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Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 8/17/2014 Definitions Relative Positional Precision Explained in Everyday Language Todd W. Horton, PE, PLS • 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. February 2015 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 2011 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. • In 1962, 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. 2011 ALTA/ACSM Land Title Survey PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE • 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 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE A brief history of ALTA/ACSM Standards • 1962 - First version published • 1986 - Minimum angle & distance requirements • 1988 - Optional Table A Items added • 1992 - Miscellaneous revisions • 1997 - Positional Tolerance requirements • 1999 - Table A and other miscellaneous revisions • 2005 - Effective since January 1, 2006 • 2011 - Effective since February 23, 2011 Measurement Standards 8/17/2014 Changes in the 2011 Standards 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 23, 2011) 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 2 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: – – • 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 ACCURACY Accuracy versus Precision IMPROVING PRECISION PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 2 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE New Accuracy Standards for NGS Datasheets (http://fgdc.er.usgs.gov/standards/status/swgstat.html) • Local Accuracy: • Network Accuracy: • • • • adjacent points 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 8/17/2014 Order & Class Codes LC1766 *********************************************************************** LC1766 CBN - This is a Cooperative Base Network Control Station. LC1766 DESIGNATION - ALEXANDER 2 LC1766 PID - LC1766 LC1766 STATE/COUNTY- IL/PIATT LC1766 USGS QUAD - SEYMOUR (1970) LC1766 LC1766 *CURRENT SURVEY CONTROL LC1766 ___________________________________________________________________ LC1766* NAD 83(1997)- 40 06 20.45303(N) 088 29 42.90776(W) ADJUSTED LC1766* NAVD 88 231.3 (meters) 759. (feet) GPS OBS LC1766 ___________________________________________________________________ LC1766 X - 128,287.167 (meters) COMP LC1766 Y - -4,883,624.381 (meters) COMP LC1766 Z - 4,087,096.112 (meters) COMP LC1766 LAPLACE CORR-0.58 (seconds) DEFLEC99 LC1766 ELLIP HEIGHT199.36 (meters) GPS OBS LC1766 GEOID HEIGHT-31.94 (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 2 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(2011) POSITION- 40 06 20.45302(N) 088 29 42.90701(W) ADJUSTED LC1766* NAD 83(2011) ELLIP HT- 199.346 (meters) (06/27/12) ADJUSTED LC1766* NAD 83(2011) EPOCH - 2010.00 LC1766* NAVD 88 ORTHO HEIGHT - 231.3 (meters) 759. (feet) GPS OBS LC1766 ______________________________________________________________________ LC1766 LC1766 FGDC Geospatial Positioning Accuracy Standards (95% confidence, cm) LC1766 Type Horiz Ellip Dist(km) LC1766 ------------------------------------------------------------------LC1766 NETWORK 0.77 2.21 LC1766 ------------------------------------------------------------------LC1766 MEDIAN LOCAL ACCURACY AND DIST (039 points) 1.00 2.67 69.56 LC1766 ------------------------------------------------------------------- Mistake ≠ Error • • Mistake - Blunder in reading, recording or calculating a value. Error - The difference between a measured or calculated value and the true value. PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE Error Types 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 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 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 8/17/2014 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 Nature of Random Errors • an error that has a random distribution and can be attributed to chance. • without definite aim, direction, or method • Positive and negative errors will occur with the same frequency. – Poorly adjusted tribrach – Inexperienced Instrument operatorInaccuracy in equipment • 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. PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 4 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 8/17/2014 Error sources in leveling • Systematic • Random – Earth curvature & refraction – Maladjusted instrument – Temperature effect on level rod – Level rod not plumb – – – – – – Instrument not level Bad rod reading Poor turning point Parallax Heat wave effects Wind effects GNSS Errors Summary • Orbital error – Predicted versus as-flown trajectories – Dilution of precision • Atmospheric error • Clock synchronization error • Multipath error GNSS Errors: Clock Sync GNSS Errors: Orbit Data • Problem: – Receiver clocks are not as accurate as satellite clocks AS-FLOWN – Satellite and receiver clocks are not synchronized • 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) – 2.5 cm GPS orbit accuracy • Final ephemeris (12 to 14 day latency from IGS) – 2.5 cm GPS orbit accuracy – 5 cm GLONASS orbit accuracy Good PDOP GNSS Errors: Atmospheric Delay 12,500 mi Ionosphere 125 mi 31 mi Troposphere A PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 5 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 8/17/2014 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 Large Errors 68% 95% • Minor errors will occur more often than large ones. • Very large errors will rarely occur. • The area within one standard deviation (s) of the mean is 68.3% of the total area. • The total area within 2s of the mean is 95% of the sample population. PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 6 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 8/17/2014 2s MEAN Histograms, Sigma, & Outliers 1s 2s 1s 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 1 s : 68% of residuals must fall inside area 2 s 95 % of residuals must fall inside area 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Figure 3.5 Residuals Measurement Components Determining Uncertainty • • All measurements consist of two components: the measurement and the uncertainty statement. Uncertainty - the positive and negative range of values expected for a recorded or calculated value, i.e. the ± value (the second component of measurements). 1,320.55 ft ± 0.05 ft • 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 2’ high brush. Environment: Sunny and warm. Personnel: You and me. PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 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 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 8/17/2014 Test Data Set Measured distances: 99.96 100.04 100.00 100.02 99.98 Averages • 100.02 100.00 99.98 100.00 100.00 – • • • Averages • • Most commonly used is Arithmetic Mean Considered the “most probable value” mean • • • • n = number of observations Mean = 1000 / 10 Mean = 100.00’ Residuals • Calculating Residuals (mean = 100.00’): Readings 99.96’ 100.02’ 100.04’ 100.00’ 100.00’ 99.98’ 100.02’ 100.00’ 99.98’ 100.00’ residual -0.04 +0.02 +0.04 0 0 -0.02 +0.02 0 -0.02 0 Sn2 = residual2 0.0016 0.0004 0.0016 0 0 0.0004 0.0004 0 0.0004 0 0.0048 PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE The value within a data set that tends to exist at the center. Arithmetic Mean Median Mode Residuals • meas. n “Measures of Central Tendency” 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 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. 8 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 8/17/2014 Standard Deviation Standard Deviation Formula Standard deviation (σ ) n 2 n 1 • • s • Standard Deviation is a measure of…. 0.0048 0.023' 9 Standard Deviation of the Mean • Standard Deviation is a comparison of the individual readings (measurements) to the mean of the readings, therefore… 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 …. PRECISION! Standard Deviation of the Mean Standard Error of the Mean (sm) sm 0.023 10 s n 0.007' Distance = 100.00’±0.007’ (1s Confidence level) ACCURACY! 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” 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 (1.6449s) E50= (0.6745)s E90m PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE E90 n E 95 (1.96s ) E95m E95 n 9 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 8/17/2014 95% Probable Error Meaning of E95 “If a measurement falls outside of E 95 (2s ) (2 0.023) 0.046' E 95m • two standard deviations, it isn’t a random error, it’s a mistake!” E 95 0.046 0.015' n 10 Francis H. Moffitt Distance = 100.00’±0.015’ (2s Confidence Level) Random Error Propagation How Errors Propagate • Errors in a Series • Errors in a Sum • Error in Redundant Measurement • • • Error in a Sum (Esum) = ±(E12 + E22 + E32 + ….. + En2)1/2 Error in a Series (Eseries) = (n)1/2) Error in Redundant Measurement (Ered.) = ±(E / (n)1/2) ±(E Esum E1 2 E2 2 E3 2 ... En 2 Eseries E n Ered.meas. E n Error in a Series Error in a Sum Eseries E n Esum ( E12 E 2 2 E 3 2 ... En 2 ) • Describes the error of multiple measurements with identical standard deviations, such as measuring a 1000’ line with using a 100’ chain. PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE • 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 10 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 8/17/2014 Exercise for Errors in a Sum 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. • 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? Instrument Specifications Impact of Equipment Precision Specifications Instrument Specifications PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE • Angle Measurement: – – – Stated Accuracy vs. Display What is DIN 18723? What is the True Accuracy of a Measured Angle? Instrument Specifications 11 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 8/17/2014 Instrument Specifications • Distance Measurement l sm = ±(0.01’ + 3ppm x D) l What is the error in a 3500 foot measurement? l sm= ±(0.01’+(3/1,000,000 x 3500)) = ± 0.021’ 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 Constant error RTK with single base Scalar error RTK positional error increases with distance from the base. http://trl.trimble.com/docushare/dsweb/Get/Document-140079/022543-079J_TrimbleR8GNSS_DS_1109_LR.pdf RTK Vector Errors • Vector is the line from base to rover. – 10 mm + 1 ppm horizontal error for RTK vectors is typical. – 10 mm = 0.033 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 • 0.005 ft error / 1 mile RTK vector Differential Position Errors RTK vector (miles) 1 2 4 8 16 32 E const E scalar E dist 0.033 0.033 0.033 0.033 0.033 0.033 0.005 0.011 0.021 0.042 0.084 0.169 0.033 0.034 0.039 0.053 0.091 0.172 (feet) Edist PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE (feet) (feet) 2 2 ( Econst Escalar ) 12 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE RTK relative accuracy 8/17/2014 Absolute & relative accuracy A: known B: unknown Relative accuracy = ±0.05’ Absolute accuracy = ±5’ Absolute accuracy = ±5’ A: known B: unknown Relative accuracy = ±0.05’ Base can occupy a known point or an unknown (assumed) point. Metadata Poor absolute accuracy Good relative accuracy Absolute accuracy = ±0.05’ Good absolute accuracy Good relative accuracy Absolute accuracy = ±0.07’ 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 = 2000.00 1999.90 1999.80 2000.10 2000.20 • 100 distance measurements • Mean = 2000.00 feet • Standard deviation = ±0.10 feet PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 13 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE Mean = 2000.00 8/17/2014 Standard Deviation 99% 1999.96 2000.04 95% 68% 1999.92 2000.08 • 100 distance measurements • Mean = 2000.00 feet • Standard deviation = ±0.04 feet 2D Position Quality 68% confidence 39% confidence Observed Position (address) DRMS ( E12 E22 ) PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 14 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 8/17/2014 Positional Errors Commonly Known As • HRMS HRMS = 0.03 ft – Trimble – Topcon 1000.00 ft calculated • 2DCQ – Leica 999.94 ft Observed Position (address) 1000.06 ft DRMS ( E12 E22 ) Effects of Positional Errors Weakness of GPS HRMS = 0.03 ft 1000.00 ft calculated HRMS = 0.03 ft 100.00 ft calculated 99.94 ft Any of these lines are possible. 100.06 ft Match the tool to the task 80.00ft 80.00ft 80.00ft 80.00ft 80.00ft 80.00ft • Can you stake this straight boundary line accurately with GPS? PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE Match the tool to the task • GPS methods give greater accuracy over long distances. • Total station methods give greater accuracy over shorter distances. 15 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE Confidence Levels Ground Truth 99% 95% 68% Observed Position 8/17/2014 Reported Precision • 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 2 standard deviations of the ADDRESS (mean). Trustworthy? Displayed at data collector Specified in contracts Reported in NGS datasheets 68% confidence overlap Observed Position Published Position 95% confidence overlap Strength of Figure PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 16 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 8/17/2014 Error in Redundant Measurements Redundancy • If a measurement is repeated multiple times, the accuracy increases, even if the measurements have the same value Ered.meas. Sample of Redundancy 0 1 2 3 4 5 6 7 8 9 10 Horizontal Shots 1 2 3 4 5 6 7 8 9 10 Base Line (KM) 20 22 24 26 28 30 32 34 36 38 40 12 13 14 15 16 17 18 20 21 22 23 10 11 12 13 14 15 16 17 18 19 20 9 10 11 12 13 13 14 15 16 17 18 8 9 10 11 11 12 13 14 15 16 16 8 8 9 10 11 11 12 13 14 14 15 7 8 8 9 10 11 11 12 13 13 14 7 7 8 9 9 10 11 11 12 13 13 6 7 8 8 9 9 10 11 11 12 13 0 1 2 3 4 5 6 7 8 9 10 14 16 17 18 20 21 23 24 25 27 28 Vertical Shots 1 2 3 4 5 6 7 8 9 10 40 42 44 46 48 50 52 54 56 58 60 23 24 25 27 28 29 30 31 32 33 35 20 21 22 23 24 25 26 27 28 29 30 18 19 20 21 21 22 23 24 25 26 27 16 17 18 19 20 20 21 22 23 24 24 15 16 17 17 18 19 20 20 21 22 23 14 15 16 16 17 18 18 19 20 21 21 13 14 15 15 16 17 17 18 19 19 20 13 13 14 15 15 16 16 17 18 18 19 28 30 31 33 34 35 37 38 40 41 42 n Eternal Battle of Good Vs. Evil Expected accuracy of a given number of unique observations at a given baseline length, at the 95% confidence interval and stated in mm. Base Line (KM) E Sum vs. Redundancy • Therefore, as the network becomes more complicated, accuracy can be maintained by increasing the number of redundant measurements PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE • 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. Redundancy • Check known points before, during, and after session. • Use averaged positions to improve confidence. 17 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE MOLA to RV22 10.8 Km Day 264 dh (m) Day 265 dh (m) -10.281 Importance of-10.279 -10.278 -10.270 -10.281 -10.278 Redundancy -10.291 -10.274 14:00-14:30 17:00-17:30 14:30-15:00 17:30-18:00 15:00-15:30 18:00-18:30 15:30-16:00 18:30-19:00 19:00-19:30 20:30-21:00 -10.274 -10.287 -10.279 -10.270 -10.277 -10.271 -10.277 -10.271 -10.259 -10.254 14:30-15:00 -10.274 -10.276 -10.261 -10.251 -10.270 -10.276 -10.278 -10.286 -10.278 -10.295 14:00-21:00 -10.275 14:00-21:00 -10.276 16:00-16:30 16:30-17:00 17:00-17:30 17:30-18:00 18:00-18:30 18:30-19:00 19:00-19:30 19:30-20:00 20:00-20:30 19:30-20:00 20:00-20:30 20:30-21:00 15:00-15:30 15:30-16:00 16:00-16:30 16:30-17:00 14:00-14:30 Mean dh (m) -10.280 -10.274 -10.280 -10.283 -10.274 -10.282 -10.270 -10.261 -10.274 -10.274 -10.278 -10.279 -10.269 -10.275 Mean -10.276 8/17/2014 Two Days / Same Time -10.254 > -10.253 -10.251 Spread = 0.003 m Mean = -10.276 Difference = 0.023 m Two Days/ Different Times -10.254 > -10.275 -10.295 Spread = 0.041 m Mean = -10.276 Difference = 0.001 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! PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 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 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 8/17/2014 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 ANGLE No. Meas Mean Value Rel. Corr. Corrections A 2 43 24’ 36” 4/ 4 or 4/ B 4 47 12’ 34” 2/ 4 or 2/ C 8 89 22’ 20” 1/ 4 or 17959’ 30” 7/ 4 or TOTALS Adjusted Value 7 4/ 7 X 30” = 17” 43 24’ 53” 7 2/ 7X 30” = 09” 47 12’ 43” 1/ 7 1/ 7X 30” = 04” 89 22’ 24” 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. C B PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 19 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 8/17/2014 What Least Squares Is ... Weighted Adjustments • BM “B” +7.8’, 2 mi. Elev. = 102.0 • BM “NEW” • +6.2’, 10 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 +10.0’, 4 mi. BM “A” BM “C” Elev. = 100.0’ Elev. = 104.0’ Least Squares Example 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. A point is measured for location 3 times. The measurements give the following NE coordinates: c 5,0 b. 0,5 c.5, 0 • GROUP #1 • Determine the sum of the squares from • GROUP #2 • Determine sum of the squares from X=2.5, Y=2.5 Mean X, Mean Y (1.667, 1.667) What is the best solution for an average? How can you prove it? ? a 0,0 • Arithmetic Mean Using Least squares to prove a simple arithmetic mean solution Student exercise Least Squares Example a. 0,0 • b 0,5 PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 20 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 8/17/2014 What Least Squares Isn’t ... 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 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 Examples Least Squares • Least Squares Should Be Used for The Adjustment Of: Collected By: Conventional Traverse Control Networks GPS Networks Level Networks Resections Theodolite & Chain Total Stations GPS Receivers Levels EDMs Straight Line Best Fit PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE • Straight Line Best Fit Straight Line Best Fit 21 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 8/17/2014 Straight Line Best Fit Straight Line Best Fit Least Squares “Rules” Least Squares A • • • l l Minimally Constrained Adjustment Fully Constrained Adjustment B Observed Redundancy of survey data strengthens adjustment Error Sources must be determined correctly Each adjustment consists of two “parts”: E 1st Iteration G 2nd Iteration F What happens? Iterative Process C 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. PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE Least Squares Four components that need to be addressed prior to performing least squares adjustment 1 2 3 4 Errors Coordinates Observations Weights 22 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 8/17/2014 Errors Coordinates • Blunder - Must be removed • Systematic - Must be Corrected • Random - No action needed Weights • 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.) Methods of Establishing 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 σ = 2.2 Weight = 1/2.22 = 0.21 More Influence Less Influence 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 • Observational Group Good for combining Observations from different classes of instruments. • Least Desirable Method • Example: All Angles Weighted at the Accuracy of the Total Station • Each Observation Individually Weighted Good for projects where standard deviation is calculated for each observation. • 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 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 PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 23 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 8/17/2014 Unconstrained Adjustment Flow Chart Start • Also Called • Minimally Constrained Adjustment Field Observations Setup Observation Standard Deviation Field Data Needs Editing? • Free Adjustment Yes Edit Field Data • Remove Blunders • Correct Systematic Errors Perform Unconstrained Least Squares Adjustment No • Used to Evaluate Print out Unconstrained Adjustment Statistics • Observations No • Observation Weights • Relationship of All Observations Statistics Indicate Problems Analyze Adjustment Statistics Yes Perform Constrained Least Squares Adjustment Modify Input Data Constrain Fixed Control Points • Only fix the minimum required points Print out Final Coordinate Values for All Points in Adjustment Performed by User Decision Step Least Square Adjustment Software Finish Carlson Adjustment Software Least Squares Adjustment • A Tour of the Software Package Demonstration Project Sample Network Adjustment • A Simple 2D Network Adjustment PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE Sample Network Adjustments • A 3D “Grid” Adjustment using GPS and Conventional Data 24 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 8/17/2014 Beyond Control Surveys • Least Squares Adjustment Other Uses for Least Squares Adjustments / Analysis Interpreting Results Analyze the Statistical Results Standard Deviation of Unit Weight Also Called 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 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 Observation Residuals Site 10-11-12 11-12-13 Observation 214 33’ 17.2” 174 16’ 43.8” Residual 1.7” 7.2” S Dev. 1.2” 1.9” 12-13-14 337 26’ 08.6 2.1” 1.3” 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 * Outlier -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE Flag 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 25 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 8/17/2014 Relative Errors Coordinate Standard Deviations and Error Ellipses Predicted amount of error that can be expected to occur between points when an observation is made in the network. • 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 Chi-square Test Reporting Compliance with Standards • 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) 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 PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE Coordinate Standard Deviations and Error Ellipses: Point 12 Northing 583,511.320 Easting 2,068,582.469 N SDev 0.021 E SDev 0.017 { Northing Standard Deviation } • Coordinate Standard Deviations and Error Ellipses Easting Standard Deviation 26 Kentucky Association of Professional Surveyors PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE Truth versus Address • Physical monument – Truth – Relatively stable (in most of CONUS) 8/17/2014 Error circles: 1dRMS & 2dRMS • 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. • Point coordinate – Merely an address – Contains error – Subject to change • Owing to the low probability content of the dRMS error circle, 95% is generally required for position-finding errors. 1dRMS & 2dRMS Confidence Confidence 2*dRMS @ 1*dRMS @ 2*dRMS σy/σx 1*dRMS 0.0 1.0 68.27% 2.0 95.45% 0.25 1.0308 68.15% 2.0616 95.91% 0.5 1.1180 66.29% 2.2361 96.97% 0.75 1.25 63.92% 2.5 97.87% 1.0 1.4142 63.20% 2.8284 98.16% PRESENTATION SAMPLE FOR BOARD REVIEW ONLY SUBJECT TO CHANGE 27