturated vapor pressure, PET represents potential evapotranspiration. When soil moisture S1 is less than zero, evapotranspiration is also zero. When S1 exceeds the maximum soil moisture content Smax, ET is equal to PET.

[0033] The present invention has the following advantages: The present invention replaces some water cycle processes that cannot be reasonably explained by physical formulas in the differential programming framework by using embedded neural networks. This enables the model to extract reasonable hydrological knowledge and possess process understanding and hydrological awareness similar to process-based models to improve the predictive performance of the model. The neural network is used to map the watershed attributes to the parameters of the process model, and a set of appropriate physical parameters is generated for each watershed when performing regional modeling.

[0034] The present invention has good runoff prediction capabilities. Thanks to the fact that the neural differential equations in the model structure contain physically meaningful elements, it can also infer unobserved hydrological variables. This not only provides a guarantee for the physical consistency of the model prediction results, but also broadens the application scenarios of the model. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 It is a schematic diagram of the structure of the invention. DETAILED DESCRIPTION

[0036] The present invention provides a hydrological prediction model of a hybrid deep neural network. Based on the hybrid hydrological model, the hybrid hydrological model calculates the evapotranspiration in the current evapotranspiration module and the outlet runoff ratio in the runoff response module through an embedded neural network, and couples the recurrent neural network so that the hybrid hydrological model can perform relatively accurate runoff prediction for watersheds with large attribute differences when performing regional modeling.

[0037] In this embodiment, the hybrid hydrological model includes a precipitation partition module, a snowmelt and snow accumulation module, a soil moisture and effective precipitation module, an evapotranspiration module, and a runoff response module, and complies with the water conservation law, and the relationship is as follows:

[0038]

[0039] Among them, S0 represents the amount of snow, in millimeters; t represents the time, in days; Indicates the daily snow cover status; P snow Indicates snowfall, mm / day; S melt represents snowmelt, mm / day; S1 represents soil moisture, mm; Indicates daily soil moisture status; P eff represents effective precipitation, mm / day; ET represents evapotranspiration, mm / day; Qb represents base flow, mm / day; Qs represents surface runoff, mm / day.

[0040] In this embodiment, the precipitation classification module divides precipitation into rain and snow according to temperature. When the temperature is lower than the set threshold, the precipitation exists in the form of snow; when the temperature is higher than the threshold, the precipitation exists in the form of rain; the snowmelt and snow accumulation module is used to simulate the melting process of snow, calculate the amount of snowmelt, and update the change of snow storage; the soil moisture and effective precipitation module is used to calculate the distribution process of precipitation after entering the soil, including infiltration, water storage and surface runoff generation. This module is responsible for determining the effective precipitation (the amount of water that can participate in the formation of runoff) and simulating soil moisture changes; the evapotranspiration module is used to estimate the actual evapotranspiration, using the potential evapotranspiration calculation method based on meteorological data, and considering the limiting effect of soil moisture on the actual evapotranspiration; the runoff response module is used to simulate the confluence process of surface runoff and base flow, and convert the effective water volume of each source into runoff at the outlet of the basin. The whole prediction process is: the precipitation classification module divides precipitation into rain and snow, in which snowfall is transferred to the snowmelt and snow accumulation module, and rainfall is directly transferred to the soil moisture module. The amount of snowmelt water calculated by the snowmelt and snow accumulation module is passed as input to the soil moisture and effective precipitation module to update the soil moisture state. The output of the soil moisture and effective precipitation module, including effective precipitation and soil moisture state, is passed to the evapotranspiration module to calculate the actual evapotranspiration, and is also passed to the runoff response module to generate runoff. The actual evapotranspiration calculated by the evapotranspiration module is used as feedback to reduce the soil water storage in the soil moisture module. The output of the runoff response module is the total runoff at the outlet of the basin, providing the final result of the basin hydrological response.

[0041] In this embodiment, the precipitation division module is expressed as:

[0042]

[0043] Among them, P snow Indicates snowfall, mm / day; P rain represents rainfall, mm / day; prcp represents precipitation, mm / day; tmean represents temperature, ℃; Tmin is the temperature threshold. When the average temperature is lower than the threshold, the precipitation prcp is in the form of snowfall, otherwise it is in the form of rain.

[0044] In this embodiment, the snow melting and snow accumulation module is expressed as:

[0045]

[0046] Among them, S0 represents the amount of snow, millimeters; Df is the degree-day factor, mm / day / ℃; tmean represents the air temperature, ℃; Tmax is the snowmelt temperature threshold, ℃; when tmean exceeds Tmax and the current S0 is greater than zero, the snow begins to melt. In other cases, snowmelt does not occur and Smelt is zero.

[0047] In this embodiment, the soil moisture and effective precipitation modules are expressed as follows:

[0048]

[0049] Among them, P eff represents effective precipitation, mm / day; S1 represents soil moisture, mm; FC represents field capacity, The ratio of represents soil moisture deficit; Prain represents rainfall, mm / day; β is the model parameter; S melt Indicates snowmelt, mm / day.

[0050] In this example, the runoff at the outlet of the basin is estimated based on two conceptual reservoirs. The first reservoir is used to simulate near-surface flow, and the second reservoir is used to simulate base flow. The two reservoirs represent fast and slow underground processes, respectively. The water transfer between the two reservoirs is quantified by using a constant seepage rate. The runoff response module is expressed as follows:

[0051]

[0052] Q0=Qs+Qb

[0053] Among them, Qb is base flow, mm / day; Qs is surface runoff, mm / day; Q0 ​​is the total runoff of the basin, which is the sum of base flow Qb and surface runoff Qs, mm / day; Qmax is the maximum underground flow, mm / day; f is the attenuation factor of the runoff attenuation rate; Smax is the maximum soil moisture content, mm; S1 is soil moisture, mm. When the soil moisture S1 is greater than Smax, it means that the soil is fully saturated, and the base flow is the maximum underground flow.

[0054] In this embodiment, the recurrent neural network uses the Nash efficiency coefficient as the Loss function, and the formula is expressed as follows:

[0055]

[0056] Where NSE represents the Nash efficiency coefficient, represents the observed runoff on day t, represents the simulated runoff on day t, Represents the average value of observed runoff over n days.

[0057] In this embodiment, the embedded neural network is used to replace the calculation of evapotranspiration. The traditional calculation of potential evapotranspiration is as follows:

[0058]

[0059] Among them, L day Indicates the length of the day (hours), e sat(T) represents saturated vapor pressure (kPa), T represents temperature (Fahrenheit), and potential evapotranspiration PET (mm / day) is affected by day length and temperature, so the embedded neural network is based on T and L day As input, PET as output. On this basis, the evapotranspiration in the evapotranspiration module is calculated as follows:

[0060]

[0061] Where ET stands for evapotranspiration, e sat represents saturated vapor pressure, PET represents potential evapotranspiration. When soil moisture S1 is less than zero, evapotranspiration is also zero. When S1 exceeds the maximum soil moisture content Smax, ET is equal to PET.

[0062] In this embodiment, the EXP-HYDRO-HBV process model and the recurrent neural network are used as the backbone, and each module in the process can be flexibly replaced, that is, by using a neural network instead of the physical formula calculation, the part with relatively large error in the optimization process model is optimized. Specifically, the calculation of potential evapotranspiration in the model and the calculation of the final outlet runoff in the runoff response unit are replaced by a neural network respectively. The embedded neural network that replaces the potential evapotranspiration calculation takes temperature and day length as input and potential evapotranspiration as the target variable, and the embedded neural network in the runoff response unit takes surface runoff and base flow as input and outlet runoff (total runoff) as the target variable. The two embedded neural networks do not perform loss calculation separately, and their network parameters are gradually adjusted in the recurrent neural network as the loss function is calculated.

[0063] In this embodiment, the basic principle of the hydrological model is water balance, which is a specific form of the law of conservation of mass. The general form of the water balance equation is where S is the water state of a watershed response unit (either a specific part of a watershed or the entire watershed), I(t) is the amount of water flowing into the watershed response unit at time t, and O(t) is the amount of water flowing out of the watershed response unit at time t.

[0064] In this embodiment, the daily runoff of the basin is simulated, and the structure is as follows: Figure 1As shown. First, the lumped HBV model is coupled with the EXP-HYDRO model, that is, the HBV model is combined with different modules of the EXP-HYDRO model to better generalize the runoff formation process. The different effects of different phases of precipitation (snow, sleet and rainfall) on runoff, as well as the differences in the runoff formation process under different soil conditions, are considered. Specifically, the precipitation partitioning module of EXP-HYDRO is used to partition the rain and snow phases at different temperatures. Furthermore, the snowmelt and snow accumulation module of EXP-HYDRO is used to evaluate the accumulation and melting of snowfall at different temperatures, providing a basis for further evaluating the contribution of snowmelt and rainfall to runoff. Thereafter, the soil moisture and effective precipitation modules of HBV are used to describe the maximum soil moisture content under the surface through field capacity, and snowmelt and rainfall (here the snowmelt comes from the snowmelt and snow accumulation module of EXP-HYDRO, and the rainfall comes from the precipitation partitioning module of EXP-HYDRO) are divided into two parts: one part contributes to infiltration into the soil zone; the other part contributes to surface runoff, that is, effective precipitation. At the same time, the evapotranspiration module of EXP-HYDRO is coupled to evaluate evapotranspiration under different temperatures and radiation based on soil moisture content. Finally, the runoff response module of HBV is optimized and improved to estimate the runoff at the outlet of the basin based on two conceptual reservoirs (the two conceptual reservoirs are similar to the concepts of water collection buckets and snow collection buckets in EXP-HYDRO). The first reservoir is used to simulate near-surface flow, and the second reservoir is used to simulate base flow. The two reservoirs represent fast and slow underground processes, respectively, and the water transfer between the two reservoirs is quantified by using a constant seepage rate. The calculation of evapotranspiration and the proportion of outlet runoff are replaced by embedded neural networks.

[0065] In this embodiment, the coupled recurrent neural network sets the parameters of the process model within a certain range. The specific parameters and ranges of the process model are shown in Table 1. The parameters are adjusted after each learning of the training set data. The parameters here include both the parameters of the process model and the weight parameters of the neural network. Specifically, the daily meteorological data time series and runoff observation time series data are standardized, and then the appropriate time step and window size are set for sliding learning. The purpose of this is to fully learn the training data, force the model to understand and learn the runoff series in a process-based manner during the training process, and maintain the water balance state of the basin.

[0066] Table 1 Specific parameters of the process model

[0067]

[0068]

[0069] In this embodiment, due to the addition of a parameterized channel, the parameterized neural network maps the watershed attributes to the parameters required by the process model, and the weight parameters of the parameterized neural network are continuously adjusted as the model is trained. When performing regional modeling, the attribute data of the watersheds in the region may have heterogeneity and similarity. When the model learns these attribute data, it can form a parameterized neural network suitable for the region. This neural network can map the most appropriate process model parameters for each watershed in the region based on its attribute characteristics. This enables the hybrid deep neural network hydrological model to perform relatively accurate runoff predictions for watersheds with large attribute differences when performing regional modeling.

[0070] In this embodiment, the model input needs to include two parts: meteorological forcing data and watershed attribute data. The specific information of meteorological forcing variables is shown in Table 2. Attribute data generally includes data on climate, topography, vegetation, soil and geological characteristics, and can be selected according to the characteristics of the watershed in the modeling area.

[0071] Table 2 Detailed information of meteorological forcing variables

[0072]

[0073] Although the specific implementation of the present invention is described in detail in conjunction with the drawings, it should not be understood as limiting the scope of protection of the present invention. Within the scope described in the claims, various modifications and variations that can be made by those skilled in the art without creative work still fall within the scope of protection of this patent.

Claims

1. A hydrological prediction model based on a hybrid deep neural network, characterized by: The hydrological prediction model is constructed based on a hybrid hydrological model, which divides the process of basin runoff formation into different modules for separate representation, including a precipitation division module, a snowmelt and snow accumulation module, a soil moisture and effective precipitation module, an evapotranspiration module, and a runoff response module. The hybrid hydrological model calculates the ratio of evapotranspiration in the evapotranspiration module to the outlet runoff in the runoff response module through an embedded neural network, and couples a recurrent neural network to enable the hybrid hydrological model to accurately predict runoff for basins with large attribute differences when performing regional modeling.

2. The hydrological prediction model of the hybrid deep neural network according to claim 1, characterized in that: The process of runoff formation as a whole complies with the physical law of water conservation, and the variables between different modules conform to the following relationship: Where S0 represents the amount of snow, in millimeters; t represents the time, in days; Indicates the daily snow cover status; P snow Indicates snowfall, mm / day; S melt represents snowmelt, mm / day; S1 represents soil moisture, mm; Indicates daily soil moisture status; P eff represents effective precipitation, mm / day; ET represents evapotranspiration, mm / day; Qb represents base flow, mm / day; Qs represents surface runoff, mm / day.

3. The hydrological prediction model of the hybrid deep neural network according to claim 2, characterized in that: The precipitation partition module is expressed as: Where P snow Indicates snowfall, mm / day; P rain represents rainfall, mm / day; prcp represents precipitation, mm / day; tmean represents temperature, ℃; Tmin is the temperature threshold. When the average temperature is lower than the threshold, the precipitation prcp is in the form of snowfall, otherwise it is in the form of rain.

4. The hydrological prediction model of a hybrid deep neural network according to claim 2, characterized in that: The snow melting and snow accumulation module is expressed as: Among them, S0 represents the amount of snow, millimeters; Df is the degree-day factor, mm / day / ℃; tmean represents the air temperature, ℃; Tmax is the snowmelt temperature threshold, ℃; when tmean exceeds Tmax and the current S0 is greater than zero, the snow begins to melt. In other cases, snowmelt does not occur and Smelt is zero.

5. The hydrological prediction model of the hybrid deep neural network according to claim 2, characterized in that: The soil moisture and effective precipitation modules are expressed as follows: Among them, P eff represents effective precipitation, mm / day; S1 represents soil moisture, mm; FC represents field capacity, The ratio of represents soil moisture deficit; Prain represents rainfall, mm / day; β is the model parameter; S melt Indicates snowmelt, mm / day.

6. The hydrological prediction model of the hybrid deep neural network as claimed in claim 2, characterized in that: The runoff response module is expressed as follows: Q0=Qs+Qb Where Qb represents base flow, mm / day; Qs represents surface runoff, mm / day; Q0 ​​represents total runoff in the basin, mm / day; Qmax represents maximum underground flow, mm / day; f represents the attenuation factor of runoff attenuation rate; Smax represents maximum soil moisture content, mm; S1 represents soil moisture, mm.

7. The hydrological prediction model of a hybrid deep neural network according to claim 1, characterized in that: The Nash efficiency coefficient is used as the Loss function in the recurrent neural network, and the formula is expressed as follows: Where NSE represents the Nash efficiency coefficient, represents the observed runoff on day t, represents the simulated runoff on day t, Represents the average value of observed runoff over n days.

8. The hydrological prediction model of a hybrid deep neural network according to claim 1, characterized in that: The calculation of evapotranspiration in the evapotranspiration module is as follows: Where ET stands for evapotranspiration, e sat represents saturated vapor pressure, PET represents potential evapotranspiration. When soil moisture S1 is less than zero, evapotranspiration is also zero. When S1 exceeds the maximum soil moisture content Smax, ET is equal to PET.

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