## Introduction

Analysis and adjustment of measurement error are critical skills for the land surveyor in modern practice. Despite this, I’ve always found the traditional textbook materials on error adjustment difficult to understand. I’ve struggled to wrap my brain around the concepts from statistics and probability that are part of error analysis and adjustment. A couple of things have helped me overcome (at least partly) this challenge to my professional knowledge. One thing was many hours of contemplation and consideration. The other was my slowly improving ability to write software code. I eventually realized most survey error adjustment problems could be solved using the iterative and brute force power of the desktop computer. Thinking about analysis and adjustment of measurement in error put the underlying concepts within my grasp.

This is the first article in what I hope will be a series of articles on the adjustment of surveying measurement error. The target audience for the article series is the working surveyor who would like a better understanding of measurement error and methods of error adjustment that he can apply in his or her own work. The articles won’t require a high-level background in statistics or calculus. We’ll explain all of the most complicated math with simple examples, diagrams, and computer code. (You will need a solid understanding of algebra, basic trigonometry, analytic geometry and coordinate geometry to get the most benefit from the article series.)

All of the source code we write for the article series will be in the Java and Groovy programming language. Will post all of the source code online, under and open source license. All of the text and media content for this article series will be released under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. This means you are free to distribute copies of the articles.

In this first article of the series we are going to:

1) Define a few basic terms about the quality of measurements.

2) Describe the basic parts of any survey measurement system.

3) Talk about the basic type of errors in a survey measurement system.

4) Define some goals for a basic measurement error analysis and adjustment (MEAA) software program.

## About

This is the first article in a series of articles on the adjustment of measurement error in land surveys. This article first appeared in the 2016 Fall Issue of the California Surveyor Magazine. In this article the reader is given a basic introduction to a few fundamental concepts of measurement error in land surveys.

## Measurement Precision, Accuracy, and Granularity

In this section of the article, we want to define a few terms that describe our survey measurements. This is important because these terms are often confused and used interchangeably in common language. It is also important to understand the definition of the measurement qualities because we will want to determine their values as part of our MEAA software program.

The first term we will define is precision.

Precision: The degree to which measurement values are tightly clustered. Measurement values with small differences from one another (or small difference from an average value) are said to be tightly clustered or more precise, whereas measurement values with large differences from one another are said to be loosely clustered, or less precise. As a general rule, more precise measurements are an indication of a measurement system with less error. Precision is typically the most misunderstood of our 3 measurement terms. People will say precision when they mean accuracy or will confuse precision when they mean measurement granularity. As a general rule surveyors prefer more precise measurements.

Example:

The distance measurements in the first set below would be imprecise (compared to the second set below) when considered at the hundredth of a foot level of granularity. This is because there is a relatively large spread between the distance values when compared to the second set:

101.23 Feet

101.05 Feet

100.86 Feet

100.74 Feet

The distance measurements in the second set below would be precise (compared to the first set above) when considered at the hundredth of a foot level of

granularity. This is because there is a relatively small spread between the distance values when compared to the second set:

101.232 Feet

101.231 Feet

101.230 Feet

101.233 Feet

The second term we will define is accuracy:

Accuracy: The degree to which an observed measurement value varies from the true value for a measurement. An observed value that is closer to the true value is considered to be more accurate than a value that is farther from the true value. The required accuracy values of a measurement typically depend on the way in which the spatial data created from the measurements will be used. More critical applications of spatial date require more accurate measurements. Accuracy can be difficult to determine in many situations because the true value for a measurement is unknown. As a result, precision is often used as a substitute for accuracy when considering the quality of measurements. (There are certain rules of geometry that do allow us to calculate accuracy. For example, we know that the 3 interior angles of a triangle should sum to exactly 180 degrees. We will discuss these type of geometry rules more later.)

If the true value of a distance is 100.00 feet, the second set of measurements below would be considered more accurate than the first set of measurements below:

101.01

101.02

100.98

100.99

101.32

101.31

101.30

101.29

The third term we will define is granularity.

Granularity: A description of the size of the smallest unit of measurement with which a measurement is made. A more finely grained measurement has a smaller unit of measurement than a more coarsely grained measurement. The granularity of a measurement is typically constrained by the type of instrument used to make the measurement observations.

Granularity is often confused with precision. But they are not the same thing. You can make highly precise but course grained measurements. You can also make imprecise but finely grained measurements.

Example

The following distance measurements would be considered precise but course grained:

50.1 Chains

50.0 Chains

49.9 Chains

50.2 Chains

The following distance measurements would be considered imprecise but fine-grained:

101.232 Feet

101.054 Feet

100.867 Feet

100.742 Feet

The following distance measurements would be considered precise and fine-grained:

101.232 Feet

101.231 Feet

101.230 Feet

101.233 Feet

The granularity of measurements can be visualized with a couple of examples.

In the first example, we want to measure the volume of several large glass jars. We can do this by filling each jar with marbles and counting the marbles. A larger jar will hold more marbles. If we shrink the diameter of the marbles we use to measure the volume, our measurement becomes finer grained.

In the second example, we can use a more familiar tool to most land surveyors, the engineer’s measuring scale (or ruler). A scale with ticks to the nearest inch will create a courser-grained measurement than a scale with ticks to the nearest 100th of an inch.

As a general rule, surveyors prefer finer-grained measurements to coarser grained measurements. However, there is a danger with digital technology that our hardware and software will report a more finely grained measurement value than it can practically measure. This is like reporting the volume of a glass jar to the nearest 1/100th of a marble or a distance on a map to the nearest 10th of an inch with a measuring scale only marked to the nearest inch.

## A Note on Analyzing Measurement Quality

We should make a quick note about analyzing measurement quality. As a general rule, when we talk about the precision, accuracy, and granularity of measurements we need to be comparing one set of measurements to another set. It doesn’t make sense to say that a measurement set is “highly precise”, “highly accurate” or “very fine grained” unless we are comparing it to another measurement set.

Example

Consider the list of distance measurements in chain units from above:

50.1 Chains

50.0 Chains

49.9 Chains

50.2 Chains

Are we comparing this measurement set to another set of measurements in chain units? Are we comparing this measurement set to another made in hundredths of a foot? Are we comparing measurements made with an actual chain to measurements made with an electronic distance meter? We need to compare at least two (2) measurement sets and consider their metadata before we can properly answer these questions about the level of precision, accuracy, and granularity of our measurements.

Is this measurement set highly precise? Is it highly accurate? Is it very fine grained? The answer to these questions isn’t logical unless we are comparing this set of measurements to another set. Precise in relation to what? Accurate in relation to what? Fine grained when compared to what?

We’ve just defined three important terms related to the quality of measurements. Now we want to talk about the parts of a typical survey measurement system.

## Parts of the Survey Measurement System

What are the parts of the typical survey measurement system? We need to identify and understand these parts if we are going to have a good handle on survey error analysis and adjustment. For the purposes of this article series, we can state the typical survey measurement system has 6 parts:

1) The observer.

2) The observation instrument.

3) The observation.

4) The observation environment.

5) The observation error.

6) The measurement.

7) The measurement error.

8) Calculations.

9) Calculation errors.

The observer uses the instrument to make an observation. Each observation has an observation error. This observation error is typically related to either the operator, the instrument, or the measurement environment. One or more observations can be used to calculate a measurement. Each measurement will have a measurement error that is related to errors of the observations used to make the measurement. Measurements can be combined and used in calculations. The result of calculations may be other measurements. Calculations may also have errors.

Observation errors and measurement errors can be grouped into three main categories. We will discuss those in the next section.

## Types of Errors

We can group our errors into 3 broad categories:

1) Systematic Errors are caused by problems with our measurement system. They are typically consistent and of the same absolute size or proportional size. An example of a systematic error is an EDM that consistently reads distances longer than they actually are because it is out of calibration. This type of error could be a fixed amount for each distance or a proportional amount that grows with the length of the distance being measured. Systematic errors tend to accumulate into large overall errors visible in the resulting measurements or calculations.

2) Blunders and Mistakes are usually caused by the measurement operator, although they may be caused by the measurement system or measurement environment in rare cases. Blunders and mistakes can be large. (If repeated consistently, a blunder could become a systematic error.) An example of a blunder is a surveyor that flips the digits on a level road reading when recording the reading in his/her notebook.

3) Random Errors are usually small errors caused by the granularity of the measurement, imperfections in the instrument, or variations in the conditions of the measurement environment. An example of a random error is the misreading of a level road by a hundredth or two because of heat shimmer. Random errors tend to cancel each other out. For example, you are as likely to turn an angle on a total station 5 seconds too large as you are to turn the angle 5 seconds too small.

In a future article, we will discuss methods to detect and adjust the errors in each of the three (3) categories. We will also show how random errors are distributed along the geometric shape known as the “bell curve”. Now let’s turn from our discussion of errors, and talk about goals for the MEAA software we want to create as part of our article series.

## Goals for the Survey Measurement Analysis and Adjustment Software

Now that we’ve got a basic understanding of survey measurement errors, what initial goals do we want to set for our MEAA software? Here is a short list of initial goals:

1) Detect and identify all 3 types of errors (systematic, blunders, and random).

2) Analyze measurement qualities (precision and estimated accuracy).

3) Identify observation, measurement or calculation outliers. (Data points that don’t fit well with their neighbors.)

4) Perform fixed and best fit error adjustments.

5) Create basic error analysis and adjustment reports.

## Conclusion

In this article, I defined three (3) terms related to measurement quality and talked briefly about analyzing measurement quality. I also discussed the basic parts of a survey measurement system and the three (3) main categories of survey measurement errors. I finished by sketching out some goals for our MEAA software.

In the upcoming articles for this series, I will talk briefly about methods to identify the 3 types of errors. I will also describe the differences between fixed adjustment methods and best-fit adjustment methods. Then I will show how random errors are distributed along a bell curve.

I’ve written some Java source code that corresponds to the concepts laid out in this article. This source code will serve as the basis for our MEAA software. If you’d like to see the code and learn more about the programming of the MEAA software, visit the following link: