P. L. Stocklin 341 
In light of these facts, we in ONR are seeking a fundamentally different ap- 
proach, one which builds upon the sonar equation, but which, in addition, pos- 
sesses those characteristics found lacking and leads us directly to answer 
basic questions of what processing we can do, how we can do it, and what its 
basic limitations are in terms of the physical acoustic situations with which 
we must deal. Let us recapitulate the shortcomings of the signal-to-noise ap- 
proach discussed earlier by way of listing requirements for our new approach. 
We want, then, not to be restricted to power processing; we want to include, 
directly, variation, fluctuation, and interaction in our approach; and, most im- 
portant, we want to have an effective guide to sustained improvement in our 
processing design. Finally, we want a direct relation between the acoustic field 
in its realistic complexity and the answers concerning processing limitations 
and processor design. 
18.3. SPACE-TIME DECISION THEORY APPLICATIONS 
18.3.1. ‘Introduction 
Space—time decision theory involves a direct attack upon the instantaneous 
acoustic field using decision theoretical methods. Here the point of view is that 
we have available for a given amount of time T a local volume of acoustic field 
which we must use to make decisions concerning some event or series of events 
having possibly occurred at some distance from the local volume. To get infor- 
mation from the acoustic field, we sample it in space and time, then process 
these samples, ultimately making a decision. The basic ingredients for proc- 
essor design, then, are: first, the description of our state of knowledge of the 
acoustic field in the local volume available to us; and second, the set of deci- 
sions, one of which we must make. In this view, any description of an acoustic 
event, however distant, must be in terms of the acoustic fields produced in the 
local volume. 
Now, for many situations of interest, it is abundantly clear from experience 
at sea and from model studies, such as the excellent experiments described by 
Dr. A.B. Wood in Lecture 10, that we never know and probably never will know 
the local acoustic field in exact, deterministic detail. In fact, the best descrip- 
tion of our state of knowledge appears to be a probabilistic one. I hasten to add 
that the term "probabilistic" is not taken to mean only random or stochastic 
processes—far from it. A probabilistic description properly includes, as a pos- 
sible extreme, the deterministic descriptions with which we are most familiar, 
such as the wavefront and waveform. 
A useful way in which to express a probabilistic state of knowledge is in 
terms of the conditional probability that, ifa certain event occurs, then a certain 
set of acoustic field pressures will occur in the available volume V during proc- 
essing time T. In order to make these joint probabilities both tractable and re- 
alistic in terms of our actual field sampling in space and time, it is necessary 
to develop space—time sampling theorems, analogous to the temporal sampling 
approach of Peterson, Birdsall, and Fox [1]. In this paper, two such sampling 
theorems are stated, one treating the monochromatic acoustic field and the 
