The Nature of Measurement: Part 9: The Concept of Random Error Propagation

Error theory is covered briefly in my review seminars. One multiple choice question taken from the NCEES "Typical Questions" booklet asks that if the standard error in measuring one tape length with a 100-ft. tape is ±0.02 ft., what is the standard error in measuring 1,296.88 ft. with this tape? The four answer choices are ±0.05, ±0.07, ±0.13 and ±0.26 ft. A compilation of responses from the many seminar groups, representing more than 1,000 attendees, reveals that about 92 percent choose ±0.26 as the answer, about 4 percent choose ±0.07, and the remaining 4 percent choose one of the other answers or do not respond. The correct answer is ±0.07, and it is calculated by taking 0.02 times the square root of 13 (the number of tape lengths). This is called "error in a series" and is an example of error propagation in addition.

The popular answer of ±0.26 results from treating the error in a tape length as a systematic error and indicates a failure to recognize what the ± (plus or minus) sign means, the significance of the "standard error" term or the difference between a systematic and a random error. In short, the vast majority of the people aspiring to become licensed surveyors do not understand random errors and their propagation, even on the most elementary level.

The Present "State of the Art"

One would think that licensed surveyors might do much better, but this is not the case. Problems such as the above have been used as part of group discussion at my professional development seminars on analysis of surveying measurements. More than 90 percent of the licensed surveyors attending these seminars fail to answer such questions correctly. When it comes to more complex error propagation problems, fewer than 5 percent analyze them correctly.

Occasionally a few surveyors become defensive, rationalizing that "we never need to do that in practice." Many seem to conclude that error propagation is unimportant, impractical and just a lot of theoretical nonsense. If that were the case, we might as well forget discussing concepts such as positional tolerance, positional error, designing measurement specifications and developing theoretically sound measurement standards because error propagation is a significant part of these concepts.

A thorough coverage of error propagation would require several hours of explanation or several chapters in a textbook. It can easily comprise two or three semester hours of college course work, preferably preceded by mathematics courses in calculus and statistics. Space does not permit doing more than introducing the concept here, illustrating how it works and trying to demonstrate the importance of using error propagation in the overall analysis of errors.

Definition

The Glossary of Mapping Sciences (ACSM, ASPRS, ASCE, 1994) defines propagation of error as "the effect of error in a quantity on a function of that quantity." This brief definition leaves open the possibility that error propagation deals with both systematic and random errors and therefore is a bit misleading. For example, the effect of a systematic error of 0.02 per 100-ft. tape length would indeed have an effect of 0.26 ft. over a distance of about 1,300 ft. I believe that error propagation applies to random errors, not systematic errors. The reasons why this is proper will be revealed later. I define the concept as follows:

Propagation of errors (or error propagation) is the mathematical process used to estimate the expected random error in a computed or indirectly measured quantity, caused by one or more identified and estimated random errors in one or more identified variables that influence the precision of the quantity.

A problem that contains only one random error and one variable is simple. In the above problem, any contributing error was lumped into the "per tape length" statement, with no breakdown into individual sources. This makes the problem very simple and renders it only an academic exercise to illustrate how error in a series works.

Real error propagation situations are not so simple. There are three considerations, each contributing to the complexity of the problem: 1) the number of variables involved, 2) the number of random error sources contributing to the error in each variable and 3) the mathematical complexity of the equation(s) (the functions) used to compute the desired quantity.

For instance, if we considered random errors in determining the temperature and tension on the tape and separately considered reading and end-marking errors, we would have four variables in the taping problem. Then we might consider that the temperature error has random errors of calibrating the thermometer, as well as reading it. The same is true for the tension handle. If the quantity being computed is area, not distance, it is a much more complex equation than if it were merely the error in a distance.

General Equation for Error Propagation

If y = f(x1, x2, x3, … xn); that is, "y" is a function of several variables called x1, x2, x3, and so forth, the general equation for error propagation, in somewhat simplified form, is:

Ey = [Ex12 + Ex22 + Ex32 + … + Exn2]1/2 (1)

where Ex12, Ex22, Ex32 through Exn2, are the random errors contributed by variable 1, 2, 3 through n, respectively, and Ey is the computed random error in "y" from these several contributing variables or error sources.

The mathematics above is simple. Identifying the variables and their error sources, then evaluating the resulting Ex1, Ex2, Ex3, … Exn is what creates an analysis problem. How to identify and analyze the variables involved and the random error sources contributing to the error in each variable goes beyond the scope of this article. Also, we cannot present all of the statistical procedures (beyond what was covered in Part 8 ["Basic Statistical Analysis of Random Errors," Professional Surveyor, April 1998, pp.56-58]) for arriving at a sound numeric value for each contributing error. These numbers must be carefully determined, or else the result is only "garbage." However, we can explain the mathematical procedures in evaluating each Ex once the variables, their error sources and good estimates of the individual contributing errors have been identified and determined.

One way to evaluate each Ex is through the use of differential calculus. Each Ex can be evaluated separately as follows:

Ex1 = (jy/jx1) ex1, E

x2 = (jy/jx2) ex2 and so forth (2)

In words, this says take the partial derivative of "y" with respect to each variable (x1, x2 and so forth), and multiply the result by the error in the respective variable.

Example Using Calculus

Suppose A = LW. This is the area of a rectangle, where A = area, L = length and W = width. The calculus gives us "W" as the partial derivative of "A" with respect to "L," and "L" as the partial derivative of "A" with respect to "W." Substituting these derivatives into Equation (2) yields:

EAL = W eL, EAW = L eW

where EAL is the error in the area resulting from error in "L" and EAW is the error in the area resulting from error in "W." Substituting these into Equation (1) results in:

EA = [(W eL)2 + (L eW)2]1/2

This is the formula for propagation of random errors in a product, often listed among other simple propagation formulas in textbooks.

Say that L = 300.00 ±0.04 feet, W = 200.00 ±0.03 feet. Substituting these values into the last formula yields EAL = ±8 sq. ft. and EAW = ±9 sq. ft., from which EA = ±12 sq. ft. as the propagated error in the area.

Note that the propagation equations for error in addition, error in a series and error in a mean, commonly listed in textbooks and other references, can be derived from calculus and the general formula (Equation 1) as easily as what was done here for the error in a product.

Rationale of Error Propagation

Error propagation is, in a sense, just adding the contributing errors. However, unlike systematic errors, random errors cannot be added algebraically. Systematic errors accumulate, whereas random errors tend to compensate. The sign of a systematic error remains the same; therefore, the error must accumulate. However, the sign of a random error, as likely to be positive as negative, causes the compensatory effect. Taking the square root of the sum of the squares of the errors yields an answer reflecting this effect.

In the above example, because the errors can be plus or minus and lie anywhere within the estimated range, the resulting error will be less than the absolute maximum possible but greater than zero. The probability of the total error accumulating to the absolute maximum of 17 is as improbable as the error exactly canceling to zero. The best estimate is somewhere between the two, which is dictated by the law of compensation. The square root of the sum of the squares of the individual contributing errors yields the best estimate of the propagated error in the function being determined.

The random errors in the equations can be at any level of certainty, as long as the level is consistent. That is, they can be standard deviations, 90 percent errors, 95 percent errors, error estimates and so forth. The error in the quantity being computed will be at the same level of certainty as that of the error estimates used to compute it.

Error Propagation Without Calculus

The calculus approach obviously does not work if a person does not know calculus. Another approach is available. I call it the "rate-of-change method." This approach is not inconsistent with the calculus approach. In fact, the results of both are virtually the same because differential calculus is just a way of determining rates of change.

Understanding this approach requires some simple reasoning. In the above example, the effect of an error in "L" on "A" is logically reasoned to be the difference between two values for "A," one computed with and the other without the error estimate in "L" added to "L." Similar reasoning applies for the error in "W." In this example, A = 60,000 sq. ft., error free. If the error of 0.04 ft. in "L" is added to "L," the area would be 60,008 sq. ft., which is 8 sq. ft. larger than the error-free 60,000. Had we considered the error to be negative, the erroneous area would have been 59,992 sq. ft., which is 8 sq. ft. smaller than the error-free result. Note that because we anticipate that we will be squaring the error effects, it does not matter whether we make them plus or minus. In magnitude, (-8)2 is the same as (+8).2 If we consider the effect of the error in "W," we calculate the area (holding "W" fixed), changing "W" by its error. The result is either 60,009 sq. ft. or 59,991 sq. ft., which when subtracted from the nominal 60,000 sq. ft., results in ±9 sq. ft. error in the area. It should be obvious that there is no difference in the results, with or without calculus. The propagated error in the area using either approach is the square root of the sum of 82 and 92, which is ±12 sq. ft.

Using the rate-of-change method, the procedure is always the same: compute the quantity, error-free, then re-compute it by adding (or subtracting) the error in each variable to (or from) the error-free quantity. Subtract each of these values from the error-free quantity and then take the square root of the sum of their squares of these differences.

This method may be easier to conceptualize than the calculus method. Also, it is probably no more cumbersome to use, especially if the calculations are computerized. A few simple programmed steps will yield the answers.

Graphical Analysis

Some problems, such as the area problem and others that can be sketched on a horizontal or vertical plane, lend themselves easily to graphical analysis. This approach bypasses the need for calculus and affords another way to literally see how the process works in a geometric sense. The figure below shows the 200 x 300 ft. rectangle with the effect of the errors in the two dimensions (positive errors assumed).

The cross-hatched areas are the area errors from the two error sources. It is easily seen that the area created by the error in "L" is EAL = 0.04 x 200 = 8 sq. ft. and the error caused by the error in "W" is EAW = 0.03 x 300 = 9 sq. ft. Remember that random errors are added using the square root of the sum of the squares of the contributing errors. Applying this reality to the graphical analysis, we have the same ±12 sq. ft. as the expected error in the area, with no calculus or computations of areas and erroneous areas, as with the rate-of-change method.

Dealing with Complex Problems

The more variables contributing to the errors and the more error sources in each variable, the longer the equation is for the square root of the sum of the squares. Although most problems do not lend themselves to a graphical analysis, taking partial derivatives and using the general formula or applying the rate-of-change approach works in any case. The complexity of the analysis is somewhat proportional to the complexity of the equation for the variable whose magnitude and error estimate are being determined. Let us look at a couple of examples.

Hour Angle Formula. In the hour angle method for astronomic azimuth, there are three variables: 1) latitude of the observation station, 2) declination of the celestial body and 3) hour angle. However, the equation itself has six trigonometric functions, two products, one subtraction and one quotient, and each of the variables has more than one contributing error source. The error in the latitude, if scaled from a map, would consist of the propagated error resulting from map inaccuracy, identification of the point on the ground, plotting the point on the map and the scaling error.

The error in the hour angle consists of the error in the longitude (which would have the same error sources as the latitude) plus the error in the timing. The timing error is affected by the accuracy of the time source, the starting/stopping precision in using a stopwatch and a small error in the DUT1 correction.

After all of this is considered and computed, the result is the estimated error in one solar azimuth at one point in time. To get the estimated error in the azimuth of a referenced line, we would need to consider the errors in the horizontal angles, as well as the number of repetitions used to arrive at a mean azimuth of the line. The angle errors would be affected by most of the usual sources, such as reading, pointing, ground target centering, bubble (effect on horizontal axis) centering and so forth. These errors are thoroughly analyzed and discussed in my manual, Astronomic and Grid Azimuth.

Coordinate Geometry Problems. Problems such as area by coordinates or DMDs involve equations for latitude and departure, which involve the length and directions of lines. To determine the error in such an area, one would need to consider all errors in distances and angles, perform an error propagation for each departure and latitude and then perform another error propagation for the area using the area formula itself. For more on application of error propagation to complex problems the reader is referred to my text, Surveying Measurements and their Analysis, several chapters of which are devoted to the subject.

Regardless of the complexity of the situation, the procedures are the same—identify the mathematical function to compute the desired quantity, identify the variables and error sources, do some rather thorough analysis or consult with experts to get some good values for the individual error estimates and then use either calculus or the rate-of-change method to propagate the errors.

Application

Error propagation is just one of the many aspects of analyzing errors. I used only a few examples here to illustrate, but the list is endless. Have you ever wondered what the error of closure ought to be in a given traverse, taking into consideration all of its unique geometric considerations, instruments used and overall field method?

Have you ever wanted a good estimate of the uncertainty in an elevation difference determined trigonometrically, or in a distance determined by a coordinate inverse or the law of cosines, or the positional error of a point with respect to some fixed "control point"?

How about deciding how many repetitions of a horizontal angle are needed to achieve a specified precision with a particular instrument or deciding an overall field method from among more than one choice? In all cases, the answer is found by applying the procedures outlined in this article on error propagation.

Errors are part of measurement reality. Knowing what you have in a computed quantity is part of the professional responsibility of any surveyor. Knowing how to use error propagation is an important part of the specialized body of knowledge called "surveying."

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