by A. Waddington
With the coming of cheap computing power, the days of working out the coordinates of a cave survey by hand and adjusting loops by eye are fast fading. Unfortunately, surveys are still as subject to error as ever they were and it is useful to use that computing power to detect gross errors and distribute the random error corrections around loops. Long standing readers of this august journal will have noticed that the topic of cave survey error distribution is as popular now as the subject of battery chargers was a few years ago.
The method of least squares can be used to calculate the error corrections to be applied to a survey traverse. The algorithm of Warren (1988) has now been implemented in at least two survey reduction programs, and is proving very successful. Once a correction has been calculated for a traverse, however, the question of how to distribute this correction among the survey legs arises. As Warren (1988) pointed out, there is little justification for splitting the correction up proportionately by length of survey leg, as at least part of the error is attributable to station position error. The solution both for the survey leg weighting in the initial least squares analysis and for the eventual distribution of the error along the traverse is to form an estimate of the standard deviation of the errors associated with each leg. Warren gave formulae for these variances as follows :
sx2 = dP2/3 + [x dL/L]2 +
[z sin(T) dC]2 + [y dT]2 ... (1)
sy2 = dP2/3 + [y dL/L]2 + [z cos(T) dC]2 + [x dT]2 ... (2)
sz2 = dP2/3 + [z dL/L]2 + [L cos(C) dC]2 ... (3)
for inclined legs and
sx2 = sy2 = dP2/3 + ½ [L dC]2... (4)
sz2 = dP2/3 + dL2 ... (5)
for vertical legs.
where dP, dL, dC and dT are the standard errors associated with the measurements of station position, leg length, clinometer and compass respectively. It is clear that equation (3) reduces to (5) for vertical survey legs. However, equations (1) and (2) don't reduce to equation (4). While the second and fourth terms disappear, the third term becomes meaningless as T becomes irrelevant. Making the horizontal error on a vertical leg proportional to its length is sound, and the second term of equation (4) represents a circular distribution of error around the calculated point, which is what we would expect.
A problem arises, however, with equations (1) and (2) for legs which are very nearly vertical, but not quite. To take a concrete example, the slightly off-vertical leg with a compass bearing due east will have a much smaller error estimate in the north-south direction than the vertical leg, using equations (2) and (4). This is clearly anomalous and arises because the error in the compass direction is not independant of the inclination of the leg. At half a degree off the vertical, the difference between a leg going due east and one going due north is still less than a degree of arc.
Revised equations which take this problem into account are now presented :
sx2 = dP2 /3 + [x dL/L]2 +
[y dT]2 + [ ½ +
sin2(T)cos2(C) ]·[ z dC ]2
sy2 = dP2/3 + [y dL/L]2 + [x dT]2 + [ ½ + cos2(T)cos2(C) ]·[ z dC ]2 ... (7)
sz2 = dP2/3 + [z dL/L]2 + [L cos(C) dC]2 ... (8)
An alternative method of surveying is used by the cave diving community, who use their depth gauges instead of clinometers. In this case, the uncertainties arising from station position, length and compass are very much the same as for conventional survey methods. However, depth gauge tolerances have a somewhat different effect, giving the equation :
sz2 = 2 dZ2 ... (9)
for the vertical error, where dZ is the error in each depth gauge reading.
Horizontally, the situation is a little more complex. At first sight
sx2 = dP2/3 + [x dL/L]2 + [y dT]2 + [ dD z sin(T) / D ]2 ... (10)
where D is the plan length: D = sqr( L2 - z2 ), would appear to be the error, but inspection will show that this predicts an infinite horizontal error if the length equals the depth change, and an imaginary error if the measured depth change exceeds the length of the leg (which is quite possible, on a vertical leg, given the measurement tolerances we are trying to deal with !). The simplest way out is to check if the calculated plan length is less than the errors in leg length and depth change added in quadrature, and if so, assume that the errors in both x and y directions are equal to this tolerance. ie.:
if ( D2 <= ( 2 dZ2 + dL2 )) then sx2 = sy2 = ( 2 dZ2 + dL2 ) ... (11)
else use equation (10) above.
While this is less than wholly satisfactory, it does resolve the problem in a way suitable for implementation of a closure algorithm, and warns us that near-vertical legs are not a good idea in underwater cave surveys. Unfortunately, given the layout of a cave underwater, the diver may not have a great deal of choice.