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MEASUREMENT ANALYSIS AND ADJUSTMENT Capital Project Skill Development Class (CPSD) G100398 By Jeremy Evans, P.L.S. Psomas Supplemented by Caltrans Staff Introduction “The dark side of surveying is the belief that surveying is about measurements, precisions and adjustments. It is not and never will be.” Dennis Mouland P.O.B. Magazine July, 2002 Introduction Much has been written lately about least squares adjustment and the advantages it brings to the land surveyor. To take full advantage of a least squares adjustment package, the surveyor must have a basic understanding of the nature of measurements, the equipment he uses, the methods he employs, and the environment he works in. Introduction Measurements and Adjustments: “War Stories” Class Outline Survey Measurement Basics - A Review Measurement Analysis Error Propagation Introduction to Weighted and Least Squares Adjustments Least Squares Adjustment Software Sample Network Adjustments Measure First, Adjustment Last Adjustment programs assume that: Instruments are calibrated Measurements are carefully made Networks are stronger if: They include Redundancy They have Strength of Figure Adjust only after you have followed proper procedures! Survey Measurement Basics A Review of Plumb Bob 101 Surveying (Geospatial Services?) Surveying – “That discipline which encompasses all methods for measuring, processing, and disseminating information about the physical earth and our environment.” – Brinker & Wolf Surveyor - An expert in measuring, processing, and disseminating information about the physical earth and our environment. Measurement vs. Enumeration A lot of statistical theory deals with enumeration, or counting. It’s a way to take a test sample instead of a census of the total population. The surveyor is concerned with Measurement. The true dimensions can never be known. Instrument Testing Pointing error of typical total station Instrument Specifications Instrument Specifications Instrument Specifications Distance Measurement sm = ±(0.01’ + 3ppm x D) What is the error in a 3500 foot measurement? sm= ±(0.01’+(3/1,000,000 x 3500)) = ± 0.021’ Calibration or “Don’t shoot yourself in the foot.” Leica instruments should be serviced every 18 months. EDM’s should be calibrated every six months Tribrachs should be adjusted every six months, or more often as needed. Levels pegged every 90 days Is It a Mistake or an Error? Mistake - Blunder in reading, recording or calculating a value. Error - The difference between a measured or calculated value and the true value. Blunder a gross error or mistake resulting usually from stupidity, ignorance, or carelessness. Blunder • Setup over wrong point • Bad H.I. • Incorrect settings in equipment Types of Errors Systematic Random An error is the difference between a measured value and the true value. Later we will compare this to the definition of residual Systematic an error that is not determined by chance but is introduced by an inaccuracy (as of observation or measurement) inherent in the system Systematic • Glass with wrong offset • Poorly repaired chain • Imbalance between level sightings Each measurement made with the tape is 0.1' shorter than recorded. Random an error that has a random distribution and can be attributed to chance. without definite aim, direction, or method Random • Poorly adjusted tribrach • Inexperienced Instrument operator • Inaccuracy in equipment Nature of Random Errors A plus or minus error will occur with the same frequency Minor errors will occur more often than large ones Very large errors will rarely occur (see mistake) Normal Distribution Curve #1 A plus or minus error will occur with the same frequency, so Area within curve is equal on either side of the mean Normal Distribution Curve #2 Minor errors will occur more often than large ones, so The area within one standard deviation (s) of the mean is 68.3% of the total Normal Distribution Curve #3 Very large errors will rarely occur, so The total area within 2s of the mean is 95% of the sample population Histograms, Sigma, & Outliers MEAN 2s 1s 1s 2s Histogram: Plot of the Residuals Bell shaped curve \ / Outlier \ -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1 s : 68% of residuals must fall inside area Residuals 2 s 95 % of residuals must fall inside area Measurement Components All measurements consist of two components: the measurement and the uncertainty statement. 1,320.55’ ± 0.05’ The uncertainty statement is not a guess, but is based on testing of equipment and methods. Accuracy Vs. Precision Precision - agreement among readings of the same value (measurement). A measure of methods. Accuracy - agreement of observed values with the “true value”. A measure of results. Measurement Analysis Determining Measurement Uncertainties Determining Uncertainty Uncertainty - the positive and negative range of values expected for a recorded or calculated value, i.e. the ± value (the second component of measurements). Your Assignment Measure a line that is very close to 1000 feet long and determine the accuracy of your measurement. Equipment: 100’ tape and two plumb bobs. Terrain: Basically level with 2’ high brush. Environment: Sunny and warm. Personnel: You and me. Planning the Project Test for errors in one tape length. Measure 1000 foot distance using same methods as used in testing. Determine accuracy of 1000 foot distance. Test Data Set Measured distances: 99.96 100.02 100.04 100.00 100.00 99.98 100.02 100.00 99.98 100.00 Averages “Measures of Central Tendency” The value within a data set that tends to exist at the center. Arithmetic Mean Median Mode Averages Most commonly used is Arithmetic Mean Considered the “most probable value” meas. mean n n = number of observations Mean = 1000 / 10 Mean = 100.00’ Residuals The difference between an individual reading in a set of repeated measurements and the mean Residual (n) = reading - mean Sum of the residuals squared (Sn2) is used in future calculations Residuals Calculating Residuals (mean = 100.00’): Readings residual residual2 99.96’ -0.04 0.0016 100.02’ +0.02 0.0004 100.04’ +0.04 0.0016 100.00’ 0 0 100.00’ 0 0 99.98’ -0.02 0.0004 100.02’ +0.02 0.0004 100.00’ 0 0 99.98’ -0.02 0.0004 100.00’ 0 0 Sn2 = 0.0048 Standard Deviation The Standard Deviation is the ± range within which 68.3% of the residuals will fall or … Each residual has a 68.3% probability of falling within the Standard Deviation range or … If another measurement is made, the resulting residual has a 68.3% chance of falling within the Standard Deviation range. Standard Deviation Formula n 2 Standard deviation (σ ) n 1 0.0048 s 0.023' 9 Standard Deviation Standard Deviation is a comparison of the individual readings (measurements) to the mean of the readings, therefore… Standard Deviation is a measure of…. Standard Deviation Standard Deviation is a comparison of the individual readings (measurements) to the mean of the readings, therefore… Standard Deviation is a measure of…. PRECISION! Standard Deviation of the Mean This is an uncertainty statement regarding the mean and not a randomly selected individual reading as is the case with standard deviation. Since the individual measurements that make up the mean have error, the mean also has an associated error. The Standard Deviation of the Mean is the ± range within which the mean falls when compared to the “true value”, therefore the Standard Deviation of the Mean is a measure of …. Standard Deviation of the Mean This is an uncertainty statement regarding the mean and not a randomly selected individual reading as is the case with standard deviation. Since the individual measurements that ACCURACY! make up the mean have error, the mean also has an associated error. The Standard Error of the Mean is the ± range within which the mean falls when compared to the “true value”, therefore the Standard Deviation of the Mean is a measure of …. Standard Deviation of the Mean s Standard Error of the Mean (sm) n 0.023 sm 0.007' 10 Distance = 100.00’±0.007’ (1s Confidence level) Probable Error Besides the value of s =68.3%, other error values are used by statisticians An error value of 50% is called Probable Error and is shown as “E” or “E50” E50= (0.6745)s 90% & 95% Probable Error A 50% level of certainty for a measure of precision or accuracy is usually unacceptable. 90% or 95% level of certainty is normal for surveying applications E90 (1.6449 s) E 95 (1.96s ) E90 E 95 E90 m E95 m n n 95% Probable Error E 95 (2s ) (2 0.023) 0.046' E 95 0.046 E 95 m 0.015 ' n 10 Distance = 100.00’±0.015’ (2s Confidence Level) Meaning of E95 “If a measurement falls outside of two standard deviations, it isn’t a random error, it’s a mistake!” Francis H. Moffitt How Errors Propagate Error in a Series Errors in a Sum Error in Redundant Measurement Error in a Series Describes the error of multiple measurements with identical standard deviations, such as measuring a 1000’ line with using a 100’ chain. Eseries E n Error in a Sum Esum E1 E2 E3 ... En 2 2 2 2 Esum is the square root of the sum the errors of each of the individual measurements squared It is used when there are several measurements with differing standard deviations Exercise for Errors in a Sum Assume a typical single point occupation. The instrument is occupying one point, with tripods occupying the backsight and foresight. How many sources of random error are there in this scenario? Exercise for Errors in a Sum There are three tribrachs, each with its own centering error that affects angle and distance Each of the two distance measurements have errors The angle turned by the instrument has several sources of error, including poor leveling and parallax Error in Redundant Measurements If a measurement is repeated multiple times, the accuracy increases, even if the measurements have the same value E Ered.meas. n Sample of Redundancy Expected accuracy of a given number of unique observations at a given baseline length, at the 95% confidence interval and stated in mm. Base Horizontal Base Vertical Line Shots Line Shots (KM) 1 2 3 4 5 6 7 8 9 10 (KM) 1 2 3 4 5 6 7 8 9 10 0 20 14 12 10 9 8 8 7 7 6 0 40 28 23 20 18 16 15 14 13 13 1 22 16 13 11 10 9 8 8 7 7 1 42 30 24 21 19 17 16 15 14 13 2 24 17 14 12 11 10 9 8 8 8 2 44 31 25 22 20 18 17 16 15 14 3 26 18 15 13 12 11 10 9 9 8 3 46 33 27 23 21 19 17 16 15 15 4 28 20 16 14 13 11 11 10 9 9 4 48 34 28 24 21 20 18 17 16 15 5 30 21 17 15 13 12 11 11 10 9 5 50 35 29 25 22 20 19 18 17 16 6 32 23 18 16 14 13 12 11 11 10 6 52 37 30 26 23 21 20 18 17 16 7 34 24 20 17 15 14 13 12 11 11 7 54 38 31 27 24 22 20 19 18 17 8 36 25 21 18 16 15 14 13 12 11 8 56 40 32 28 25 23 21 20 19 18 9 38 27 22 19 17 16 14 13 13 12 9 58 41 33 29 26 24 22 21 19 18 10 40 28 23 20 18 16 15 14 13 13 10 60 42 35 30 27 24 23 21 20 19 Eternal Battle of Good Vs. Evil With Errors of a Sum (or Series), each additional variable increases the total error of the network With Errors of Redundant Measurement, each redundant measurement decreases the error of the network. Sum vs. Redundancy Therefore, as the network becomes more complicated, accuracy can be maintained by increasing the number of redundant measurements Error Ellipses Used to described the accuracy of a measured survey point. Error Ellipse is defined by the dimensions of the semi-major and semi-minor axis and the orientation of the semi-major axis Assuming standard errors, the measurements have a 39.4% chance of falling within the Error Ellipse E95 = ± 2.447s Coordinate Standard Deviations and Error Ellipses Coordinate Standard Deviations and Error Ellipses: Point Northing Easting N SDev E SDev 12 583,511.320 2,068,582.469 0.021 0.017 Northing Standard Deviation { } Easting Standard Deviation Positional Accuracy vs. Precision Ratio Or, “How good is one error ellipse compared to all those others? Introduction to Adjustments Adjustment - “A process designed to remove inconsistencies in measured or computed quantities by applying derived corrections to compensate for random, or accidental errors, such errors not being subject to systematic corrections”. Definitions of Surveying and Associated Terms, 1989 Reprint Introduction to Adjustments Common Adjustment methods: Compass Rule Transit Rule Crandall's Rule Rotation and Scale (Grant Line Adjustment) Least Squares Adjustment Weighted Adjustments Weight - “The relative reliability (or worth) of a quantity as compared with other values of the same quantity.” Definitions of Surveying and Associated Terms, 1989 Reprint Weighted Adjustments The concept of weighting measurements to account for different error sources, etc. is fundamental to a least squares adjustment. Weighting can be based on error sources, if the error of each measurement is different, or the quantity of readings that make up a reading, if the error sources are equal. Weighted Adjustments Formulas: W (1 E2) (Error Sources) C (1 W) (Correction) W n (repeated measurements of the same value) W (1 n) (a series of measurements) Weighted Adjustments A = 4324’36”, 2x B = 4712’34”, 4x A C = 8922’20”, 8x Perform a weighted adjustment based on the above data C B ANGLE No. Meas Mean ValueRel. Corr. Corrections Adjusted Value A 2 43 24’ 36” 4/ 4 or 4/ 7 4/ 7X 30” = 17” 43 24’ 53” B 4 47 12’ 34” 2/ 4 or 2/ 7 2/ 7X 30” = 09” 47 12’ 43” C 8 89 22’ 20” 1/ 4 or 1/ 7 1/ 7X 30” = 04” 89 22’ 24” TOTALS 17959’ 30” 7/ 4 or 7/ 7 = 30” 180 00’ 00” The relative correction for the three angles are 1 : 2 : 4, the inverse proportion to the number of turned angles. This is the first set of relative corrections. The sum of the relative corrections is 1 + 2 + 4 = 7 , This is used as the denominator for the second set of corrections. The sum of the second set of relative corrections shall always equal 1. The second set is used for corrections. Weighted Adjustments BM “B” Elev. = 102.0 +7.8’, 2 mi. BM “NEW” +6.2’, 10 mi. +10.0’, 4 mi. BM “A” BM “C” Elev. = 100.0’ Elev. = 104.0’ Introduction to Least Squares Adjustment Simple Examples What Least Squares Is ... A rigorous statistical adjustment of survey data based on the laws of probability and statistics Provides simultaneous adjustment of all measurements Measurements can be individually weighted to account for different error sources and values Minimal adjustment of field measurements What is Least Squares? A Least Squares adjustment distributes random errors according to the principle that the Most Probable Solution is the one that minimizes the sums of the squares of the residuals. This method works to keep the amount of adjustment to the observations and, ultimately the ‘movement’ of the coordinates to a minimum. Least Squares Example Arithmetic Mean Using Least squares to prove a simple arithmetic mean solution Least Squares Example A point is measured for location 3 times. The measurements give the following NE coordinates: a. 0,0 b. 0,5 c.5, 0 c 5,0 What is the best solution for an average? ? How can you prove it? a 0,0 b 0,5 Student exercise GROUP #1 GROUP #2 Determine the sum Determine sum of of the squares from the squares from X=2.5, Y=2.5 Mean X, Mean Y (1.667, 1.667) Solution If ?= 1.667, 1.667, then Distance a-?= 2.357, b-?= 3.727, c-?=3.727 c 5,0 N= (0 + 0 +5) 3 = 1.667 E= (0 + 5 +0) 3 = 1.667 2.357² + 3.727² + 3.727² = 33.333 ? a 0,0 b 0,5 What Least Squares Isn’t ... A way to correct a weak strength of figure A cure for sloppy surveying - Garbage in / Garbage out The only adjustment available to the land surveyor Least Squares Least Squares Should Be Used for The Adjustment Of: Collected By: Conventional Traverse Theodolite & Chain Control Networks Total Stations GPS Networks GPS Receivers Level Networks Levels Resections EDMs Least Squares A B Observed E 1st Iteration 2nd Iteration G F What happens? C Iterative Process D Each iteration applies adjustments to observations, working for best solution Adjustments become smaller with each successive iteration Least Squares The Iterative Process 1 Creates a calculated observation for each field observation by inversing between approximate coordinates. 2 Calculates a "best fit" solution of observations and compares them to field observations to compute residuals. 3 Updates approximate coordinate values. 4 Calculates the amount of movement between the coordinate positions prior to iteration and after iteration. 5 Repeats steps 1 - 4 until coordinate movement is no greater than selected threshold. Least Squares Four component that need to be addressed prior to performing least squares adjustment 1 Errors 2 Coordinates 3 Observations 4 Weights Errors Blunder - Must be removed Systematic - Must be Corrected Random - No action needed Coordinates Because the Least Squares process begins by calculating inversed observations approximate coordinate values are needed. 1 Dimensional Network (Level Network) - Only 1 Point. 2 Dimensional Network - All Points Need Northing and Easting. 3 Dimensional Network - All Points Need Northing, Easting, and Elevation. (Except for adjustments of GPS baselines.) Weights • Each Observation Requires an Associated Weight Weight = Influence of the Observation on Final Solution • Larger Weight - Larger Influence Weight = 1/σ2 • σ = Standard Deviation of the Observation The Smaller the Standard Deviation the Greater the Weight σ = 0.8 Weight = 1/0.82 = 1.56 More Influence σ = 2.2 Weight = 1/2.22 = 0.21 Less Influence Methods of Establishing Weights • Observational Group • Least Desirable Method Good for combining Observations from • Example: All Angles Weighted at the Accuracy of different classes of the Total Station instruments. • Each Observation Individually Weighted • Best Method Good for projects where standard • Standard Deviation of Field Observations Used as deviation is calculated the Weight of the Mean Observation for each observation. • Combination of Types • Assigns the Least weight possible for each observation Least Squares If you remember nothing else about least squares today, remember this! Least Squares Adjustment Is a Two Part Process 1 - Unconstrained Adjustment Analyze the Observations, Observations Weights, and the Network 2 - Constrained Adjustment Place Coordinate Values on All Points in the Network Unconstrained Adjustment • Also Called Minimally Constrained Adjustment Free Adjustment • Used to Evaluate Observations Observation Weights Relationship of All Observations • Only fix the minimum required points Flow Chart Start Field Observations Edit Field Data Perform Setup Field Data • Remove Blunders Unconstrained Least Observation Needs Yes • Correct Systematic Squares Adjustment Standard Deviation Editing? Errors No Print out Unconstrained Statistics Analyze Adjustment Statistics No Indicate Adjustment Problems Statistics Yes Perform Constrain Fixed Modify Input Constrained Least Control Points Data Squares Adjustment Print out Final Least Square Coordinate Values Performed by Decision Step Adjustment for All Points in User Software Adjustment Finish Analyze the Statistical Results There are 4 main statistical areas that need to be looked at: 1. Standard deviation of unit weight 2. Observation residuals 3. Coordinate standard deviations and error ellipses 4. Relative errors A 5th statistic that is sometimes available that should be looked at: Chi-square Test Standard Deviation of Unit Weight Also Called Standard Error of Unit Weight Error Total Network Reference Factor The Closer This Value Is to 1.0 the Better The Acceptable Range Is ? to ? > 1.0 - Observations Are Not As Good As Weighted < 1.0 - Observations Are Better Than Weighted Observation Residuals • Amount of adjustment applied to observation to obtain best fit This is the residual that is being minimized • Used to analyze each observation • Usually flags excessive adjustments (Outliers) (Star*net flags observations adjusted more than 3 times the observations weight) • Large residuals may indicate blunders Observation Residuals Site Observation Residual S Dev. Flag 10-11-12 214 33’ 17.2” 1.7” 1.2” 11-12-13 174 16’ 43.8” 7.2” 1.9” * 12-13-14 337 26’ 08.6 2.1” 1.3” Outlier -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Coordinate Standard Deviations and Error Ellipses • Coordinate standard deviations represent the accuracy of the coordinates • Error ellipses are a graphical representation of the standard deviations • The better the network the rounder the error ellipses • High standard deviations can be found in networks with a good standard deviation of unit weight and well weighted observations due to effects of the network geometry Relative Errors Predicted amount of error that can be expected to occur between points when an observation is made in the network. Chi-square Test noun: (ki'skwâr) a statistic that is a sum of terms each of which is a quotient obtained by dividing the square of the difference between the observed and theoretical values of a quantity by the theoretical value In other words: A statistical analysis of the statistics. 10 coins 6 to 4 (6-5) or 100 coins 60-40 (60-50) Least Squares Examples Straight Line Best Fit Straight Line Best Fit Straight Line Best Fit Straight Line Best Fit Straight Line Best Fit Least Squares “Rules” Redundancy of survey data strengthens adjustment Error Sources must be determined correctly Each adjustment consists of two “parts”: Minimally Constrained Adjustment Fully Constrained Adjustment Star*Net Adjustment Software A Tour of the Software Package Star*Net Sample Network Adjustment A Simple 2D Network Adjustment Star*Net Sample Network Adjustments A 3D “Grid” Adjustment using GPS and Conventional Data Star*Net Beyond Control Surveys Other Uses for Least Squares Adjustments / Analysis Questions & Discussion Systematic vs. Random Error Systematic Error - An error whose magnitude and algebraic sign can be determined or corrected by procedure. Example: temperature correction for steel tape or balanced level distances. Random Error - An error whose magnitude and algebraic sign cannot be determined. They tend to be small and compensating. Measurement Analysis is the study of random errors. What Least Squares Is... Adjustment report provides details of survey measurements A TOOL to be used by the Surveyor to complement his knowledge of measurements Random Error Propagation Error in a Sum (Esum) = ±(E12 + E22 + E32 + ….. + En2)1/2 Error in a Series (Eseries) = ±(E (n)1/2) Error in Redundant Measurement (Ered.) = ±(E / (n)1/2) Esum E1 2 E2 2 E3 2 ... En 2 Eseries E n E Ered.meas. n Instrument Specifications Angle Measurement: Stated Accuracy vs. Display What is DIN 18723? What is the True Accuracy of a Measured Angle?