Thursday, April 24, 2014
Finding a Position Based on an Amplitude Function
The search for the black boxes of Flight 370 managed to pick up some pings on a number of occasions but were unable to get good location for them. The depth and decaying batteries were complicating factors. Under favorable conditions one can use the amplitude of the detected pings to estimate a source location.
On April 14th I came up with a method that located a source using just four sets of observed amplitudes and positions. In lieu of measured amplitudes computed values for an assumed amplitude function were used instead but the actual location of the source was not used to find a best fit. The bottom of the water was assumed to be constant so that the depth would be known and a value was assumed for the source strength. The situation is defined by the figure below. Complex numbers were used to minimize the number of terms in the amplitude function with d being the depth and s the distance of a surface vessel from the point directly over the source. The amplitude, a, was assumed to follow an inverse square law.
A bar above a complex number such as z indicates its complex conjugate which is found by replacing the imaginary component with its negative value. For an ordinary least squares fit we need the amplitude deviations, δ, or the differences between the computed values and the "measured" values but that didn't work very well. So instead I tried fitting the relative amplitude deviations, ρ, found by dividing a calculated deviation by its measured amplitude at a position. The formulas can also be simplified by using relative positions, ζ, and the source strength at the surface directly above the source, Φ0.
The objective function which is what one tries to minimize is the sum of the squares of the relative amplitude deviations. If the measurements are taken for a small number of points on a grid one can estimate the center, ζ0, of the amplitude function. One does not have to be directly over the source to locate its position. One of the data points was used as a reference position.
The fit found the position that was assumed for the source. The assumed value below are unprimed while the estimates have a prime on them. From the fit position one can compute heading and range from the reference position to the source location.
The geometry of the ocean bottom can act to focus the sound waves of the pings and alter their amplitudes. Changes in the density of sea water with depth can deflect the path of the sound waves much like the change in density of the air with altitude can alter the index of refraction at a given height and alter the observed position of a star. Under more complicated conditions a least squares fit would be more difficult but it still may be possible if one can collect more information of the ocean floor. One also needs to know the hazards in the neighborhood of the pingers if one wants to move about and collect objects from the ocean floor for investigation.
If the source strength is unknown the range of the source will be uncertain but one might still be able to use a grid search at well separated locations to get the relative headings and use intersecting arcs to estimate an actual position for the source.