## Ranging

The best way to reduce data from passive ranging is suggested

Electromagnetic waves and sound waves are widely used for the determination of the location of objects. Familiar examples are radar, sonar, GPS positioning, Polaroid cameras, police speed meters, and many others. Most are echo devices, generating a wave and interpreting its echo from the object of interest. GPS is a cooperative system, in which the receiver observes timing signals from sources of known location, and locates itself in reference to them. In what follows, we will consider systems based on sound waves in water, locating in two dimensions only, and in the simple case of a uniform, isotropic speed of propagation. This will allow us to clarify the fundamentals free of unnecessary complication. For any practical system, all the complicating factors must be taken into consideration.

The problem of determining the location of an object will be called ranging here. There are three kinds of ranging systems, which we will call echo, cooperative and passive. Echo ranging, illustrated in the Figure, is by far the most common method in practice. The observer at point 1 emits a wave at a certain time. When the outgoing wavefront strikes the object 0, a scattered wavefront is launched, which is detected at point 1 a certain time interval Δt later. The distance r from 1 to 0 is then r = cΔt/2, where c is the speed of sound. Two such stations can locate the point 0 by comparing the distances they obtain. More commonly, the observer is able to determine the direction of the received wave, and in that case the object can be located from the location 1 alone. This, of course, is the principle of radar. The object observed is passive, and need take no part in the procedure.

In cooperative ranging, the object takes an active part. In one form, the object emits waves that contain timing information. Two fixed stations with clocks synchronized with that of the object note the time delays of the received wave, and from the two distances thereby determined can locate the object, and perhaps radio back its position. In the second form, the object receives waves from two fixed stations, and determines its position from the two distances. Of course, in three dimensions, three distances are necessary. The GPS system uses the second plan, with the remarkable circumstance that the 'fixed' stations are actually moving in orbits, but their positions can be accurately predicted at any time. Cooperative ranging places great demands on clocks, especially when carried out with electromagnetic waves. Timing sound waves is much less critical.

Passive ranging makes use of a wave generated by the object to be located, but timing information is not necessary. Therefore, the wave used may be generated in the normal actions of the object, and not necessarily for location purposes. An example might be the location of a boat from its propeller noise, or a whale from its singing. A minimum of three fixed stations is necessary, which we assume can detect a sound only, not determine its direction. The fundamental data is the relative times, the time delays, at which the signals are received at the stations. Let us assume all delays have been expressed as distances by multiplying them by the speed of sound. In the Figure, b and c are the delays at stations 2 and 3, relative to station 1. A circular wavefront must pass through point 1, and be tangent to the circles drawn from stations 2 and 3. The centre of this wavefront is then the location of the object.

It was this diagram that caught my attention in the Reference, since the problem of constructing a circle passing through a given point and tangent to two given circles is a challenging one that I have not seen before. It happens that the two circles cannot be given arbitrarily, and the wavefront must be convex at the points of tangency. A means of solution with compass and straightedge did not suggest itself, so I turned to algebra. The triangle 023 can be solved by the law of cosines when a guess for a is used. Then, the distance 01 can be computed and compared with the guess for a. If they are not equal, a new guess for a is chosen in the direction making them more equal, and soon a consistent solution arises. Of course, a computer is necessary for the tedious calculations. It became obvious that certain input values were inconsistent, and that various cases existed that would complicate the matter. However, this could be worked into a practical means of solution.

A different attack was based on the familiar problem of different times of arrival at two stations. If the wavefront arrives simultaneously, the source is somewhere on the perpendicular bisector of the line joining the observation points. If the delay d in arrival is equal to the distance between the two points 2a, then the source is located somewhere on the line through the two points, excepting the segment joining them. For intermediate values of delay, the point must be on a branch of a hyperbola whose foci are the observation points, since the hyperbola is the locus of points whose distances to two fixed points, the foci, have a constant difference. The problem now is to take the three observation points as two pairs, determine the hyperbolas, and find their point of intersection. This is also possible, but extremely tedious.

One notices that the hyperbolas lie relatively close to their asymptotes at even moderate distances from the foci. If the hyperbolas are replaced by their asymptotes, the intersections of these straight lines are relatively easy to determine. The intersection of the asymptotes for two pairs will give an approximate location of the point. Now we apply a procedure to correct this approximate location to a more accurate one. To do this, the time delays predicted from the approximate location are subtracted from the observed time delays to obtain an error at each observation point, and a small displacement of the approximate location is made that reduces the sum of the squares of the errors to a minimum. Least squares also handles the possiblity of errors of measurement, and the use of redundant additional observations. Iteration will reduce the error to as small a figure as is consistent with the accuracy of the observations. This, I believe, is the best way to reduce the data for passive ranging.

### References

C. V. Drysdale, Submarine Signalling and the Transmission of Sound Through Water, in C. V. Drysdale, et. al., The Mechanical Properties of Fluids (London: Blackie and Son, 1925).