 Welcome to Caltrans LSIT LS exam preparation course. One aid in your preparation for California licensure examinations. A word of caution. Don't use this course as your only preparation. Devise and follow a regular schedule of study which begins months before the test. Work many problems in each area. Not just those in this course's workbook, but problems from other sources as well. This course is funded by Caltrans, but you and I owe a profound thanks to others, the courses instructors from the academic community, the private sector, other public agencies, and from Caltrans as well. We wish you well in your study toward becoming a member of California's professional land surveying community. Welcome to Unit 3 of the Caltrans LS, LSIT video exam preparation course. For the next couple of hours, we will be discussing basic survey measurements. My name is Jeremy Evans, and I am a land surveyor licensed in California since 1983. I work in the Costa Mesa office of Somersen Associates, where I'm the technical manager of the survey department. This unit will serve two purposes. First of all, for those of you studying for the land surveyor and training exam, this unit should prepare you for about two-thirds of the material you will find on the exam. For those of you studying for the land surveyors exam, the information presented here concerning measurement analysis and error theory will be helpful. There is a second benefit to this tape. The information presented here is a bit dry and not real exciting. This allows you to use this tape as a sheer cure for insomnia. Can't sleep? Just pop this tape in the VCR and I guarantee you that you'll be asleep by the middle of precision versus accuracy. Okay, seriously, again, Philip Kissam in his book Survey Practice defines surveying as the art, science, and technology of making such measurements as are necessary to determine the relative positions of points above, on, or beneath the surface of the earth, or to establish such points in a specified position. The key terms in the above definition are art, science, technology, and most importantly to us, measurements. The science and technology of surveying are easy to recognize. The science includes the laws of mathematics and physics that we use every day while making measurements. Technology includes the instrumentation, calculation devices, and mapping technologies, better known as computer aided drafting and design, or CAD, that have become a necessity in current survey practice. The art of surveying is a little harder to define but includes the use of judgment gained through experience that allows us to choose the technologies and procedures to do a project correctly and efficiently. Measurements are the cornerstones upon which the surveyor builds his experience. Without a thorough understanding of the basic survey measurements, a surveyor cannot expect to move on to the more complicated technical issues nor onto the professional issues. This unit of the training program will deal with the basic survey measurements of distance, direction, and elevation. We will also spend time dealing with errors and how they affect our measurements. I would suggest that you take a few minutes now to review the learning objectives and key terms that will be presented in this unit of study. We'll now begin our study of measurements by reviewing the basic principles of measurement. In surveying, we deal with five basic types of measurement. They include horizontal angles indicated by the angle at O, subtended from A to B. Horizontal distances indicated by the lines from O to A, O to B, and A to B. Vertical angles indicated by the angle at O, subtended from A to C. Vertical distances indicated by the line from A to C. And last but not least, slope distances indicated by the line from O to C. Again, these five basic survey measurements that you must understand how to measure or calculate before you can move on to the higher technical issues. In California, the basic units in surveying are feet and hundreds of a foot for linear measurement or length. This can be horizontal distances, vertical distances, or slope distances. Degrees, minutes, and seconds for angular measure, which includes both horizontal angles and vertical angles. Areas expressed in square feet or acres, while volume, generally used in excavation, is expressed in cubic yards. Another basic mathematical concept you should be familiar with is significant figures. In recording measurements, an indication of the accuracy attained is the number of digits or significant figures recorded. The number of significant figures in any measurement includes the certain or definite digits plus one and only one estimated digit, which is questionable. If a number is recorded as 129.85 feet, as shown here in the sixth numerical value, the measurement has five significant figures. Of which the first four are certain and the last digit is estimated, making it questionable. In the field, it is important that measurements be recorded with the correct number of significant figures. If fewer significant figures are recorded, then the accuracy is lessened and the time spent acquiring the data is wasted. Too many significant figures leads to false accuracies and problems with future computations. The number of significant figures is often confused with the number of decimal places. While decimal places may have to be used to maintain the correct number of significant figures, they do not themselves indicate the number of significant figures as can be seen here in the number 0.000213, which has only three significant figures. Exact numbers, such as 5,280 feet in a mile, have an infinite number of significant figures. In survey computations, it is important that calculations be consistent with measured values. Again, we do not want to indicate false accuracies in our answers. In the addition problem shown here, the answer must be rounded to the rightmost column that has significant figures for all the values being added. In other words, the four in the value 1.4. The answer must be rounded to the 10th place since this is the rightmost column that has a significant figure for each value being added. In the multiplication problem shown, the answer must be rounded to the least number of significant figures found in any factor. In this case, the answer must be rounded to three significant figures since this is the number of significant figures in the factor 2.16. At this point, you may want to stop the tape and try the significant figure problems found in your workbook. One last comment about significant figures. It is common practice in survey computations to carry one more digit than is required in the answer and then round to the correct number of significant figures. Measurements can be direct or indirect. Direct measurements are simply that. They are made directly on the ground. They include taping or measuring a distance with an EDM, measuring an angle with a theodolite, or determining an elevation with a level. On the other hand, indirect measurements involve some type of mathematical computations. Examples include area calculations from measured field data or intersection calculations like a bearing-bearing intersection used to determine the center of a section in public land surveying. Many indirect measurements are made in surveying. Therefore, a thorough knowledge of geometry and trigonometry is essential. This brings us to one of the most important and yet most overlooked topics in surveying, the analysis of measurements, more commonly called error theory. Now don't let the word theory scare you. The analysis of measurements is not that difficult to understand and the mathematics are relatively simple. You must understand that above everything else a surveyor does, he is an expert in measurement. As a matter of fact, in a court of law, the only area of expertise that a surveyor can testify to is that of measurements. While the surveyor may be the best available source of knowledge on boundary determination, he is not considered an expert on the subject since the ultimate decision on boundary disputes lies with the court itself. Getting back to measurements, the best way I know to define measurement is to compare it with counting. Counting is an absolute. I have five fingers on my hand. I have 32 cents in my pocket. These two values are not approximate values. They are absolutes. Measuring on the other hand is an estimate of a value and always contains some error. I can estimate that I am approximately 15 feet from the camera. This distance is an estimate and is obviously not very exact. It does contain error. If I used a tape to measure my distance to the camera, I would get a better estimation of the distance, but it would still be an estimate and there is still some error in the measurement. No matter how carefully you measure something, it will always be an estimate. Measurements are not exact. All measurements contain some error. Any surveyor who tells you that his measurements are right on really doesn't understand what he is saying. Being an expert in measurement means not only being able to make a measurement in the field or a calculation in the office, but also understanding the quality of the measurement made and being able to determine the equipment and procedures necessary to achieve a required accuracy. Without understanding these basic principles, a surveyor cannot move on to the higher technical issues in surveying. As I mentioned before, all measurements contain some error and therefore no measurement is exact. Also, the true value of any measurement is never known and therefore the exact error present is always an unknown. With this in mind, let's look at the formula for determining error. The error in any measurement is equal to the measured or observed value minus the true value. Now, I just stated that the true value is never known, so how can it be used in this equation? Our job as surveyors is to estimate a most probable value for this true value. This most probable value is usually the mean of a series of measurements. We'll get back to most probable values in a few minutes. What I'd like to talk about now is the sources of error in surveying. Natural errors include the effects of wind, temperature, humidity, refraction, gravity, and magnetic declination. An example would be heat waves that cause a target to jump while looking through an instrument on a hot day. Instrumental errors are caused by the imperfections in the construction of a survey instrument. The graduations on the circle of a theodolite may not be spaced evenly, which can cause an error in reading the instrument. Most instrument errors can be eliminated by correct survey procedures, like measuring all angles twice, once with the telescope of the instrument in the direct position, and once with the telescope in the plunged or reversed position. Personal errors arise principally from the limitations of the human senses of sight and touch. Examples include the ability to estimate a value on a scale or not being able to hold a rod vertically. Each of these will introduce a small error to the measurement. Now that we know the sources of errors in our measurements, let's discuss the type of errors. There are actually only two error sources in survey measurements, systematic errors and random errors. Systematic errors follow mathematical and physical laws. Their magnitude and algebraic sign can be determined, and therefore systematic errors can be removed from survey measurements by applying corrections or using field procedures that remove the error or applying corrections to the measurements after they are made. Examples include removing the effect of temperature from taping measurements by using a correction formula or removing the effect of an instrumental error by measuring an angle twice with the telescope in both the direct and plunged positions. Random errors are the errors that remain after systematic errors are removed from the measurement. Remember that no measurement is exact. It will always contain some error. Random errors are caused by factors beyond the control of the surveyor and follow the laws of probability. The magnitudes and algebraic signs of random errors are a matter of chance. The errors tend to be small and compensating in nature, and while they cannot be eliminated from the measurements, with some experience, the surveyor can estimate their value. Our discussion of measurement analysis is a study of these random errors. There is another type of error that is really not an error at all, but still must be mentioned here. This is the blunder or mistake. These are personal in nature and include writing down the wrong measurement or writing down the measurement incorrectly. Transposition of figures is probably the most common mistake a surveyor makes. Mistakes can be detected by making repeated measurements of the same value. When a mistake is detected, the measurement should be repeated. One last area of confusion we need to clear up before we get to measurement analysis is the difference between accuracy and precision. The best way to define these two is to use the classical example of a bull's eye. Suppose you take your gun to the shooting range for some practice. You aim very carefully at the bull's eye, take a deep breath, exhale, and then take the 10 shots shown here. Your shots are very close together, but they miss the bull's eye. Your shots are very precise, in other words, close together, but not very accurate. They miss the bull's eye. Your gun has some type of aiming problem. This problem can be fixed so it can be considered a systematic error. Now suppose you have your gun fixed and go back to the range. This time, you're not quite as careful with your aiming before you take your 10 shots. The result is shown here. The shots are scattered around the target but are centered around the bull's eye. These shots are not as precise as before but are considered more accurate. As you have seen, precision does not necessarily equate with accuracy. You may tape a distance very carefully and get very precise results, but if you fail to take into account the tape is short by five hundredths of a foot, you will not be accurate. The goal of the surveyor, and this is very important, is to always get accurate results with the precision necessary for the project at hand. Your precision does not have to be as high for say, topo work as it does for boundary control work. Again, let me emphasize that your work must always be accurate or you have wasted your time but the precision can vary depending on the project at hand. This brings us to the study of error theory. Before we can begin looking at a series of measurements to define the precision and accuracy, all systematic errors and mistakes or blunders must be eliminated from the measurements. Remember that systematic errors have magnitudes and algebraic signs that can be determined and therefore can be removed from the measurements. Mistakes or blunders must be detected and removed and then the measurement should be made again. Let's look at a group of measurements and begin our study of random errors. Shown here is a set of 12 measurements of the same angle. In other words, we've measured the angle 12 times. As you can see, the values for the individual measurements differ slightly. All systematic errors and mistakes have been eliminated. Our first task is to determine the most probable value for the angle. In surveying, we generally use the mean as the most probable value. The mean is defined as the summation of all the measurements divided by the number of measurements. In our example, the number of measurements would be 12. At this point, I would like for you to stop the tape and determine the mean of the 12 measurements. Since all the measurements have the same value for the degree and minute portion, I would suggest you work only with the seconds portion of the measurement. Let's look at the results of determining the mean. As you can see, I've added up the second portion of each measurement and with the result being a value of 540. When I divide this number by the number of measurements, in this case, 12, I get an answer of 45. The mean of this set of measurements is 56 degrees, 23 minutes, 45 seconds. This is the most probable value for the set of angles. Remember that the true value can never be actually determined. The mean is our best estimation of this value. Our next step is to determine the precision of the set of measurements. To determine the precision, we must first determine the residual for the 12 individual measurements. A residual is defined as the difference between the value of each individual measurement and the mean of all the measurements as shown by the formula. Residuals are theoretically identical to errors, except that the residual can be calculated and errors cannot. Take time now to determine the residual for each measurement. In our example, and note them in the column under the small r. Here is the result of calculating the residual for each measurement. For measurement number one, the observed value minus the mean equals zero. For measurement number two, the residual equals one and so on down the line. If you total the residuals for each measurement, the answer should be approximately zero. In our example, the sum actually is zero. This indicates that the residuals or errors are in fact compensating and tend to be small values. At this time, it's usually a good idea to create a histogram from the information we have calculated so far. A histogram gives us a graphical representation of the precision of our work. In the histogram, we plot the value of the residual shown along the bottom line versus the frequency of each value, here shown along the left side. The negative two residual occurs one time. The negative one value three times, and so on. When you connect the location of each value with a smooth curve, the result is the standard probability curve. This graphical representation will highlight any values that might contain a systematic error or a blender that were not removed prior to determining the mean. These will appear as spikes or valleys in the histogram. If these are present, they should be removed and the mean and residuals recalculated. Steep-sided curves indicate high precision. The residuals are grouped closely together. Shallow-sided curves indicate low precision. The residuals are more spread out. After you have completed this course, you may want to take some survey data and determine the mean and residuals and then plot the histogram to compare precision. To accurately determine the precision of the data, we must next determine the standard deviation or standard error as it's sometime called for our measurement data. Here is the formula for calculating the standard deviation. The standard deviation or error equals plus or minus the square root of the sum of the residuals squared divided by the number of measurements minus one. The result is plus or minus because the algebraic sign of a random error cannot be determined. The sum of the residuals squared is determined by squaring each residual then adding up the results. This formula is a standard formula used to determine the precision of any group of measurements and is developed from statistical analysis. Take time now to determine the standard deviation for the example. Let's see how you did. Here I've determined the residuals squared for each residual. I've totaled these values and gotten 14. By inserting this value along with the number of measurements into the formula for standard deviation, I've come up with a value of plus or minus 1.13 seconds which can be rounded to plus or minus one second. Remember that all random errors have a plus or minus value since the algebraic sign cannot be determined. The standard deviation establishes the limits within which measurements are expected to fall 68.3% of the time. Understand that this 68.3% value is a constant established by the laws of probability and we will accept it without any further discussion because if I tried to explain how it was determined we'd be here for the next three weeks. Therefore, in our example of 12 measurements it would be expected that eight of the 12 measurements would fall within the range of 56 degrees, 23 minutes, 44 seconds, to 56 degrees, 23 minutes, 46 seconds while four of the measurements would be outside the limits. In our example, 10 of the measurements actually fall within the plus or minus one second range. Another interpretation could be that if one or more measurements were made it would have a 68.3% chance of falling within the standard deviation range. A third interpretation would be that the true value, and remember that we cannot actually measure the true value, has a 68.3% chance of falling within the standard deviation range. You should be able to see that the standard deviation is a comparison of the mean to the individual measurements that it was established from. By comparing the mean to the individual measurements we have determined the precision of our measurements. This is the most important thing to remember about standard deviation. It is a measure of the precision of the measurement we are making. As a surveyor, you probably wouldn't want to tell your client that you're only 68.3% certain about the results of your measurements. A 90% or 95% level of confidence would be much better. You can determine different levels of confidence by multiplying the standard deviation by a constant that, again, has been determined by the laws of probability. Shown here are the three most common levels of confidence that a surveyor should be familiar with. The 50% error or E sub 50 is the so-called probable error. It establishes the limits within which the measurement should fall 50% of the time. In other words, a measurement has the same chance of falling within the limits as it has falling outside the limits. The 50% error is calculated by multiplying the standard deviation by 0.6745. In the past, the 50% error was used extensively in the discussion of random error, but is seldomly used today. However, it does occasionally show up on the LS or LSIT exam. The 90% error or E sub 90 establishes the limits within which the measurement should fall 90% of the time. It is calculated by multiplying the standard deviation by 1.6449. In our example, the 90% error would be plus or minus 1.13 seconds times 1.6449, which equals plus or minus 1.86 seconds, which is then rounded to plus or minus two seconds. Referring back to our example, you will see that all the residuals fall within this limit. The 90% error is the standard used today in the survey industry. Another statistical value that we should be aware of is the standard error of the mean. The mean or most probable value has some error with respect to the true value of a measurement. This difference is determined by calculating the standard error of the mean. In this formula, the standard deviation or standard error is divided by the square root of the number of measurements. Again, this is a plus or minus value since the sign of a random error cannot be determined. If the standard deviation is not required, you can calculate the standard error of the mean directly from the measurements by taking the square root of the sum of the residual squared divided by the number of measurements times the number of measurements minus one. Of course, this is a plus or minus value. In our problem, the standard deviation, which is 1.13, is divided by the square root of the number of measurements and is equal to plus or minus 0.33 seconds. Since we are comparing the mean to the true value here, it can be said that the mean value in our example has a 68.3% chance of falling within plus or minus 0.33 seconds of the true value of the angle. By using the 90% error in the standard error of the mean equation, we will get the 90% standard error of the mean value. Comparing the mean to the true value is a measure of accuracy. Therefore, the most important thing to remember here is that the standard error of the mean is a measure of accuracy of the figure being measured. Again, I must emphasize that if any systematic error or mistake exists in the measurement data, then these calculations of the standard deviation and the standard error of the mean are invalid. So far, we have been discussing the error found in a single value that has been measured numerous times. Seldomly, if ever, is a survey project made up of a single measurement. Our projects are usually made up of numerous measurements of distance, direction, and elevation. Let's now see how the errors within the individual measurements propagate through a series of measurements. Sometimes, a series of similar quantities, such as the angles of a traverse, are read with the same care. Each measurement will be an error by about the same amount. The total error in the sum of all measured quantities of such a series is called the error of the series and is designated by the E-sub-series. If the error E is the same in each measurement and there are N number of measurements, then the error in the series will be plus or minus E times the square root of N. Suppose that when a party chief and his chain man use 100 foot tape, there is a plus or minus 0.02 foot error from all the error sources. What would be the error if this team had to measure a 5,000 foot distance? In this case, the number of tape lengths in this 5,000 foot distance is 50 and the error of this series would be equal to plus or minus the error, which is 0.02 feet, times the square root of the number of measurements, which is 50. The answer would be plus or minus 0.14 feet. Using a similar technique for each of the 50 tape lengths in a 5,000 foot distance will produce an error of plus or minus 0.14 feet. This answer illustrates that random errors tend to be compensating. If each tape length definitely contains 0.02 feet of error, then the total error in 5,000 feet would be one foot. This is the product of 0.02 times the 50 lengths. However, in random errors, errors have a plus or minus value and therefore they tend to compensate over the 5,000 foot distance. Therefore, the error in the 5,000 foot distance is less than the absolute maximum error of one foot and in this case is plus or minus 0.14 feet. One other point needs to be made here. If you want to determine the precision of your 5,000 foot measurement, then you use the standard deviation or 90% error for the value of E in the equation. If you want to determine the accuracy of your 5,000 foot distance, then you should use the standard error of the mean or the 90% standard error for the value of E. It is important to identify what it is you're trying to determine, either precision or accuracy before you do your calculations so that the correct formulas and procedures are used. A special type of error in a series is the error found in electronic distance measurement or EDM measurement. As you all are aware, EDMs are used for most distant measurements today. We'll discuss the specifics of EDMs a little later on. Right now we'll discuss the error found in EDM measurements. Here we have the formula for determining the error in EDM measurement. In the specifications that were sent with the EDM, there is a value given for the accuracy of the instrument. For the geotometer 410, this value is plus or minus five millimeters, which is equal to 0.016 feet plus five parts per million. This equation indicates that there is an error plus or minus five millimeters in every measurement, regardless of the length of the line, plus an additional error, which is dependent on the length of the line. This additional error is a proportion to the value of five parts in a million. What this means is that if we could measure one million feet with this EDM, it would create a five-foot error. By proportioning this ratio to the length of the line being measured, we can determine the total error. In this example shown here, we have measured a line one mile, or 5,280 feet long with an EDM. The total error for this line would be equal to the plus or minus 0.016 foot error found in every measurement, plus five parts times the ratio of 5,280 feet. This proportion is equal to 0.026 feet. The total error would therefore be plus or minus 0.016 feet, plus 0.026 feet, which equals plus or minus 0.042 feet for the one mile line. This error is a standard error of the mean value and is therefore a measure of accuracy. The next type of error propagation that you should be familiar with is the error in a sum. In this case, we are determining the error of a group of measurements that do not have the same error for each measurement. Our formula shows that the error in a sum is equal to plus or minus the square root of the error in measurement A squared plus the error in measurement B squared plus the error in measurement C squared, et cetera. Suppose that we measure each angle in a triangle under varying conditions. The error in angle A is plus or minus 10 seconds. The error in angle B is plus or minus two seconds and the error in angle C is plus or minus five seconds. The total error for the angles in this triangle is equal to plus or minus the square root of 10 squared plus two squared plus five squared which is equal to plus or minus 11.4 seconds. This answer demonstrates that the total error in a group of measurements having varying errors is only as strong as its weakest link or angle in this case. Measuring the other two angles more precisely than the weakest angle does not improve the overall precision of the sum of the errors. You now have the tools necessary to begin determining the error in most of your survey measurements. Take time now and try to solve the point location problem in your workbook. Let's see how you did in determining the positional error in point C. First, we'll determine the error in the distance measurement. We measured a distance of 4,350 feet using an EDM that had an accuracy of plus or minus 0.016 feet plus five parts per million. Substituting this data into the formula for EDM measurement, we get plus or minus 0.016 feet plus five times the ratio of 4,350 over one million which works out to be plus or minus 0.038 feet. Now, solving for the error in the angle, we can use simple trigonometry. The linear error value is equal to the tangent of 10 seconds which is the error in our angle times the length of the line 4,350 feet which equals plus or minus 0.211 feet. Remember that the tangent of an angle is equal to the opposite side which is the error amount divided by the adjacent side which is the length of the line. By rearranging this formula, we get the error at point C equals the tangent of the angular error times the distance of the line. Now, since we have two error sources with different values, we can use the error in a sum to determine the positional error of point C with respect to point A. This is equal to plus or minus the square root of 0.038 squared plus 0.211 squared which equals plus or minus 0.214 feet. Remember that if we want to determine the precision of our measurements from A to C, then the angular error and distance error must be the standard deviation or 90% error. If we want to determine the accuracy of point C, then the angular error and the distance error must be the standard error of the mean or the 90% error of the mean. Let's take this idea of error propagation one step farther. You can use the study of errors to predetermine the outcome of a survey. Take time now to try the estimated traverse closure problem in your workbook. In this problem, we know the positional error for each side of the traverse and we can therefore determine the positional error for the entire traverse or as it's more commonly known, the error of closure. Here I have calculated the positional error for each side of the traverse based on the data given. Since each of the values is different, we must use the error of a sum to calculate the total error for the traverse. This is equal to plus or minus the square root of 0.212 squared plus 0.176 squared plus 0.144 squared plus 0.230 squared, which is equal to plus or minus 0.387 feet. This would be the expected closure of the traverse based on the data given. If you calculate the actual error of closure, it is 0.311 feet. Since the actual closure is better than the estimated closure, the work was done correctly and there probably are no mistakes in the field data. You can also use this estimated error to determine the equipment and procedures necessary to have a survey conform to a classification of accuracy. In your workbook, you'll find the classifications of accuracy from the Caltran Survey Manual. In table 4-04-B, there is an accuracy requirement for second order modified traverse that states that the positional closure after azimuth adjustment will not exceed 1.67 feet times the square root of the length of the traverse in miles or one part in 10,000, whichever is smaller. If you are planning a traverse that is two miles long, then by the requirements mentioned above, the survey must have an error of closure of not greater than 1.06 feet. By knowing this and the approximate dimensions of the sides of the traverse, you can determine the equipment and procedures necessary to achieve this requirement. For a complete study of this subject of error propagation and estimating survey requirements, get a copy of Ben Buckner's book, Survey Measurements and Their Analysis, which is noted in the bibliography in your workbook. The last topic we need to touch on before we move on to different types of survey measurements is the different types of adjustments we can use to ensure that our surveys are mathematically correct. In some types of surveys, the sum of several measurements must equal a fixed value. For example, the sum of the interior angles of a triangle has to total 180 degrees. In practice, therefore, the measured angles of a triangle are adjusted to make them add up to the required total. Correspondingly, distances may be altered slightly to meet certain requirements. In making these adjustments, the principles of probability are most important. The most common type of survey adjustments are the proportional or linear adjustment. This is an adjustment used in simple level circuits and traverses. In adjusting the survey data, the adjustment is simply a proportion of the total error total distance ratio. For example, if a simple level circuit has a closure error of 0.25 feet and is five miles long, then a benchmark at the one mile mark will get one-fifth of the adjustment or 0.05 feet. The compass rule and transit rule adjustments used in the adjustment of traverse data are also proportional adjustments. Level adjustments and traverse adjustments are dealt with in detail in their appropriate unit of this review course. The most important thing to remember about these adjustments is that the errors are considered to be equal for each leg of the level circuit or traverse. There cannot be any other error considerations with proportional or linear adjustments. It is evident that some measurements are more precise than others because of better equipment, improved techniques, and superior field conditions. In this situation, it is better to assign relative weights to the individual measurements. When this is done, the stronger measurements receive the least amount of adjustment, while the weaker measurements receive the greater amount of adjustments. This is the theory behind using weighted means and least squares adjustment. Least squares is an adjustment where the sum of the residuals squared for the set of measurements is minimized. It is a statistical analysis where each measurement is weighted and all measurements are adjusted simultaneously. The result is considered a superior adjustment to proportional or linear adjustments. The study of weighted means and least squares is a class all by itself, and we won't go any farther here. It should be realized, however, that any adjustment beyond the very simplest should be done by a least squares adjustment. With the new least squares adjustment packages on the market today, which make the input and adjustment relatively simple, least squares adjustment is becoming a very common method of adjustment of almost all survey measurements. It is time now to begin our study of the different types of measurements we have in surveying. We'll begin our study of survey measurements with that of distance. To review the three types of distances that we are interested in are horizontal distances, shown here by the lines A to O, B to O and A to B, vertical distances shown here by the line AC, and slope distances shown here by line OC. Distance measurement is the basis of all surveying. In plain surveying, the distance between two points means the horizontal distance. If the points are at different elevations, the distance is the horizontal length between the two vertical lines of the points. The horizontal distance that relates to the slope distance OC is OA measured to the vertical line AC. There are several different methods of measuring horizontal distances. These methods include pacing, odometer readings, optical range finders, tachyometry, better known here as stadia, subtense bars, taping, and electronic distance measurement. We'll discuss taping and EDM measurements in detail in a few minutes. Let's briefly discuss the others now. Pacing distances are sufficiently accurate for many purposes in surveying. Determining the general location of a monument from a found monument is a common example. Once you're in the general area of the monument, you can use a metal detector or similar device to actually locate the monument. Pacing can also be used to detect blunders and tape distances. Pacing is simply counting the number of steps or paces in a required distance. Before this is done, you must determine the length of your pace or the number of paces per a certain distance like 100 feet by walking with natural steps back and forth several times over a level measured course. With practice, you should be able to pace within a foot over a distance of 100 feet if the terrain is open and reasonably level. An odometer converts the number of revolutions of a wheel of known circumference to a distance. The odometer in your car is suitable for some preliminary work in route location work. More common is the measuring wheel that is pushed by hand and the distance then read on the odometer connected to the wheel. Accuracy with this type of device on level ground is approximately two feet per 100 feet. Optical range finders operate on the same principle as the range finder in your single lens reflex camera. The operator looks through the lens and adjusts the focus until a distance object viewed is focused in coincidence, whereupon a distance reading is obtained. Accuracy is about two feet per 100 feet up to a distance of 1,000 feet. Beyond this distance, the accuracy drops rapidly. Techeometry or STATIA as it is more commonly known is a surveying method used to quickly determine the horizontal distance to and the elevation of a point. STATIA measurements are obtained by siding through a telescope equipped with two or more horizontal crosshairs, commonly called STATIA hairs, set at a known spacing. In the United States, the crosshairs are usually set at a ratio of one to 100. When looking at a leveling rod held vertically at a distance of 100 feet from the instrument, the distance between the two STATIA crosshairs will be one foot. When the distance between the two STATIA hairs is 1.59 feet, the rod is 159 feet away. STATIA is still used in many parts of the United States for doing topographic surveying. Our field crews at SOMAS are all trained in STATIA just in case one of their electronic total stations fails on a critical topo job that is far away from the office. And if this should happen, they can use the backup the out-of-light they carry and use STATIA to continue until a replacement instrument arrives. STATIA is accurate to one foot and 500 feet for horizontal measurement. The substance bar can be used along with a precise the out-of-light to determine distance. The bar is usually made of in-bar and targets at the end of the bar are a known distance apart, usually two meters. By setting the substance bar on a tripod at one end of a line and measuring the angle subtended by the bar with a the out-of-light at the other end of the line, the horizontal distance can be computed by trigonometry. An accuracy of one foot and 3000 feet can be easily obtained with this setup. The substance bar method of distance measurement was often used in the past to obtain distances over inaccessible courses, for example, over bodies of water. EDM devices have now almost totally replaced this procedure. Before we get into our study of tape distances, let's take a break here and get something cold to drink. I'll see you in about 10 minutes. Yeah, welcome back. Let's now discuss distance measurement by taping. Many of you may be wondering why we are discussing taping when most of the distance measuring we do today is done with electronic distance measuring devices and global positioning systems. This is a valid question since very little taping actually is done nowadays. I have two reasons for studying taping. One is that taping for distances under approximately 300 feet over level ground with few obstructions is still more accurate than using an EDM. At Somus, we use taping to pull street intersection ties so we can accurately replace centerline monuments. These tie distances are generally under 100 feet and taping is a more accurate procedure. We also always use a tape when making measurements that involve high rise construction. High rise construction utilizes steel beams that are pre-cut to plus or minus one quarter of an inch. The only way to guarantee that the steel will fit together correctly is to use a tape for measurement. The second reason to study taping has to do with our continuing study of measurements and their associated errors. There are several taping errors that we will discuss in the next few minutes. For the person new to surveying, these errors are easier to visualize than some of the errors associated with angular measurement or leveling. This allows the student a chance to further understand how errors affect our measurements and how the surveyor must constantly analyze his or her measurements. So the study of taping not only introduces us to the subject, it also improves our understanding of the basics of measurement and the effect errors have on these measurements. Now down off my soapbox and back to the subject at hand, taping consists of applying the known length of a graduated tape directly to a line and number of times. Two types of situations arise. First is measuring an unknown distance between fixed points, such as hubs in the ground. And the second is laying out a known or required distance with only the starting mark in place. Taping is performed in six steps. First is lining in or making sure the surveyor at the lead end of the tape is online. This is done by the rear chainment. The next is applying the correct amount of tension, either by using a tension handle for a precise measurement or by estimating the tension from experience. Next is plumbing with the plumb bobs if the tape must be held off the ground to avoid obstacles online. Fourth is marking the tape lengths on the ground and making sure the number of measured tape lengths is accurately kept. Next is reading the, is the reading of the tape at the end of the line by reading directly if the tape is on the ground or sliding the plumb bob string along the tape until the plumb bob is directly over the point. And finally, recording the distance correctly in your field notes. When using a tape, a surveyor usually employs procedure that result in a horizontal distance being measured. On level ground, horizontal taping is relatively simple. When the ground slopes, the procedure of breaking tape can be employed. This involves measuring sections of the total line, usually in the downhill direction by holding the end of the tape on the hub in the ground and measuring as much of the 100 foot tape length as possible until plumbing becomes too difficult, usually at shoulder height. The rear chain then then moves up to this point and the next portion of the line is measured in the same manner. When breaking tape is impractical, a slope distance can be measured and the vertical angle or difference in elevation of the two points can be used to calculate the horizontal distance. We'll talk more about slope distance reduction when we talk about EDMs. In the United States, there are two basic types of tapes, also called chains after the historical chains that were used well into the 20th century. These are the ad tape, which is what I grew up with in surveying, and the graduated throughout tape. The ad tape is calibrated from zero to 100 by full feet in one direction and has an additional foot beyond the zero end, graduated from zero to one foot in tenths or more commonly in tenths and hundreds in the other direction. This makes the complete tape 101 feet long. When the full graduation is held by the rear chainment at a hub, shown here as being at the 81 foot graduation, the ad end should straddle the point being measured to and the chainment reads the additional length. Again, shown here as six tenths of a foot. The total distance between the hubs is therefore 81 plus six tenths of a foot for a total of 81.6 feet. When the graduated throughout tape, the rear chainment simply holds zero on the point being measured from while the chainment reads both the whole foot and the portion thereof. On our example, the distance is 41.4 feet. Many servers prefer this type of tape because they feel less errors are made if only one person reads the tape. Whichever method isn't used, care must be exercised in recording the distances in the field notes. One blunder here can cancel many hours of very precise work. There are three fundamental sources of error in taping. First is the actual tape length differing from its nominal or marked length due to defects in the manufacture of the tape. This is an instrumental error. Next is the variation in tape length due to temperature, wind, and the weight of the tape itself. These are natural errors. Finally are the errors due to the chainman's ability or inability to read and manipulate the tape. These are personal errors. Let's see how these errors affect our measurement of distances by tape. The first error source we will consider is the incorrect length of tape. This is a systematic error. Tape manufacturers do not guarantee steel tapes to be exactly their nominal length, for example, 100 feet, or provide a standardization certificate unless requested and paid for as an extra. The actual length of the tape is determined by comparing it with a tape of known length. The National Bureau of Standards will make such a comparison for a small fee and will certify the actual distance between the ends of the tape under given conditions of temperature, tension, and type of support. In the area where I work, the County of Orange will do a comparison of your tape to a tape that has been tested at the Bureau of Standards. In our office, we have a tape that has been standardized that we keep specifically for comparing with other tapes. The standardized tape is never sent out in the field. An error due to incorrect length of tape occurs each time the tape is used. If the actual length of the tape is known and is different than the nominal length, the correction can be determined and applied from the formula shown, where the correction to be applied, noted here as C sub L, is equal to the actual tape length minus the nominal tape length divided by the nominal tape length times the measured length of line. This correction is then added to the measured length of line to arrive at the actual length of the line. Make sure you note the algebraic sign of the correction so it is applied correctly to the measured distance. Here we see graphically the situation where a tape is actually shorter than the indicated or nominal length by three hundredths of a foot. If the actual distance between the hubs is 100.00 feet, then this tape will read 100.03 feet. This can also be shown by applying the correction formula. Here the actual tape length is three hundredths short, which equals 99.97 feet. The nominal length is 100 feet and the measured distance is 100.03 feet. The correction is calculated to be negative 0.03 feet and when added to the measured distance equals 100 feet, the actual distance between the hubs. From a practical standpoint, the effect of any error in taping, whether it be due to temperature, tension, sag, et cetera, which we'll talk about in a minute, will make the tape length incorrect. Note that the actual distance equals the measured distance plus a correction and the proper algebraic sign is built in. This is true for all taping corrections. However, you should still try to reason whether a certain condition makes a tape too long or too short and apply the correction accordingly. Drawing a sketch like I have here is a great help in helping you reason or visualize an answer. One of the most common corrections made is that for temperature other than the temperature at which the tape was standardized. Steel tapes will expand and contract depending on the temperature of the tape at the time of making the measurement. The coefficient of thermal expansion and contraction of steel normally used in tapes is approximately 6.5 times 10 to the negative six per unit length per degree Fahrenheit. The actual length of a line measured at a temperature other than standard is equal to the measured length plus the product of the coefficient of the thermal expansion and contraction of steel times the tape temperature at the time of observation minus the tape temperature at the time of standardization, which is usually 68 degrees Fahrenheit, times the measured length of the line. Take some time now to work the temperature correction problem in your workbook. Let's check your results. I've worked the right side of the equation first. The value for the thermal expansion and contraction of steel is multiplied by the sum of the difference between the temperature of the tape at the time of the measurement, 45 degrees Fahrenheit, and the temperature of the tape at the time of standardization, which was 68 degrees Fahrenheit. And this product is multiplied by the distance measured, 850.44 feet. This equals a negative 0.127 feet. This correction is then added to the measured distance, again, 850.44 feet, which equals 850.313 feet, which is then rounded to 850.31. Another correction you should be familiar with is the correction for inconsistent pull. When a steel tape is pulled with a tension greater than its standard, the tape will stretch and be longer than its standard length. Conversely, if less than standard pull is used, the tape will be shorter than its standard length. Accurate tension, or pull, is measured with a spring balance. With experience, you will be able to pull within a few pounds of a specific tension without the spring balance. The modulus of elasticity of steel of the tape regulates the amount that it stretches. The actual length of a line measured with a tension other than standard is equal to the measured length of the line plus the difference of the tension applied at the time of measurement and the tension applied at the time of standardization times the measured length of the line divided by the product of the cross-sectional area of the tape and the elasticity of steel, which is usually 29 million pounds per square inch. Try the correction of tension problem in your workbook now. Let's work through the problem. Again, I'll work the right side of the equation first. First, I computed the difference between the tension used at the time of measurement, 20 pounds, and the tension at which this tape was standardized, 12 pounds, giving me a result of 8. This value is then multiplied by the quotient of the length of the line measured 650.45 feet divided by the product of the cross-sectional area of the tape, 0.005 square inches, and the elasticity of steel, 29 million pounds per square inch, giving us a correction of positive 0.036 feet. When this correction is added to the measured length of the line, the actual length of the line is 650.486 feet, which is rounded to 650.49 feet. The effect of tension is usually relatively small and is only needed for the most precise measurements. Another systematic taping error is that of SAG, a steel tape when not supported throughout, SAG's in the form of a cantonary, the same as the cable on a suspension bridge. SAG shortens the horizontal distance between the ends of a tape. SAG can be diminished by greater tension, but cannot be eliminated unless the tape is supported throughout. The actual SAG of the tape is not important. The reduced distance between the ends is critical. The formulas shown here are used to compute the SAG correction. There are two formulas because the effect of SAG is different for a full tape length as compared with a partial tape length. The first formula would be used for a partial tape length, and the second formula would be used for the full tape length. For the partial tape length, the correction for SAG is equal to negative the weight of the tape per foot squared times the length of the unsupported tape cubed divided by 24 times the tension applied at the time of measurement squared. For a full tape length, the correction for SAG is negative the total weight of the tape squared times the unsupported length of the tape divided by 24 times the tension applied squared. The formulas are a negative value because the effect of SAG always reduces the length of the tape. The sum of the corrections due to SAG are then added to the measured length of line to give the actual length of the line. Try the correction for SAG problem in your workbook. Let's look at the results for the correction for SAG. The correction for each of the three 100 foot lengths would be negative the weight of the tape, 1.5 pounds squared times the unsupported tape length, which is 100 feet, divided by 24 times the tension applied being 12 pounds squared. This equals negative 0.065 feet per 100 foot tape length. The effect of SAG for the 50.42 foot portion of the measurement is equal to negative the weight of the tape per foot squared. The weight per foot was determined by taking the total weight, 1.5 pounds, and dividing by 100 times the length of the unsupported tape cube, this being the 50.42 foot portion, divided by 24 times the tension applied squared, the tension being 12 pounds. This is equal to negative 0.008 feet. The sum of the corrections for SAG is determined by multiplying the correction for the full tape length, negative 0.065 feet times the number of full tape lengths being three, and subtracting the SAG correction for the partial tape length, which is 0.008 feet. This total correction is negative 0.203 feet, which is then added to the measured length of line, resulting in the actual length of line being 350.22 feet. As you can see, the effect of SAG can be rather large. It is important to note whether your tape was standardized, supported throughout, or supported at the ends only. If this is known, then the effect of SAG can be properly corrected. The last systematic error in taping that I would like to discuss is the correction for poor alignment. If one end of a tape is offline, or the tape is snagged on an obstruction, an error occurs. When the end of a tape is set offline by 1.4 feet, the error in the measurement will be 1.00th of a foot. If the center of 100 foot tape is caught on brush and is one foot offline, the error will be 2.00ths of a foot in the 100 foot length. Errors resulting from poor alignment always make the recorded length longer than the actual length of the line. With practice, the rear chainmen should have little trouble keeping the chainmen well within a foot of the correct course. The formula to correct for poor alignment is negative the distance a tape is offline squared divided by two times the length of tape. The actual length of the line is equal to the measured length of the line plus the sum of the individual alignment corrections. Try the problem for correction for poor alignment in your workbook now, shown here is the result for the problem on correction for poor alignment. The correction for the first 100 foot tape length is negative the distance the tape is offline, 1.26 feet squared divided by two times the tape length, which is 100 feet. The answer is then negative 0.008 feet. The calculations for the remaining three tape lengths are shown with the results being negative 0.005 feet, negative 0.032 feet, and negative 0.027 feet. The sum of the corrections is negative 0.072 feet and when this correction is applied to the measured length of line, the result is 350.49 feet for the actual length of the line. We have seen the results of several types of errors in taping. Remember that these errors for temperature, incorrect tension, sag and poor alignment are systematic and can therefore be removed from the measurement either by applying certain methods and techniques in the field or by calculating corrections after the measurements have been made. All of these taping errors result in the fact that a nominal 100 foot tape is either longer or shorter than 100 feet. There are only two types of taping tasks. An unknown distance between two fixed points can be measured or a required distance can be laid out from one fixed point. Since the tape may be too long or too short for either task, there are four possible types of taping problems. They are measuring with a tape that is too long, measuring with a tape that is too short, laying out with a tape that is too long and laying out with a tape that is too short. The solution to a particular taping problem is always simplified and verified by drawing a sketch. Say we measure between points A and B and later find that the tape is too long. In other words, the distance between the zero and 100 foot marks on the tape is actually 100.03 feet. Then the first tape length would extend to a point, would extend to point one, 0.03 feet farther than the actual 100 foot location. The next tape length would extend to point two and the third to point three. Since the distance remaining from three to B is less than the correct distance from the actual 300 foot location to point B, the length recorded in your fieldbook is too small and must be increased by a correction value. If the tape had been too short, then the recorded distance would be too large and the correction must be subtracted. In laying out a required distance from one fixed point, the reverse is true. Suppose now that we want to lay out point B from a known point A with a tape that is too long. If we lay out a plan length with a tape that is too long, we will end up beyond point B. Therefore, we must subtract our correction to set point B correctly. Conversely, the correction must be added for tapes that are too short. A simple sketch like the one here makes clear whether the correction should be added or subtracted for any of the four cases. Let's now move on to the final type of distance measurement. We're going to discuss that being electronic distance measurement or EDMs. A major advance in surveying occurred with the development of EDMs in the 1950s and the 1960s. EDMs make the actual measurement of distances a very simple and fast task. In modern EDMs, the measured distance is automatically displayed in digital form in feeder meters on a display screen on the instrument and many have built-in microcomputers that calculate both horizontal and vertical components of the measured slope distance. EDMs are now being incorporated with electronic theodolites having automatic angle readout capabilities to create so-called total stations. These instruments can simultaneously and automatically measure both distances and angles. When the total stations are equipped with data collectors, they can record field notes electronically and transmit them to computers, plotters, and other equipment for processing. These so-called field-to-finish systems are revolutionizing the practice of surveying. Let me try to explain now how an EDM works. In general, the EDM measures a distance by comparing a line of unknown length to the known length of transmitted energy, the wavelength shown here by the Greek letter lambda. This is similar to relating an unknown distance to the calibrated length of a steel tape. The EDM centered over the point A transmits to point B a carrier signal of electromagnetic energy, usually infrared light on modern EDMs, on which a reference frequency has been superimposed or modulated. The signal is returned from B to the EDM, so its travel path is double the slope distance of A to B. This modulated electromagnetic energy is represented by a series of sine waves, each having the wavelength lambda. One full wavelength is equal to 360 degrees phase shift. EDMs operate by measuring this phase shift of the wavelengths of the light. Let's say, for example, that the wavelength is 10 feet long. When the light returns to the EDM, the instrument measures the amount of phase shift of the light wave. Suppose the phase shift is equal to 180 degrees. This is equal to one half of 360, therefore the length of that portion of the wave is 5.00 feet. Once this distance is computed, the EDM changes the wavelength to 100 feet. The instrument then measures the phase shift of this light wave. The amount of this phase shift would identify the number in the tens column of our measurement. The wavelength is then changed to 1,000 feet and the procedure is repeated for the hundreds column and so forth until the entire distance is determined. This entire procedure takes a fraction of a second and the entire process is repeated numerous times before a value. The mean of all the measurements is displayed on the instrument's screen. It is important to remember that EDM always measures a slope distance. This distance must be reduced to a horizontal distance by one of two methods. The most common method of determining the horizontal distance is to measure the zenith angle from the instrument station to the target at the far end of the line. It should be noted here that all modern theodolites measure a zenith angle rather than a vertical angle as the older transits did. The horizontal distance is then calculated by multiplying the slope distance times the sine of the zenith angle. Another method is to determine the difference in elevation between the instrument station and the target station by one of the leveling procedures. The horizontal distance can then be calculating using Mr. Pythagoras theorem, the horizontal distance equals the square root of the slope distance squared minus the difference in elevation squared. We've already discussed how the accuracy of EDM distance is determined when we review the error in a series. Let's now discuss some of the error sources in EDM measurements. Personal errors include inaccurate setup over control points and blenders in reading the displayed data. Instrumental errors are generally very small if the EDM is maintained in top repair. Occasionally the EDM will become maladjusted. This can be detected by taking the EDM to a calibrated baseline on a regular basis. Another instrumental error occurs when the offset in the prism is misidentified. Natural errors relate mostly to atmospheric considerations. The speed of electromagnetic energy wave can be affected by the temperature and humidity of the air. Most newer EDMs have microprocessors that allow the input of the atmospheric data and this automatically calibrates the instrument correctly. One other type of distance measuring device that is being covered in another unit of this Vaneo training series is global positioning systems or GPS. This technology, which provides points positioning by using satellites is revolutionizing the way we survey much the same way that EDMs did in the 1960s and the 1970s. Before too long, a study of basic survey measurements will be covering GPS the way we cover tapes and EDMs in this session. This concludes our discussion of distance measurement. Another unit of this video training series is covering leveling and the determination of elevations. The only discussion I'd like to cover here is sources of error in leveling. If you are unfamiliar with the basic leveling equipment and procedures, I would suggest you review that unit now. The primary source of error in leveling, as with all survey measurements, is personal and usually revolves around mistakes in recording the field data. In this case, the rod readings. Other personal errors include not holding the rod vertical and not keeping the instrument level. The major instrumental error, the line of sight not being horizontal, can be compensated for in the field by balancing the length of the sights. You can see here that the error created when the line of sight is not perfectly horizontal, indicated by the small letter A, is the same on both sides if the length of the sight is balanced. When this happens, the error at each rod is compensated for and can be ignored. If the sights are unbalanced, as shown by the rod on the far right, then an error is introduced to the measurements as shown. Keeping your shots balanced while leveling has always been an important practice. Let's move on now to the study of direction. Determining locations of points and orientation of line frequently depends on the measurement of angles and directions. In surveying, direction is given by bearings and azimuths. Angles measured in surveying are classified as horizontal, shown here as the angle O, subtended from A to B, and vertical, shown as the angle at O, subtended from A to C. Horizontal angles are the basic measurements needed for determining bearings and azimuths. Vertical angles and their complement, zenith angles, are used in trigonometric leveling, stadia, and for reducing soap distances to horizontal distances. Angles are most often measured directly in the field by the theodolite. There are three basic requirements for determining a horizontal angle. As shown here, they are the reference or starting line, line OA, the direction of turning, in this case to the right from A towards B, and the angular distance, which is the value of the angle. Vertical angles are slightly different. They start with the telescope horizontal, here the horizontal line from O to A. The direction of turning is in the vertical plane, and is measured positively up or negatively down, and the angular distance is the same as measuring a horizontal angle. As mentioned at the beginning of this unit, the unit of measure for direction in the United States, and many other countries is based on degrees, minutes, and seconds, with the last unit further divided decimally. Let's now discuss the direction of a line. Bearings represent one system for designating the directions for lines by means of an angle in quadrant letters. The bearing angle of a line is the acute horizontal angle between a reference meridian and the line. Meridians are the north-south reference line from which the bearing angle is determined. There are several types of meridians. Astronomic meridians are based on observations of the stars, primarily Polaris, which are used to determine astronomical or true north. Magnetic meridians are based on the direction of the needle in a compass. Assumed meridians are very common in surveying. An example of an assumed meridian is taking the direction of a street to be north. State-playing coordinates are based on grid meridians. Now back to bearings. The angle is measured from either the north or south toward the east or west to give a reading smaller than 90 degrees. The popular quadrant is identified by a capital letter N or S preceding the angle and a capital letter E or W following. In our figure, the bearing in the northeast quadrant is measured clockwise from the meridian. Thus, the line O to A has a bearing of north 83 degrees east. The bearing in the southeast quadrant is measured counterclockwise from the meridian. Therefore, the bearing of line OB is south 37 degrees east. Similarly, the bearing OC is south 50 degrees west and the bearing OD is north 40 degrees west. Another system used to identify the direction of a line is azimuths. Azimuths are angled measured clockwise from any reference meridian. In plain surveying, azimuths are generally measured from the north but astronomers, the military, and the National Geodetic Survey have used south as the reference direction. As shown here, azimuths can range from zero degrees to 360 degrees and do not require a letter to designate the quadrant. The azimuth from O to A is 65 degrees. From O to B is 140 degrees. From O to C is 236 degrees. And from O to D is 340 degrees. Just like bearings, azimuths may be based on true, magnetic, assumed, or grid meridians. They may also be forward or back azimuths. Forward azimuths, which are shown here, are converted to back azimuths and vice versa by adding or subtracting 180 degrees. For example, the azimuth from O to A is 65 degrees. Then the back azimuth, or the azimuth from A to O, is 65 degrees plus 180 degrees, which equals 245 degrees. The azimuth from O to C is 236 degrees. The back azimuth, or the azimuth from C to O, is 236 degrees minus 180 degrees, which equals 56 degrees. Bearings can be computed from azimuths by noting the quadrant in which the azimuth falls and then converting it. An azimuth in the northeast quadrant is the same as the bearing. An azimuth of 65 degrees will be a bearing of north 65 degrees east. An azimuth in the southeast quadrant is subtracted from 180 degrees to get the bearing. Thus, the azimuth of 140 degrees is equal to a bearing of 180 degrees minus 140 degrees, or south 40 degrees east. In the southwest quadrant, 180 degrees is subtracted from the azimuth to determine the bearing. An azimuth of 236 degrees minus 180 degrees equals a bearing of south 56 degrees west. Finally, in the northwest quadrant, the azimuth is subtracted from 360 degrees. An azimuth of 340 degrees is subtracted from 360 degrees, which equals a bearing of north 20 degrees west. Many types of surveys, in particular those for traverses, require calculation of bearings or azimuths. Traverses are covered in detail in another video training session. They are basically a series of distances and angles, or distances and bearings, or distances and azimuths connecting successive points. Computation of the bearing of the line is always simplified by drawing a sketch similar to the one shown here. I'd like to take a minute here to get back on my soapbox again. I've been teaching survey classes at a local junior college in Southern California for almost 10 years now. I've taught classes in both basic survey measurements and classes in more advanced survey calculations. I've always encouraged my students to visualize a problem before they attempt to solve it. One of the most important tools you can use in visualizing a problem is drawing a simple sketch of the situation. Many times, if you can't remember a specific formula, you can still solve the problem by drawing a sketch and then working the problem out using simple mathematics. I've noticed over the years that the students who do well in class are those who take the time to draw a sketch before trying to solve a problem. Not only does it help them see the problem, but it gives them a sense of what order to do their calculations in. Conversely, the students who have not done very well seldomly draw a sketch. They get the problem and immediately begin trying to solve it, usually by trial and error, usually incorrectly. This is something I can't stress enough. Always draw a sketch of the situation before you try to solve the problem. It will help you see the problem and help you solve it correctly. Okay, back to the problem at hand. Here the problem is to determine the bearing from B to C. The bearing of a line, AB is known, and has a value of north 41 degrees 25 minutes east. This means that the angle of point B from the meridian to point A is 41 degrees 25 minutes. When we then add the angle of 130 degrees, 11 minutes to 41 degrees 25 minutes, giving us a value of 171 degrees 36 minutes, and also placing the line BC in the northwest quadrant. Remember that a bearing cannot exceed 90 degrees, so we can subtract this value from 180 degrees, giving us a bearing of north 8 degrees 24 minutes west. Now let's determine the bearing of line CD. The bearing of line BC is north 8 degrees 24 minutes west, so the angle at C from the meridian of the line, 2 point B is 8 degrees 24 minutes. The angle at C from B to D is 88 degrees 17 minutes, which places the line in the southwest quadrant. Since bearings in the southwest quadrant are measured from the meridian going south, the bearing of line CD is simply the value of the angle, 88 degrees 17 minutes minus the bearing of line BC, which is 8 degrees 24 minutes, equaling a bearing of south 79 degrees 53 minutes west. I hope you can see how a sketch for this type of problem can ultimately save you time and a lot of frustration in determining the solution. Determining azimuth from measured angles is a little easier than determining bearings. Here we want to determine the azimuth from B to C. The azimuth from line A to B is 41 degrees 25 minutes. The back azimuth, which equals the azimuth from B to A, is then 41 degrees 25 minutes plus 180 degrees, which equals 221 degrees 25 minutes. We simply add the measured angle, 129 degrees 30 minutes, to this which results in an azimuth from B to C of 350 degrees 55 minutes. Remember to draw your sketches. A special type of direction determination that is of interest to the surveyor is magnetic bearings and azimuths. As I mentioned before, magnetic bearings and azimuths are based on a meridian that is determined by the direction that a needle in a compass points. The magnetic declination is the horizontal angle from a true meridian to the magnetic meridian. An east declination exists when the magnetic meridian is east of true north. A west declination occurs if it is west of true north. The declination at any location can be determined by establishing a true meridian by astronomical observations and then reading the compass while sighting along this true meridian. This is what the early surveyors did with their compasses while they were surveying the public lands. Today we must be able to follow the township and section lines established back in the late 1800s and early 1900s and the knowledge of magnetic bearings is essential. This map of California shows the magnetic declination throughout the state. A map showing magnetic declination is called an isogonic map and the lines showing the declination are called isogonic lines. As you can see in California, the magnetic north is east of true north. This was the magnetic declination in 1985. As you are aware, the magnetic pole is constantly shifting which causes a constant change in the magnetic declination. The lines on the map show the rate of change and the direction each year. In the LA Basin where I live, the magnetic declination in 1985 was close to 14 degrees east. Since then it has changed at a rate of one minute east per year. This means that the magnetic declination today in 1992 is 14 degrees seven minutes east. Here you can see how to determine a true bearing for magnetic bearing and magnetic declination. The magnetic declination is 14 degrees seven minutes east and the magnetic bearing established from this meridian is south 60 degrees 52 minutes east. Since the magnetic meridian extends into the southwest quadrant, we must subtract the magnetic declination from the magnetic bearing to determine a true bearing which in this case is south 46 degrees 45 minutes east. Again, a sketch helps makes this type of problem easy to calculate. This brings us to our final topic of discussion which is the instruments used to determine direction and their use. The Adelites are the most universal survey instruments. Even though their primary use is for measuring or laying out horizontal and vertical angles, they are also commonly used for a wide variety of other tasks, such as determining horizontal and vertical distances by stadia, prolonging straight lines, and occasionally low order leveling. The theodolite consists of a telescope with crosshairs, horizontal and vertical circles made of glass with graduation lines and numerals etched on the circle surfaces which are encased within the instrument. Also includes a reading system consisting of a microscope and other optics which allow detailed viewing of the glass circles, rotation about the vertical axis on a system of precision ball bearings, tribracks that allow the instrument to be removed and targets attached without setting up again, and an optical plummet that allows great accuracy in the setup. There are two types of theodolite repeating and directional. Repeating instruments like the Wild T-16, arguably the most popular theodolite ever built, are equipped with a double vertical axis that allows horizontal angles to be repeated any number of times, and the results of these repeated measurements can be accumulated on the horizontal circle. The angle is then determined by dividing this accumulated angle by the number of repetitions. In other words, the angle is the mean of the accumulated measurements. Directional instruments like the Wild T-2, another very popular instrument, does not have the ability to repeat angles. Directions instead of angles are red. The initial direction are then subtracted from each subsequent direction to determine the angle. Directional instruments tend to be more precise than the repeating instruments. Recently, the electronic digital theodolite has been introduced that shows the angle measured directly on a display, thus eliminating the procedure of reading the scale on the repeating and directional instruments. This feature eliminates one of the main personal error sources inherent in the mechanical instruments, that of interpreting the value on the scale. The system used to automatically measure and display the angles is similar to the barcoding now used for automatic checkout at grocery stores. Another advantage of electronic digital theodolites is the ability to be attached to data collectors so that the angles can be recorded electronically and thus eliminating another source of personal error recorded in the wrong value for a measurement. Adding an EDM to the electronic theodolite creates a total station that we have discussed before. We don't have time in this training session to go into detailed operation of each type of theodolite, repeating, directional, and electronic. This is best left for your field training when the time can be spent practicing setting up and operating the theodolite. What we will talk about here are some general rules to remember when using the theodolite in the field. As mentioned before, horizontal angles should be measured with each type of theodolite at least twice, once with the telescope in the direct position, and once with the telescope in the reversed or plunged position. Again, this procedure eliminates all instrumental errors except the one associated with the level of aisle being out of adjustment. Another procedure that many surveyors recommend is that of closing the horizon. Closing the horizon is the process of measuring the angles around a point to obtain a check on the sum, which should equal 360 degrees. As shown here, if only angles A and B are required, angle C is also measured to close the horizon and provide that added check that will catch a mistake in reading an angle or citing an incorrect target. One of the most common activities done with the theodolite is prolonging a straight line. On route surveys, straight lines may be continued from one hub through several others by the practice of double centering. To prolong a straight line from a backside, the backside is sited and the telescope is then plunged and a point or points are set on the line. If the instrument is in maladjustment, the point C prime will be set. The instrument is then rotated back to the backside. Once again, the telescope is plunged and another point is set on line, this point, this time point C double prime. The bisector of the line between point C prime and C double prime is actually on the prolongation of the line AB. Once again, using the theodolite in both the direct and plunged position will eliminate most of the instrumental errors. Another common procedure that is done with the theodolite is establishing a point on a line between two points that are not intervisible. Here, it is necessary to establish point X on the line between points Y and Z, which are not intervisible. This is called balancing in or wriggling in. Both points X, both points Y and Z can be seen from X. Trial point X prime is located as close to the line as possible and the instrument is set up over it. Point Y is then cited and the telescope is plunged. If the line of sight does not pass through point Z, then the instrument has moved the distance of X prime X estimated from the proportional distance Z prime Z. The procedure is repeated until the instrument is located correctly. The final adjustment is usually done by moving the instrument on the tripod head a small amount. The modern theodolite measures zenith angles instead of vertical angles. As I mentioned before, the zenith angle is measured from the zenith. In other words, the telescope is pointed vertically when the scale reads zero and is measured through 360 degrees. Horizontal is at 90 degrees and 270 degrees. Zenith angles should also be measured in both the direct and plunged position and the total of the two measurements should equal 360 degrees. If they don't, the difference between the actual reading and 360 is divided by two and either added to the direct reading if the total is less than 360 degrees or subtracted if the total is greater than 360 degrees. Instrumental error sources in the measurement of angles are mostly related to the alignment of the various axis within the instrument. The axis of sight must be perpendicular to the horizontal axis. The horizontal axis must be perpendicular to the vertical axis, et cetera. As I have mentioned numerous times before, these are eliminated by sighting in both the direct and plunged positions. Other errors are caused by the level vials being out of adjustment. Natural errors include the effects of wind, heat, which causes heat waves on hot days and also affects the level vial. Refraction and the settling of the tripod. As is the case with all survey operations, personal errors account for a majority of the error sources. These include not setting up directly over a point, level bubbles not centered correctly, poor focusing, careless setting of sights, et cetera. Mistakes include setting up over the wrong point, recording an incorrect reading. One of my favorites is reading the wrong circle and using poor field procedures. Well, this brings us to the conclusion of Unit 3 of the Caltrans LS, LSIT video training program. In conclusion, I'd like to discuss some exam taking techniques that were suggested to be by my friend and fellow surveyor Jack Sands of the San Diego County Surveyors Office. These ideas worked very well for me when I took the LS exam back in 1982. First of all, don't bother studying the night before the exam. If you're not ready by then, it doesn't make any difference. Make sure you know exactly how to get to the exam location, even if this means driving to the site ahead of time. The last thing you want on exam day is getting lost. Finally, when you get ready to go to bed the night before the exam, plan out exactly what you are going to do from the time you get up until you arrive at the test site. Decide what you are going to wear, what you're going to eat for breakfast, the route you're going to take to the test site, et cetera. Then, when you get up in the morning, try to follow your plan from the night before. This will do two things for you. Number one, since you have this plan in the back of your mind, you will be more relaxed and be able to sleep better the night before. And by following the plan in the morning, you'll be sharpening your concentration before the test begins. And you won't start cold when they pass the exam out. I hope these suggestions and this training video will help you succeed in passing the LS or LSIT exam and help make you a better surveyor. Good luck.