Relative Positional Precision Explained in Everyday Language. Measurement vs. Enumeration. ALTA / ACSM Standards and Kentucky Standards of Practice

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1 Relative Positional Precision Explained in Everyday Language Todd W. Horton, PE, PLS February 015 Definitions 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. ALTA / ACSM Standards and Kentucky Standards of Practice 011 ALTA/ACSM Land Title Survey The American Land Title Association (ALTA) and the American Congress on Surveying and Mapping (ACSM) are two organizations that represent the Title Insurance Industry and the Land Surveying Industry respectively. 011 ALTA/ACSM Land Title Survey In 196, ALTA and ACSM came together for the first time to develop a survey product that would meet the needs of the title insurer to delete the standard survey exceptions from their title policy. The product that was developed is titled an ALTA/ACSM Land Title Survey, and the land surveyor s responsibilities are outlined in the Minimum Standard Detail Requirements for ALTA/ACSM Land Title Surveys. 1

2 IMPROVING ACCURACY Kentucky Association of Professional Surveyors A brief history of ALTA/ACSM Standards Changes in the 011 Standards First version published Minimum angle & distance requirements Optional Table A Items added Miscellaneous revisions Positional Tolerance requirements Table A and other miscellaneous revisions Effective since January 1, Effective since February 3, 011 American Land Title Association (ALTA) and American Congress on Surveying and Mapping(ACSM) MINIMUM STANDARD DETAIL REQUIREMENTS FOR ALTA/ACSM LAND TITLE SURVEYS (Effective February 3, 011) Measurement Standards Measurement Standards E. Measurement Standards - The following measurement standards address Relative Positional Precision for the monuments or witnesses marking the corners of the surveyed property. E. Measurement Standards - The following measurement standards address Relative Positional Precision for the monuments or witnesses marking the corners of the surveyed property. i. Relative Positional Precision means the length of the semi-major axis, expressed in feet or meters, of the error ellipse representing the uncertainty due to random errors in measurements in the location of the monument, or witness, marking any corner of the surveyed property relative to the monument, or witness, marking any other corner of the surveyed property at the 95 percent confidence level (two standard deviations). Relative Positional Precision is estimated by the results of a correctly weighted least squares adjustment of the survey. v. The maximum allowable Relative Positional Precision for an ALTA/ACSM Land Title Survey is cm (0.07 feet) plus 50 parts per million (based on the direct distance between the two corners being tested). It is recognized that in certain circumstances, the size or configuration of the surveyed property, or the relief, vegetation or improvements on the surveyed property will result in survey measurements for which the maximum allowable Relative Positional Precision may be exceeded. If the maximum allowable Relative Positional Precision is exceeded, the surveyor shall note the reason as explained in Section 6.B.ix below. Accuracy versus Precision Accuracy: agreement of observed values with the true value. A measure of results. Precision: agreement among readings of the same value (measurement). A measure of methods. IMPROVING PRECISION

3 New Accuracy Standards for NGS Datasheets ( Local Accuracy: adjacent points Network Accuracy: relative to CORS Numeric quantities, units in cm (or mm) Both are relative accuracy measures Will not use distance dependent expression Order/Class codes will no longer be used Order & Class Codes LC1766 *********************************************************************** LC1766 CBN - This is a Cooperative Base Network Control Station. LC1766 DESIGNATION - ALEXANDER LC1766 PID - LC1766 LC1766 STATE/COUNTY- IL/PIATT LC1766 USGS QUAD - SEYMOUR (1970) LC1766 LC1766 *CURRENT SURVEY CONTROL LC1766 LC1766* NAD 83(1997) (N) (W) ADJUSTED LC1766* NAVD (meters) 759. (feet) GPS OBS LC1766 LC1766 X - 18, (meters) COMP LC1766 Y - -4,883, (meters) COMP LC1766 Z - 4,087, (meters) COMP LC1766 LAPLACE CORR (seconds) DEFLEC99 LC1766 ELLIP HEIGHT (meters) GPS OBS LC1766 GEOID HEIGHT (meters) GEOID99 LC1766 LC1766 HORZ ORDER - B LC1766 ELLP ORDER - FOURTH CLASS I Relative Accuracy LC1766 *********************************************************************** LC1766 CBN - This is a Cooperative Base Network Control Station. LC1766 DESIGNATION - ALEXANDER LC1766 PID - LC1766 LC1766 STATE/COUNTY- IL/PIATT LC1766 COUNTRY - US LC1766 USGS QUAD - SEYMOUR (1970) LC1766 LC1766 *CURRENT SURVEY CONTROL LC1766 LC1766* NAD 83(011) POSITION (N) (W) ADJUSTED LC1766* NAD 83(011) ELLIP HT (meters) (06/7/1) ADJUSTED LC1766* NAD 83(011) EPOCH LC1766* NAVD 88 ORTHO HEIGHT (meters) 759. (feet) GPS OBS LC1766 LC1766 LC1766 FGDC Geospatial Positioning Accuracy Standards (95% confidence, cm) LC1766 Type Horiz Ellip Dist(km) LC LC1766 NETWORK LC LC1766 MEDIAN LOCAL ACCURACY AND DIST (039 points) LC Error Types Mistake 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. Setup over wrong point Bad H.I. Incorrect settings in equipment 3

4 Surveying errors Systematic errors Can be measured / quantified Have a positive or negative value Can be determined or corrected by procedure Random errors Cannot be measured / quantified Tend to be small and compensating Systematic Error Error that is not determined by chance but is introduced by an inaccuracy (as of observation or measurement) inherent in the system. Prism with wrong offset Poorly repaired tape Imbalance between level sightings Random Error an error that has a random distribution and can be attributed to chance. without definite aim, direction, or method Poorly adjusted tribrach Inexperienced Instrument operatorinaccuracy in equipment Nature of Random Errors Positive and negative errors will occur with the same frequency. Minor errors will occur more often than large ones. Very large errors will rarely occur. Error management Error Sources Some amount of error is acceptable. Acceptable error is determined by the intended use of the measurement. Good surveying procedures are designed to minimize systematic and random errors. 4

5 AS-FLOWN Error sources in leveling GNSS Errors Summary Systematic Earth curvature & refraction Maladjusted instrument Temperature effect on level rod Level rod not plumb Random Instrument not level Bad rod reading Poor turning point Parallax Heat wave effects Wind effects Orbital error Predicted versus as-flown trajectories Dilution of precision Atmospheric error Clock synchronization error Multipath error GNSS Errors: Clock Sync Problem: Satellite and receiver clocks are not synchronized Receiver clocks are not as accurate as satellite clocks GNSS Errors: Orbit Data Broadcast ephemeris (almanac file from satellite) 100 cm GPS orbit accuracy PREDICTED Ultra-rapid ephemeris (6-hour latency from IGS) 5 cm GPS orbit accuracy Rapid ephemeris (13-hour latency from IGS).5 cm GPS orbit accuracy Final ephemeris (1 to 14 day latency from IGS).5 cm GPS orbit accuracy 5 cm GLONASS orbit accuracy Good PDOP 1,500 mi 15 mi Ionosphere GNSS Errors: Atmospheric Delay 31 mi Troposphere A 5

6 GNSS Error: Multipath Error Computations = EXTRA DISTANCE 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. Normal Distribution Positive and negative errors will occur with the same frequency. Area under curve is equal on either side of the mean. Minor Errors 68% Large Errors 95% Minor errors will occur more often than large ones. The area within one standard deviation (s) of the mean is 68.3% of the total area. Very large errors will rarely occur. The total area within s of the mean is 95% of the sample population. 6

7 MEAN Histograms, Sigma, & Outliers s 1 s 1 s s Histogram: Plot of the Residuals \ Bell shaped curve / Outlier \ s : 68% of residuals must fall inside area s 95 % of residuals must fall inside area -0.5 Residuals Figure 3.5 Measurement Components All measurements consist of two components: the measurement and the uncertainty statement. 1,30.55 ft ± 0.05 ft 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). The uncertainty statement is not a guess, but is based on testing of equipment and methods. 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 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. 7

8 Test Data Set Measured distances: 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 mean n n = number of observations Mean = 1000 / 10 Mean = meas. 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 (Sn ) is used in future calculations Residuals Standard Deviation Calculating Residuals (mean = ): Readings residual residual Sn = 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. 8

9 Standard Deviation Formula Standard Deviation Standard deviation ( σ) n n 1 Standard Deviation is a comparison of the individual readings (measurements) to the mean of the readings, therefore Standard Deviation is a measure of. s ' PRECISION! Standard Deviation of the Mean Standard Deviation of the Mean 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. ACCURACY! Standard Error of 0.03 sm 0.007' 10 the Mean ( sm) Distance = ±0.007 (1s Confidence level) s n 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 E 50 E 50 = (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 E (1.6449s) E95 (1.96s ) 90m E 90 n E 95m E 95 n 9

10 95% Probable Error E95 (s ) (0.03) 0.046' E n E95m 0.015' Meaning of E 95 If a measurement falls outside of two standard deviations, it isn t a random error, it s a mistake! Francis H. Moffitt Distance = ±0.015 (s Confidence Level) How Errors Propagate Errors in a Series Errors in a Sum Error in Redundant Measurement Random Error Propagation Error in a Sum (E sum ) = ±(E 1 + E + E E n ) 1/ Error in a Series (E series ) = ±(E (n) 1/ ) Error in Redundant Measurement (E red. ) = ±(E / (n) 1/ ) Esum E1 Eseries E E red.meas. n E E n E3... En Error in a Series Error in a Sum Eseries E n E sum ( E E E... E 1 3 n ) Describes the error of multiple measurements with identical standard deviations, such as measuring a 1000 line with using a 100 chain. E sum 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 10

11 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. 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. How many sources of random error are there in this scenario? Impact of Equipment Precision Specifications Instrument Specifications Angle Measurement: Stated Accuracy vs. Display What is DIN 1873? What is the True Accuracy of a Measured Angle? Instrument Specifications Instrument Specifications 11

12 Instrument Specifications Distance Measurement l s m = ±( ppm x D) l What is the error in a 3500 foot measurement? l s m = ±(0.01 +(3/1,000,000 x 3500)) = ± 0.01 Calibration Total station 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 Trimble R8 Accuracy Specs RTK with single base Constant error Scalar error RTK positional error increases with distance from the base. RTK Vector Errors Vector is the line from base to rover. 10 mm + 1 ppm horizontal error for RTK vectors is typical. 10 mm = ft = constant error 1 ppm = scalar error (distance dependent) 1 part error to 1,000,000 parts measurement 1 mm error / 1 km RTK vector ft error / 1 mile RTK vector Differential Position Errors RTK vector (miles) E const (feet) E scalar (feet) E dist (feet) E dist ( E const E scalar ) 1

13 RTK relative accuracy Absolute & relative accuracy A: known B: unknown Relative accuracy = ±0.05 Poor absolute accuracy Good relative accuracy Absolute accuracy = ±5 Absolute accuracy = ±5 Base can occupy a known point or an unknown (assumed) point. A: known B: unknown Relative accuracy = ±0.05 Absolute accuracy = ±0.05 Absolute accuracy = ±0.07 Good absolute accuracy Good relative accuracy Metadata Observed positions and errors Data describing data creation and data quality Key measure of GPS data reliability Position quality expressed in terms of standard deviations Positions lose their credibility without error estimates. Mean = distance measurements Mean = feet Standard deviation = ±0.10 feet 13

14 Mean = Standard Deviation 99% 95% 68% distance measurements Mean = feet Standard deviation = ±0.04 feet D Position Quality 68% confidence 39% confidence Observed Position (address) DRMS ( E 1 E ) 14

15 Commonly Known As HRMS Trimble Topcon DCQ Leica Positional Errors ft calculated HRMS = 0.03 ft Observed Position (address) DRMS ( E 1 E ) ft ft Effects of Positional Errors ft calculated HRMS = 0.03 ft Weakness of GPS ft calculated HRMS = 0.03 ft Any of these lines are possible ft ft Match the tool to the task 80.00ft 80.00ft 80.00ft 80.00ft 80.00ft 80.00ft Match the tool to the task GPS methods give greater accuracy over long distances. Can you stake this straight boundary line accurately with GPS? Total station methods give greater accuracy over shorter distances. 15

16 Confidence Levels Reported Precision Ground Truth 99% 95% Observed Position 68% 68% confidence = 68% probability that the TRUTH falls within 1 standard deviation of the ADDRESS (mean). 95% confidence = 95% probability that the TRUTH falls within standard deviations of the ADDRESS (mean). Displayed at data collector Specified in contracts Reported in NGS datasheets Trustworthy? 68% confidence overlap Observed Position Published Position 95% confidence overlap Strength of Figure 16

17 Redundancy Error in Redundant Measurements If a measurement is repeated multiple times, the accuracy increases, even if the measurements have the same value E red.meas. E 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. Shots Horizontal Shots Vertical Base Base Line Line (KM) (KM) 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 Redundancy Check known points before, during, and after session. Use averaged positions to improve confidence. 17

18 MOLA to RV 10.8 Km Day 64 dh (m) Day 65 Importance of dh (m) Mean dh (m) 14:00-14: :00-17: :30-15: :30-18: :00-15: :00-18: :30-16:00Redundancy :30-19: :00-16: :00-19: :30-17: :30-0: :00-17: :00-0: :30-18: :30-1: :00-18: :00-15: :30-19: :30-16: :00-19: :00-16: :30-0: :30-17: :00-0: :00-14: :30-1: :30-15: Mean 14:00-1: :00-1: Two Days / Same Time > Spread = m Mean = Difference = 0.03 m Two Days/ Different Times > Spread = m Mean = Difference = m Field Methods for Compliance with Standards Least Squares Adjustment Basic Concepts 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! 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 18

19 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. Formulas: Weighted Adjustments W (1 E ) (Error Sources) C (1 W) (Correction) W n (repeated measurements of the same value) W (1 n) (a series of measurements) Weighted Adjustments A C A = , x B = , 4x C = 89 0, 8x Perform a weighted adjustment based on the above data B ANGLE No. Meas Mean ValueRel. Corr. Corrections Adjusted Value A / 4 or 4 / 7 4 / 7 X 30 = B / 4 or / 7 / 7 X 30 = C / 4 or 1 / 7 1 / 7 X 30 = TOTALS / 4 or 7 / 7 = The relative correction for the three angles are 1 : : 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 = 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. 19

20 Weighted Adjustments What Least Squares Is... BM B Elev. = , mi. BM NEW +6., 10 mi , 4 mi. 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 BM A Elev. = BM C Elev. = 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 GROUP #1 Determine the sum of the squares from X=.5, Y=.5 Student exercise GROUP # Determine sum of the squares from Mean X, Mean Y (1.667, 1.667) 0

21 Solution What Least Squares Isn t... If?= 1.667, 1.667, then Distance a-?=.357, b-?= 3.77, c-?=3.77 c 5,0? N= ( ) 3 = E= ( ) 3 = ² ² ² = 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 a 0,0 b 0,5 Least Squares Least Squares Examples Least Squares Should Be Used for The Adjustment Of: Collected By: Straight Line Best Fit Conventional Traverse Control Networks GPS Networks Level Networks Resections Theodolite & Chain Total Stations GPS Receivers Levels EDMs Straight Line Best Fit Straight Line Best Fit 1

22 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 : l Minimally Constrained Adjustment l Fully Constrained Adjustment Observed 1st Iteration nd Iteration What happens? Iterative Process Least Squares D Each iteration applies adjustments to observations, working for best solution Adjustments become smaller with each successive iteration A G E F C B The Iterative Process Least Squares 1 Creates a calculated observation for each field observation by inversing between approximate coordinates. 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 components that need to be addressed prior to performing least squares adjustment 1 Errors Coordinates 3 Observations 4 Weights

23 Errors Coordinates Blunder - Must be removed Systematic - Must be Corrected Random - No action needed Because the Least Squares process begins by calculating inversed observations approximate coordinate values are needed. 1 Dimensional Network (Level Network) - Only 1 Point. 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 Methods of Establishing Weights Each Observation Requires an Associated Weight Weight = Influence of the Observation on Final Solution Larger Weight - Larger Influence Weight = 1/σ σ = Standard Deviation of the Observation The Smaller the Standard Deviation the Greater the Weight σ = 0.8 Weight = 1 / 0.8 = 1.56 σ =. Weight = 1 /. = 0.1 More Influence Less Influence Good for combining Observations from different classes of instruments. Good for projects where standard deviation is calculated for each observation. Observational Group Least Desirable Method Example: All Angles Weighted at the Accuracy of the Total Station Each Observation Individually Weighted Best Method Standard Deviation of Field Observations Used as the Weight of the Mean Observation Combination of Types Assigns the Least weight possible for each observation 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 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 - Constrained Adjustment Place Coordinate Values on All Points in the Network 3

24 Unconstrained Adjustment Flow Chart Also Called Minimally Constrained Adjustment Free Adjustment Used to Evaluate Observations Observation Weights Relationship of All Observations Only fix the minimum required points Start Field Observations Setup Observation Standard Deviation Perform Constrained Least Squares Adjustment Print out Final Coordinate Values for All Points in Adjustment Field Data Needs Editing? Print out Unconstrained Adjustment Statistics Constrain Fixed Control Points Performed by User Edit Field Data Remove Blunders Yes Correct Systematic Errors No Statistics No Indicate Problems Yes Least Square Decision Step Adjustment Software Perform Unconstrained Least Squares Adjustment Analyze Adjustment Statistics Modify Input Data Finish Least Squares Adjustment Carlson Adjustment Software A Tour of the Software Package Demonstration Project Sample Network Adjustment A Simple D Network Adjustment Sample Network Adjustments A 3D Grid Adjustment using GPS and Conventional Data 4

25 Beyond Control Surveys Other Uses for Least Squares Adjustments / Analysis Least Squares Adjustment Interpreting Results Analyze the Statistical Results There are 4 main statistical areas that need to be looked at: 1. Standard deviation of unit weight. 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? > Observations Are Not As Good As Weighted < Observations Are Better Than Weighted Observation Residuals 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 Site Observation Residual S Dev. Flag * Outlier

26 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 (60-50) Reporting Compliance with Standards 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. E 95 = ±.447s Coordinate Standard Deviations and Error Ellipses Coordinate Standard Deviations and Error Ellipses: Point Northing Easting N SDev E SDev 1 583,511.30,068, Northing Standard Deviation{ } Easting Standard Deviation 6

27 Truth versus Address Physical monument Truth Relatively stable (in most of CONUS) Point coordinate Merely an address Contains error Subject to change Error circles: 1dRMS & drms Contrary to one-dimensional statistics, there is no fixed probability level for this error measure. The confidence level depends on the ratio of standard deviations. Owing to the low probability content of the drms error circle, 95% is generally required for position-finding errors. 1dRMS & drms σ y /σ x 1*dRMS 1*dRMS Confidence *drms % % % % % % % % % % 7