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
R$$_{i} during the $i$th period, as a linear
function of the
observed precipitation. From hydrological considerations the expected runoff
depends on the precipitation during that period and the previous two
periods.
So the model for expected runoff is R$$_{i}=b_{0}p_{i
}+b_{1}p_{i1}+b_{2}p_{i2},
where $p$_{i} equals precipitation during the
$i$
th period; and $b),\; b$_{1},
b_{2} are the coefficients that are required to be estimated.
These
coefficients have to satisfy the following constraints from hydrological
considerations: $b$_{0}+b_{1}+b_{2}=1,
where $b$_{0}> b_{1}> b_{2}> 0.
Obtain the
best estimates for $b$_{0}, b_{1}, b_{2},
if the objectives are
 to minimize the sum of absolute deviations
sum$$_{i}R_{i}b_{0}p_{i}b_{1}
p_{i1}b_{2}p_{i2}; and
 to minimize the maximum absolute deviation
max$$_{i}R_{i}b_{0}p_{1}b_{1}
p_{i1}b_{2}p_{i_2}.
Period
 1
 2
 3
 4
 5
 6
 7
 8
 9
 10
 11
 12

Precipitation (inch hours)
 3.8
 4.4
 5.7
 5.2
 7.7
 6.0
 5.4
 5.7
 5.5
 2.5
 0.8
 0.4

Runoff (acre feet)
 0.05
 0.35
 1.0
 2.1
 3.7
 4.2
 4.3
 4.4
 4.3
 4.2
 3.6
 2.7
