2023 Higher Education Society Cup National Mathematical Modeling Competition (E Question) In-depth Analysis | Yellow River Water and Sand Detection

Question 1

Data sorting :
- Obtain data including water level, water flow, sediment content and time from Appendix 1. Ensure that the data is arranged in chronological order and perform data preprocessing, including handling missing values and outliers.
Data visualization (optional) :
- Data visualization tools such as Matplotlib can be used to graph sediment content, water level, and water flow over time to initially observe the relationship between them.
Build a mathematical model :
- Assuming that there is some relationship between sediment content and time, water level and water flow, a multiple linear regression model can be chosen. The model can be expressed as: sand content = β0 + β1 * time + β2 * water level + β3 * water flow + ε
- Here, β0, β1, β2 and β3 are the parameters of the model, and ε is the error term.
Model parameter estimates :
- Use the linear regression analysis method to fit the model based on the data and estimate the parameters β0, β1, β2, and β3 in the model.
- The estimated parameters will reflect the extent to which time, water level, and water flow affect sediment content.
Model evaluation :
- Use the model to predict the training data and obtain the predicted value.
- Calculate the goodness of fit of the model (usually using the R-squared value) to evaluate the quality of the model's fit. The closer the R-square value is to 1, the better the model fits.
Estimates of annual total water flow and annual total sediment discharge :
- Use the model to predict water level, water flow and time data in the past 6 years to estimate the sediment content.
- Calculate the total annual water flow and total annual sediment discharge in these 6 years, and sum up the estimated sediment content.
Results interpretation and analysis :
- Explain the parameters of the model, for example, parameters β1, β2 and β3 represent the degree of influence of time, water level and water flow on sand content.
- Analyze the estimation results of annual total water flow and annual total sediment discharge to understand the hydrological conditions of the Yellow River in the past six years, as well as the correlation between sediment content, water level, water flow, and time.

Question 2

The characteristics of mutation, seasonality and periodicity of water and sediment flux at this hydrological station in the past 6 years were analyzed, and the changing rules of water and sediment flux were studied.

1. Data sorting :

Collect water level, water flow and sediment content data in the past 6 years to ensure that the data is arranged in chronological order.

2. Data preprocessing :

Missing data and outliers may need to be handled.
Divide time series data over time, such as by month or season.

3. Test for mutation :

Use statistical methods (e.g., change point test) to test whether there is a mutation point in the water and sediment flux time series. Abrupt points may represent significant hydrological events such as floods or droughts.
A commonly used method is a mutation test based on mean or variance, such as CUSUM (cumulative sum) or Z test.

4. Seasonal analysis :

Use seasonal decomposition methods (such as STL decomposition) to decompose water and sediment flux time series to identify seasonal components (such as seasonal fluctuations).
The amplitude, phase and frequency of each seasonal component can be calculated.

5. Periodic analysis :

Frequency domain analysis methods such as Fourier transform or wavelet transform are used to detect whether there is a periodic component in the water and sediment flux time series.
The dominant periods and amplitudes can be estimated and the spectra of the periodic components plotted.

6. Build a mathematical model :

Time series analysis methods, such as ARIMA (Autoregressive Integrated Moving Average Model) or Prophet, can be used to establish a prediction model for water and sediment fluxes.
For example, an ARIMA model can be expressed as: $Y_t = \phi_1 Y_{t-1} + \theta_1 \varepsilon_{t-1} + \ldots + \phi_p Y_{tp} + \theta_p \varepsilon_{tp} + \varepsilon_t$
Among them, Yt is the time series observation value, εt is the white noise error, ϕ and θ are the model parameters, and p is the order.

7. Model evaluation :

Use the model to make predictions on the data, and then evaluate the model's goodness of fit, such as the root mean square error (RMSE) or R-squared value.
Techniques such as cross-validation can be used to verify the stability and accuracy of the model.

8. Interpretation and analysis of results :

Interpret the parameters and results of the model, e.g., detected mutation points, properties of seasonal and periodic components.
Analyze the changing patterns of water and sediment flux, paying special attention to factors that may cause sudden changes or seasonal changes, such as rainfall, snowmelt, etc.

9. Data visualization :

Use visual tools such as charts and spectra to present the results of mutation, seasonality and periodicity analysis, making it easier to understand and communicate to decision-makers.

Specifically, ARIMA (Autoregressive Integrated Moving Average Model) is a statistical model commonly used in time series analysis and forecasting to describe autocorrelation and seasonal components in time series data. In question 2, the ARIMA model can be used to analyze the changing patterns of water and sediment flux, especially to detect autocorrelation and seasonal components to reveal the trend and periodic characteristics of water and sediment flux.

Here are the steps and explanations for using the ARIMA model in question 2:

1. Data preparation :

First, data on water levels, water flow, and sediment content in the past six years were collected to ensure that the data were arranged in chronological order.

2. Differential operation :

ARIMA models usually require that the time series is stationary, that is, the mean and variance remain constant over time. Therefore, the water and sediment flux time series are first differentiated to eliminate the trend and seasonal components. Differential operations can be implemented by calculating the difference between the current value and the previous value.

3. Model fitting :

Choose an appropriate ARIMA model: An ARIMA model usually consists of three parts, namely the autoregressive (AR) part, the difference (I) part, and the moving average (MA) part. Choosing the appropriate model order (p, d, q) requires looking at the autocorrelation function (ACF) and partial autocorrelation function (PACF) plots.

4. Model fitting and parameter estimation :

Using the selected ARIMA model order, time series data are used to estimate the parameters of the model. These parameters include autoregressive coefficients (AR part), difference orders (I part), moving average coefficients (MA part), etc.

5. Model diagnosis :

Run a diagnostic on the model to check whether the residuals are white noise. This can be checked using autocorrelation function (ACF) and partial autocorrelation function (PACF) plots of the residuals.

6. Model prediction :

Use the trained ARIMA model to predict future water and sediment fluxes. This can be used to estimate the trend of water and sediment fluxes in the future.

7. Interpretation and analysis of results :

Explain the parameters in the ARIMA model, such as autoregressive coefficients, difference orders, moving average coefficients, etc., to understand their impact on water and sediment fluxes.
Analyze the prediction results of the ARIMA model to understand the future change trend and periodicity of water and sediment fluxes.
It should be noted that in practical applications, the ARIMA model may need to continuously adjust the order of the model to obtain the best fit.

Question three

According to the change pattern of water and sediment flux at the hydrological station, predict and analyze the changing trend of water and sediment flux at the hydrological station in the next two years, and formulate the optimal sampling monitoring plan for the hydrological station in the next two years (number of sampling monitoring and specific time etc.), so that it can not only grasp the dynamic changes of water and sediment flux in time, but also minimize monitoring cost resources.

1. Data preparation :

Collect historical time series data, ensuring the data includes timestamps and corresponding observations.
Check whether the data is stationary. If not, a differential operation is required to make it stationary.

2. Model selection :

Use graphical analysis of the autocorrelation function (ACF) and partial autocorrelation function (PACF) to select the order (p, d, q) of the ARIMA model.
The three parts of the ARIMA model represent the order of autoregression (AR), difference (I) and moving average (MA) respectively.
The AR part is used to capture the autocorrelation of the sequence, and the MA part is used to capture the autocorrelation of random errors.

3. Model fitting :

According to the selected ARIMA model order, fit the model to the historical data.
Use a statistical tool such as Python's statsmodels library to perform the fitting.

4. Model diagnosis :

Check the model's residual series to make sure it is white noise. It can be checked using methods such as Ljung-Box test.
If the residuals are not white noise, you may need to retune the model or try a different model.

5. Model prediction :

Use the fitted ARIMA model to predict values at future time points.
When predicting, you need to provide the number of predicted time points, and the model will return the predicted value and the corresponding confidence interval.

6. Interpretation of results :

Forecast results can be visualized, including time series plots of observed and predicted values, as well as confidence intervals.
Analyze forecast results to understand trends, seasonality, and cyclical characteristics of future time series.

7. Model evaluation:

Use the model to make predictions on the data, and then evaluate the model's goodness of fit, such as the root mean square error (RMSE) or R-squared value.
Techniques such as cross-validation can be used to verify the stability and accuracy of the model.
8. Interpretation and analysis of results:

Interpret the parameters and results of the model, e.g., detected mutation points, properties of seasonal and periodic components.
Analyze the changing patterns of water and sediment flux, paying special attention to factors that may cause sudden changes or seasonal changes, such as rainfall, snowmelt, etc.
9. Data visualization:

Use visual tools such as charts and spectra to present the results of mutation, seasonality and periodicity analysis, making it easier to understand and communicate to decision-makers.
Specifically, ARIMA (Autoregressive Integrated Moving Average Model) is a statistical model commonly used in time series analysis and forecasting to describe autocorrelation and seasonal components in time series data. In question 2, the ARIMA model can be used to analyze the changing patterns of water and sediment flux, especially to detect autocorrelation and seasonal components to reveal the trend and periodic characteristics of water and sediment flux.

Here are the steps and explanations for using the ARIMA model in question 2:

1. Data preparation:

First, data on water levels, water flow, and sediment content in the past six years were collected to ensure that the data were arranged in chronological order.
2. Differential operation:

ARIMA models generally require that the time series is stationary, that is, the mean and variance remain constant over time. Therefore, the water and sediment flux time series are first differentiated to eliminate the trend and seasonal components. Differential operations can be implemented by calculating the difference between the current value and the previous value.
3. Model fitting:

Choose an appropriate ARIMA model: An ARIMA model usually consists of three parts, namely the autoregressive (AR) part, the difference (I) part, and the moving average (MA) part. Choosing the appropriate model order (p, d, q) requires looking at the autocorrelation function (ACF) and partial autocorrelation function (PACF) plots.
4. Model fitting and parameter estimation:

Using the selected ARIMA model order, time series data are used to estimate the parameters of the model. These parameters include autoregressive coefficients (AR part), difference orders (I part), moving average coefficients (MA part), etc.
5. Model diagnosis:

Run a diagnostic on the model to check whether the residuals are white noise. This can be checked using autocorrelation function (ACF) and partial autocorrelation function (PACF) plots of the residuals.
6. Model prediction:

Use the trained ARIMA model to predict future water and sediment fluxes. This can be used to estimate the trend of water and sediment fluxes in the future.
7. Interpretation and analysis of results:

Explain the parameters in the ARIMA model, such as autoregressive coefficients, difference orders, moving average coefficients, etc., to understand their impact on water and sediment fluxes.
Analyze the prediction results of the ARIMA model to understand the future change trend and periodicity of water and sediment fluxes.
It should be noted that in practical applications, the ARIMA model may need to continuously adjust the order of the model to obtain the best fit.

Question 4

Based on the changes in water and sediment flux and river bottom elevation of the hydrological station, the actual effect of "water and sediment regulation" carried out in Xiaolangdi Reservoir from June to July every year is analyzed. If "water and sediment regulation" is not carried out, what will happen to the river bottom elevation of the hydrological station in 10 years?

1. Data collection and preparation :

Hydrological station data is collected from June to July every year, including inlet and outlet flow, sediment content and other information, as well as river bottom elevation data during the same time period.
Clean and preprocess data to ensure data quality and consistency.

2. Analysis of water and sediment regulation effects :

Analyze hydrological station data from June to July each year to calculate changes in sedimentation volume and river bottom elevation during this period.
Use statistical methods to calculate statistics such as the mean and standard deviation of sedimentation, as well as the change range of river bottom elevation.

3. Long-term simulation of river bottom elevation :

Use a mathematical model to simulate the changes in river bottom elevation within 10 years, including two situations: "water and sediment regulation" and no situation.
Choose a river sedimentation model, such as the Einstein-Brown equation of motion, to simulate the movement of suspended solids and the sedimentation process.
The estimation of model parameters can be based on historical data and relevant literature.
The model is implemented as a computer program and time-stepping simulations are performed using numerical methods.

4. Result analysis and comparison :

Analyze the simulation results and compare the river bottom elevation change trends within 10 years under the two conditions.
Evaluate the effect of "water and sediment regulation", including reduced sedimentation, maintained river bottom elevation, etc.
Use visualization tools to graph simulation results for easier interpretation and communication.

5. Conclusions and recommendations :

Based on the analysis results, suggestions on "water and sediment regulation" are put forward, including whether to continue the implementation and how to improve the operation methods.
Provide predictions of changes in river bottom elevation at hydrological stations over the next 10 years to support decision-making.
Consider model uncertainty and sensitivity analysis to understand how trustworthy the model results are.

Specific explanation:

Einstein-Brown equation of motion : This equation describes the movement and deposition of particles in fluids. It includes parameters such as particle deposition speed and diffusion speed. By estimating these parameters, particle movement and deposition in rivers can be simulated.
Numerical simulation method : Use the discrete time step method to convert the Einstein-Brown equation into a difference equation to simulate changes in time. These equations are usually solved using numerical calculation tools, such as the finite element method or the finite difference method.
Model parameter estimation : By estimating parameters such as particle size, density, and initial concentration based on historical data, as well as environmental parameters such as water flow velocity and bed friction coefficient.
Simulation result analysis : The simulation results will provide river bottom elevation data at each time step, and annual or quarterly statistical data, such as mean, standard deviation, and maximum values, can be calculated for comparison and analysis.

Modeling ideas :

Analysis on the effect of water and sediment regulation :
- First, we collect hydrological station data from June to July each year, including inlet and outlet flow ($Q_{in}$ and $Q_{out}$), sediment content ($C_{in}$ and $C_ {out}$) and other information.
- We calculate the sedimentation amount during this period ($Sediment_{in}-Sediment_{out}$), where $Sediment_{in}$ represents the product of the inlet water sediment content and the flow rate, and $Sediment_{out}$ represents the outlet water content. The product of sand volume and flow rate.
- We also calculate the change in river bottom elevation ($Elevation_{end}-Elevation_{start}$), where $Elevation_{end}$ represents the river bottom elevation at the end of July each year, and $Elevation_{start}$ represents the river bottom elevation at the beginning of June each year. bottom elevation.
Long-term simulation of river bottom elevation :
- We use a mathematical model to simulate the changes in river bottom elevation within 10 years, considering two situations: "water and sediment regulation" and no situation.
- During the simulation, we use a fluvial sedimentation model, which includes the following formula:
  - Sedimentation velocity formula: $V_{sediment}= k \cdot (C{in} - C_{\text{out}}) \cdot Q_{\text{in}} \cdot Q_{\text{out}}$ Among them, $V_{sediment}$ represents the deposition velocity, and $k$ is the deposition velocity coefficient.
  - The river bed elevation change formula: $Elevation_{t+1} = Elevation_{t} + V_{\text{sediment}} \cdot \Delta t$ . Among them, $Elevation_{t+1}$ represents the riverbed elevation of the next time step, and $\Delta t$ represents the time step.

Formula explanation :

The sedimentation rate formula describes the sedimentation rate of suspended particles in water, which is related to the difference in sand content of the incoming and outgoing water and the flow rate. The deposition velocity coefficient $k$ can be estimated based on actual conditions and literature.
The river bed elevation change formula represents the elevation change of the river bed in each time step, which depends on the deposition rate and time step $\Delta t$. Through continuous simulation, we can estimate the river bed elevation at the end of July each year.

Analysis process :

Using the above models and formulas, we analyze the hydrological station data from June to July each year, and calculate the sedimentation volume and river bottom elevation changes of the "water and sediment regulation" measures, as well as the simulation results in the next 10 years.
Compare the "water and sediment regulation" situation with the situation without this operation, and analyze its impact on the river bottom elevation, including the reduction of sedimentation and the maintenance of the river bed elevation.
Based on the analysis results, conclusions and suggestions are drawn about the actual effects of "water and sediment regulation" and the changes in river bottom elevation of hydrological stations in the next 10 years.
Model uncertainty and sensitivity analysis can further be considered to assess the credibility of the model results.

Code:

Modeling ideas :

We will use a one-dimensional sedimentation model to simulate changes in river bottom elevation. The model is based on the following assumptions:

The sedimentation rate depends on the difference in sediment content of the incoming and outgoing water and the flow rate.
The bed elevation is updated at each time step based on the sedimentation rate.

For the use of professional hydrological models such as HEC-RAS or MODFLOW to simulate complex water and sediment fluxes and river bottom elevation changes, corresponding model software and geographic information data are required. Here I will demonstrate how to use Python's pyhecraslibrary to interact with HEC-RAS and simulate changes in river bottom elevation.

Please note that a real HEC-RAS model usually requires input data and settings, including channel geometry, boundary conditions, hydrological data, etc.

Prerequisites :

Install pyhecrasthe library:pip install pyhecras

For the complete code + ideas, check out my column:

2023 Higher Education Society Cup National Mathematical Modeling Competition (E Question) In-depth Analysis | Yellow River Water and Sand Detection-CSDN Blog