The Accumulation Of Survey Errors

As you survey around a loop, the errors slowly accumulate and this effects how well the loop closes. It is important to understand how survey errors accumulate, because it allows us to make predictions about how well each loop should close. If you know how a loop should close, you can assess the quality of the survey measurements and the accuracy of the map as a whole.

The Strange Way Errors Add. Most people think that survey errors simply add together, one-by-one as you work your way through the cave. This leads to many misconceptions. For example, since every shot adds more errors to a loop, you would think that more shots mean more errors, but in fact, just the opposite is true. This is because random errors can be both positive and negative and they actually have the tendency to cancel each other out.

To understand this more clearly, let’s look at a simple example. Let's say that you have a series of ten-foot shots and you have a ruler that is only accurate to plus or minus one foot. That means that sometimes you will read nine feet, sometimes ten feet and sometimes eleven feet. Here is a table showing the results from three such measurements, each one labeled A, B and C:

This results in a range of errors that runs from -1 to 0 to +1 and it very similar to the kind of errors you might have in a tape measure. If you take two shots and combine all the combinations of this range of errors something very interesting happens. Since there are three possible error values for each shot, there are a total of nine combinations:

A	+	A’	=	-1	+	-1	=	-2
A	+	B’	=	-1	+	0	=	-1
A	+	C'	=	-1	+	+1	=	0
B	+	A'	=	0	+	-1	=	-1
B	+	B'	=	0	+	0	=	0
B	+	C'	=	0	+	+1	=	1
C	+	A'	=	1	+	-1	=	0
C	+	B'	=	1	+	0	=	1
C	+	C'	=	1	+	+1	=	2

As we add together all the possible errors from two shots, the errors begin to spread out. We now have five possible error combinations ranging from -2 to +2. Now we want to find out how many times the error falls into each of our five slots:

-2		-1	0
		-1	0	+1
			0	+1	+2
-------------------------------------
1	2		3	2	1

It is important to understand that we are not adding up errors, but counting the number times an error falls into a particular slot. Notice how the results tend to concentrate in the middle and thin out toward the edge. If you combine enough shots in this way, you will get the familiar "bell shaped" curve. This also gives us the probability of various total errors at the end of two shots:

This kind of information is very useful for determining the quality of a loop. Iif you get a loop error of minus three for the example survey, the chances are less than one in nine that it is a random error. Thus, it is very useful to be able to predict the kind of errors you should find in a loop.

Three Dimensions. So far, we have only looked at errors in one dimension. If we add in the error caused by the compass and inclinometer reading, the error becomes three-dimensional. Even in three dimensions, there is a pattern to the error that has the same properties as the bell curve.