The theory of error propagation is derived in topics such as Mikhail (1976) and Wolf and Ghilani (1997), and is presented concisely in matrix form as
where
Error propagation involves calculus and is used to answer the question “If something is computed on the basis of a measurement and the measurement contains uncertainty, how is the computed quantity affected?” In a trivial case (Y = X), there is a direct correspondence between the two, and the error in the result is the same as the error in the measurement. In another simple case—the volume of a tank and its height—the relationship between the measurement and the computed result is linear: volume = (area of base) * (height). Other cases—the volume of a tank and its radius—are more complex: volume = (π R2) * (height). Here the relationship between the volume and radius is exponential. However, even more complexity arises when several measurements contribute simultaneously to the quantity being computed. Equation 3.23 handles all cases from the trivial to the most complex.
Equation 3.16 could be used to compute approximate changes in volume based upon changes in radius and height, but that approach is somewhat limited. Equation 3.23 is used to answer the specific question “What is the standard deviation of the volume if the standard deviation of the radius and the standard deviation of the height are both known?” Admitting that standard deviations are estimates, the answer will still be an estimate. But, unlike equation 3.16 (which is accurate only for “small” values of ΔΗ and ΔR), equation 3.23 is a definitive procedure that is statistically reliable, and the approximation is in the standard deviation of the measurements (the user’s responsibility) and not in the equation. A simplified list of steps for performing error propagation is as follows:
1. Identify the variables (i.e., the measurements), and determine their standard deviations on the basis of repeated measurements, computations, or professional judgment.
Measurements and standard deviations in the tank example are as follows:
R = 50.0 meters +/- 0.07 meters h = 10.00 meters +/- 0.02 meters
2. Formulate the equations that will be used to compute the result.
3. Take the partial derivatives, one variable at a time.
4. Build the matrices as shown in equation 3.23.
5. Perform the matrix operations. Computers make this task much easier.
6. Interpret the results.
A. The standard deviation of the volume = square root of the variance = 553.36 meters3. Realistically, this answer has no more than two significant figures. At the 68 percent confidence level the standard deviation of the volume is 550 m3, and at the 95 percent confidence level the standard deviation of the tank volume is 1,106.7 meters3 (1,100 meters3).
B. The answer in step two really has only three significant figures and could be reported as 78,500 meters3 +/- 550 meters3 (or, at two sigma, +/- 1,100 meters3).
FIGURE 3.15 Survey Measurements of a Tank
C. In this case, there is no correlation between the measurements of height and radius. Had there been, the Σχχ matrix off-diagonals would be nonzero.
What happens if correlation is present? The following example may help. A total station surveying instrument was used to measure the size of the tank, as shown in Figure 3.15. Admittedly, this might not be the best way to measure a tank, but this procedure was chosen to show how correlation is included. Measurement of the radius and height is derived from independent observations of slope distances and zenith directions. This example assumes a standard deviation of 0.10 meters for each slope distance and 20 seconds of arc for each zenith direction. It could also be appropriate to make other assumptions about standard deviations for the observations.
To find radius and height from the field measurements, the functional model equations are
The partial derivatives of the radius are
The partial derivatives of the height are
The Jacobian matrix (transposed for ease of printing) of partial derivatives is
The covariance matrix of the observations is
Note that each slope distance was assumed to have a standard deviation of 0.10 meters and that each zenith direction has a standard deviation of 20 seconds of arc. Given that 20 seconds squared in radians is 9.401755*10-9, the elements of the observation covariance matrix are
The covariance matrix of the derived radius and height is then computed as
Note that the covariance matrix in equation 3.47 is almost the same as in equation 3.27 except that equation 3.47 contains covariance data for the derived measurements of radius and height. The standard deviation of the radius is 0.0707 m (not 0.07 m), and the standard deviation of the height is 0.0208 m (not 0.02 m). In the first example, radius and height were independent measurements (observations). In the second example, radius and height were both computed from the independent slope distance and zenith direction observations. Radius and height are not independent, and the covariance matrix contains correlation. The variance of the computed volume is now computed (the same procedure as in equation 3.28) as
The correlated standard deviation of the computed volume is 547.75 m3.
When the results in equation 3.47 are compared to those in equation 3.28, the correlated results are somewhat smaller. However, in this case, when significant figures are taken into account, the overall reported answer is the same in each case. The tank volume is 78,500 m3 +/- 550 m3.
When does correlation make a significant difference? Each user must answer that question for him or herself. As technology permits observations to be made with greater and greater precision and as smaller tolerances are imposed upon the computed result, correlated measurements will need to be considered. The important point here is that the independent observations must be identified and that the equations (models) used to compute spatial data components will need to be used properly to compute the correlated covariance matrices.
As shown in the tank example, equation 3.23 is very powerful. Specifically, matrix tools were used to illustrate using both correlated and uncorrelated measurements. Many surveying measurements are uncorrelated, and, as shown here, even correlated measurements may give the same answer. In the past, the nonmatrix form of equation 3.23 has been quoted as the special law of propagation of variance, which is used without correlations as
where U=f (X,Y,Z…) and X/Y/Z are independent variables.
If the variables really are independent, equation 3.49 can be applied to simple equations to give error propagation equations listed in various memorized by many as
Even when used without correlations, the error propagation equation is a powerful tool that has been underutilized. But, the matrix form of the error propagation, equation 3.23, is even more powerful in that it utilizes the power of matrices to handle systems of complex equations and it handles any and all correlations that may be part of a problem, simple or complex.
