In the simplest applications, solutions can be most easily found in the frequency domain. When complications arise, it is better to use the time domain, to directly apply the convolution operator and the method of least squares.
A first complicating factor in the frequency domain is a required boundary in the time domain, such as that between past and future, or requirements that a filter be nonzero in a stated time interval. Another factor that attracts us to the time domain rather than the frequency domain is weighting functions.
Weighting functions are appropriate whenever a signal or image amplitude varies from place to place. Much of the literature on time-series analysis applies to the limited case of uniform weighting functions. Such time series are said to be ``stationary.'' This means that their statistical properties do not change in time. In real life, particularly in the analysis of echos, signals are never stationary in time and space. A stationarity assumption is a reasonable starting assumption, but we should know how to go beyond it so we can take advantage of the many opportunities that do arise. In order of increasing difficulty in the frequency domain are the following complications:
We will not have difficulty with any of these complications here, because we will stay in the time domain and set up and solve optimization applications by use of the conjugate-direction method. Thus we will be able to cope with great complexity in goal formulation and get the right answer without approximations. By contrast, analytic or partly analytic methods can be more economical, but they generally solve somewhat different applications than those given to us by nature.