The battle of the water calibration networks (BWC): Roughness and demand estimation based on weighted least squares (WLS) method

The weighted least squares (WLS) method has been widely used for parameter estimation and state estimation in water distribution system (WDS). The problem is formulated as a minimization of error between the field measurements and simulation model outputs by adjusting parameter values, which is the common procedure as other calibration approaches (e.g., trial and error or optimization algorithm). Instead of random searching approach, however, the WLS updates the unknown parameters iteratively using Jacobian matrix, which is the sensitivity of the measurement vector to changes in the input parameters, forming the basis of the WLS scheme. A Taylor series expansion provides an approximation of the non-linear vector function expressing the relationship between unknowns and measurements (or model outputs). The optimal estimate is updated iteratively until the target errors are small enough or the magnitudes of the estimate corrections become smaller than a given tolerance level.