# Model 27: Hydrological Model

In constructing a hydrological model using the data in the table given at the bottom, it is required to obtain the expected runoff, denoted by Ri during the $i$-th period, as a linear function of the observed precipitation. From hydrological considerations the expected runoff depends on the precipitation during that period and the previous two periods. So the model for expected runoff is Ri=b0pi +b1pi-1+b2pi-2, where $p$i equals precipitation during the $i$ -th period; and $b-\right), b$1, b2 are the coefficients that are required to be estimated. These coefficients have to satisfy the following constraints from hydrological considerations: $b$0+b1+b2=1, where $b$0> b1> b2> 0. Obtain the best estimates for $b$0, b1, b2, if the objectives are
1. to minimize the sum of absolute deviations sumi|Ri-b0pi-b1 pi-1-b2pi-2|; and
2. to minimize the maximum absolute deviation maxi|Ri-b0p1-b1 pi-1-b2pi_2|.
 Period Precipitation (inch hours) Runoff (acre feet) 1 2 3 4 5 6 7 8 9 10 11 12 3.8 4.4 5.7 5.2 7.7 6 5.4 5.7 5.5 2.5 0.8 0.4 0.05 0.35 1 2.1 3.7 4.2 4.3 4.4 4.3 4.2 3.6 2.7