Geoscience Reference
In-Depth Information
(a)
X
X
Parameter 2
(b)
X
X
X
Parameter 2
(c)
X
X
Parameter 2
Figure 7.2
More complex response surfaces in two parameter dimensions: (a) flat areas of the response
surface reveal insensitivity of fit to variations in parameter values; (b) ridges in the response surface reveal
parameter interactions; (c) multiple peaks in the response surface indicate multiple local optima.
Another problem is parameter interactions. In simple cases this can lead to “ridges” in the response
surface (Figure 7.2b), with different pairs of parameter values giving very similar goodness of fit. In such
cases a hill-climbing technique may find the ridge very easily but may find it difficult to converge on
a single set of values giving the best fit. Again, different starting values may give different final sets of
parameter values.
There may also be a number of different peaks and ridges in the response surface (Figure 7.2c)
giving rise to a number of
local optima
. One of these will be the
global optimum
but there may be a
number of local optima that give similar goodness of fit. The response surface may also be very irregular
or jagged (see the work of Blackie and Eeles (1985) for a good two-parameter example and also the
