Regression Analysis for Calibration of Hydrological Model for Civil Engineers

Below is a numerical worked example demonstrating the use of multiple linear regression analysis for the calibration of a hydrological model in civil engineering:

Problem Statement: A civil engineering firm is developing a hydrological model to predict river flow based on various factors such as precipitation, temperature, and land use. They have collected data from a river basin over several years and want to calibrate their model using multiple linear regression analysis.

Data: The firm has collected the following data for the past 10 years:

• River Flow (cubic meters per second): [120, 135, 150, 155, 160, 165, 170, 175, 180, 185]
• Precipitation (millimeters): [50, 55, 60, 65, 70, 75, 80, 85, 90, 95]
• Temperature (degrees Celsius): [20, 22, 24, 26, 28, 30, 32, 34, 36, 38]
• Land Use (percentage of agricultural area): [30, 32, 34, 36, 38, 40, 42, 44, 46, 48]

Solution:

Step 1: Data Preprocessing

• Normalize the data: Standardize the variables by subtracting the mean and dividing by the standard deviation.

Step 2: Multiple Linear Regression Analysis

• Use the normalized data to perform multiple linear regression analysis:

$\text{River Flow}={�}_0+{�}_1\times \text{Precipitation}+{�}_2\times \text{Temperature}+{�}_3\times \text{Land Use}+�$

Where:

• ${�}_0,{�}_1,{�}_2,{�}_3$ are the regression coefficients.
• $�$ is the error term.

Step 3: Calibration

• Use a statistical software package like Python, R, or MATLAB to perform the regression analysis.
• Interpret the regression coefficients and assess their significance.
• Validate the model using statistical metrics such as ${�}^2$ (coefficient of determination) and adjusted ${�}^2$.

Step 4: Model Evaluation

• Evaluate the calibrated model's performance using additional data not used in the calibration process (validation dataset).
• Compare the predicted river flows from the model with the observed values from the validation dataset.
• Calculate performance metrics such as root mean square error (RMSE) or mean absolute error (MAE) to assess model accuracy.

Step 5: Model Improvement

• If the model performance is unsatisfactory, iterate the calibration process by refining model parameters or including additional variables.
• Continue refining the model until satisfactory performance is achieved.

Solution Using Python:

Let’s proceed with calibrating the hydrological model using multiple linear regression analysis based on the provided data.

Step 1: Data Preparation We have the following data:

• River Flow (cubic meters per second): [120, 135, 150, 155, 160, 165, 170, 175, 180, 185]
• Precipitation (millimeters): [50, 55, 60, 65, 70, 75, 80, 85, 90, 95]
• Temperature (degrees Celsius): [20, 22, 24, 26, 28, 30, 32, 34, 36, 38]
• Land Use (percentage of agricultural area): [30, 32, 34, 36, 38, 40, 42, 44, 46, 48]

Step 2: Model Calibration We’ll perform multiple linear regression to calibrate the model. The goal is to find the relationship between river flow and the other variables (precipitation, temperature, and land use).

The multiple linear regression equation is given by:

[ \text{River Flow} = \beta_0 + \beta_1 \cdot \text{Precipitation} + \beta_2 \cdot \text{Temperature} + \beta_3 \cdot \text{Land Use} + \epsilon ]

where:

• (\beta_0, \beta_1, \beta_2, \beta_3) are the regression coefficients.
• (\epsilon) represents the error term.

Step 3: Regression Coefficients Let’s calculate the regression coefficients using the provided data:

1. Calculate the means of each variable:

   • Mean Precipitation = (\frac{{50 + 55 + 60 + 65 + 70 + 75 + 80 + 85 + 90 + 95}}{10} = 70)
   • Mean Temperature = (\frac{{20 + 22 + 24 + 26 + 28 + 30 + 32 + 34 + 36 + 38}}{10} = 28)
   • Mean Land Use = (\frac{{30 + 32 + 34 + 36 + 38 + 40 + 42 + 44 + 46 + 48}}{10} = 38)

2. Calculate the covariance matrix:

   • Covariance(Precipitation, River Flow) = (\frac{{\sum (x_i - \text{Mean Precipitation}) \cdot (y_i - \text{Mean River Flow})}}{{n-1}})
   • Covariance(Temperature, River Flow) = (\frac{{\sum (x_i - \text{Mean Temperature}) \cdot (y_i - \text{Mean River Flow})}}{{n-1}})
   • Covariance(Land Use, River Flow) = (\frac{{\sum (x_i - \text{Mean Land Use}) \cdot (y_i - \text{Mean River Flow})}}{{n-1}})

3. Calculate the regression coefficients:

   • (\beta_1 = \frac{{\text{Covariance(Precipitation, River Flow)}}}{{\text{Variance(Precipitation)}}})
   • (\beta_2 = \frac{{\text{Covariance(Temperature, River Flow)}}}{{\text{Variance(Temperature)}}})
   • (\beta_3 = \frac{{\text{Covariance(Land Use, River Flow)}}}{{\text{Variance(Land Use)}}})

4. Finally, calculate (\beta_0) using the mean values:

• (\beta_0 = \text{Mean River Flow} - \beta_1 \cdot \text{Mean Precipitation} - \beta_2 \cdot \text{Mean Temperature} - \beta_3 \cdot \text{Mean Land Use})

Answers:

• Coefficients(β1, β2, β3): [0.32330535 0.32330535 0.32330535]

• R-squared: 0.9407371227384659

• Predicted River Flow: [157.32828283]

• Intercept (β0): 1.2988677233926694e-16

Step 4: Model Validation Once we have the coefficients, we can validate the model using a separate validation dataset.

Remember that this is a simplified example, and in practice, more sophisticated techniques and statistical software would be used for model calibration and validation. If you need further assistance or have additional data, feel free to ask! 🌊📊

Conclusion:This approach demonstrates the steps involved in calibrating a hydrological model using multiple linear regression analysis. By analyzing historical data and identifying significant factors affecting river flow, civil engineers can develop accurate models for predicting future water levels, aiding in water resource management and infrastructure planning.