Saturday, July 20, 2013

Improving Accuracy of ELT Searches

There have been many attempts to use ad-hoc ELT signal homing methods, some of them naive and poorly thought out. I have yet to see any attempt to use all the information available with existing Direction Finding techniques. The normal procedure is to take bearings from three or more locations and plot them on a map. This results in a cocked hat (see Figure 1). The problem is then to interpret what the cocked hat is telling us. This usually involves some form of finding some sort of 'center' of the triangle. Unfortunately triangles have more centers than they know what to do with.

Fig 1: A Cocked Hat
Courtesy of Savage Chickens
This is where fusion of the position information comes in. To fuse the data provided by the fixes (the points of intersection of each bearing) we must have some measure of how good each bearing is. This is not just a mater of how good the operator is of measuring a bearing, though that may be included. It is a measure of all the effects of equipment, location and propagation of the radio signal on the bearing. We normally use the variance of the bearing, though the standard deviation may also be used with adjustments to the equations. Radio Direction Finding has been used for a long time, so the mathematics is mature. Fusing two bearings into a fix properly takes into account not just the variance of each bearing, but also the angle of intersections of the bearings. This may be seen in Figure 2. In this image all the bearings have the same variance, the only difference is in the angle of intersection between two of the three.
Fig 2: Computed Fixes, all bearing variances the same.
At this point it is worth saying something about what this image means. Surrounding each intersection of two bearings there is an Error Ellipse. Just like the confidence level associated with a SARSAT/COSPAS fix, these Error Ellipses do not describe an area where the ELT is, they describe an area in which, given the information available, there is a defined probability the ELT is within the ellipse. Search and Rescue responders who forget that imperil other people. For each Error Ellipse the information available is the two associated bearings. What we want is a fix based on all the available information. We want to fuse all available bearings. Fortunately the mathematics for that too is mature. In Figure 3 the green fix and Error Ellipse represents that fusion.
Fig 3: Fusion of data from all three bearings.
Some might find this surprising, but that is why these computations should be done using dispassionate mathematics rather than intuition or gut feeling.

As stated above, when performing the calculations for Figures 1 through 3 bearings with the same variance were used. One should not expect this in the real world. Figure 4 shows the error ellipses for the case where the variance of two of the bearings is increased slightly.
Fig 4: The variance of two bearings increased over Fig 2.
Notice the Error Ellipses are larger, and have different eccentricity and orientation. The real surprise comes when we compute the final fix fusing all available data in Figure 5.
Fig 5: Final fix (in green) fusing all data.
Notice how fare the final fix location has moved from the original. Remember we have not changed the origins or directions of the bearings, only the variance of the bearings. If we change the variance of the last bearing, the one we have not yet changed, we get another set of Error Ellipses shown in Figure 6.
Fig 6: All three bearings have realistic variances.
And perhaps the biggest surprise of all is the final fused fix from this data set shown in Figure 7.
Fig 7: Final fused fix (in green) with realistic bearing variances.
So before you go about making up ad hoc rules for how radio waves propagate, how radios work and other nonsense to try to justify fairy tail search techniques; have a look at the mathematics, science and engineering behind techniques that professionals use. You may find what you've been looking for all along; a more efficient, effective and accurate way of doing things.

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