Statistics for Environmental Engineers

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Notice that taking the derivatives has restored the material balance constraints that inflows equal outflows.

The nine linear equations are solved simultaneously for the flows that satisfy the material balance. The adjusted values are given in Table 48.1. The calculated values of Я1 and Я2 have no physical meaning and are not reported. The adjustments range from about 10% for Q3 to a fraction of a percent for Q7.

This example assumed that all flow measurements were equally precise. It might well happen that the inflows are less reliably estimated than the outflows. This could be reflected in the calculation by assigning weights. If the metered quantities Q4 and Q7 are known four times more precisely than the other quantities, the assigned weights would be w4 = 4 and w7 = 4, with all other weights being equal to 1.0. This would cause the flows Q4 and Q7 to be adjusted less than is shown in Table 48.1, whiles flows 1, 2, 3, 5, and 6 would be adjusted by larger amounts.


The method of least squares gives the smallest possible adjustments that make the measured survey data conform to the conservation of mass constraints. Also, it agrees with common sense in that unreliable measurements are adjusted more than precise measurements. The solution for linear constraining relations is particularly simple, even for a problem with many variables, because the set of equations can be written on inspection and are easily solved using standard matrix algebra. For nonlinear constraining equations, the least squares computations are manageable, but more difficult, because the partial derivatives are nonlinear. A nonlinear programming algorithm must be used to solve the problem.

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