Horizontal Positional Accuracy (HPA) How is it calculated? Gord Gamble and Peter Goodier, Practice Advisory Department

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1 Horizontal Positional Accuracy (HPA) How is it calculated? Gord Gamble and Peter Goodier, Practice Advisory Department In the December 2014 issue of The Link, the Practice Advisory Department published an article entitled Georeferencing and Plan Preparation Frequently Asked Questions. That article discussed some of the salient points of Horizontal Positional Accuracy (HPA), but it did not examine the different scenarios for calculating HPA. The aim of this article is to take a closer look at calculating HPA in a number of different georeferencing scenarios. As a reminder, the GSI Rules define Horizontal Positional Accuracy as the network horizontal accuracy of all the georeferenced points in the survey. Also; Network Horizontal Accuracy means the absolute accuracy of the coordinates for a point with respect to the adopted British Columbia Geo-Spatial Reference to a 95% confidence level, which is dependent on the network accuracy of the known point(s) used to derive the coordinates of the legal survey and the relative accuracy of the connection(s) to the known point(s). There are several important points to consider when we talk about the HPA of the georeferenced points: The HPA is a measure of absolute accuracy; This accuracy must be with respect to the adopted BC Geospatial Reference as published in Circular Letter #463; The HPA is expressed for your georeferenced points (i.e. points which have occupied by GNSS equipment) and does not necessarily apply to other points in your survey. In order to determine an HPA for other points in your survey, you will have to ensure that rigorous methods are followed for field observations and processing these observations (e.g. least squares adjustment) to be confident about the HPA of such derived points. Calculating Horizontal Positional Accuracy The approach one takes for calculating the horizontal positional accuracy for georeferenced points depends on the georeferencing method used. This article looks at several different georeferencing situations and discusses approaches for calculating HPA. In each approach, we emphasize the following: To the maximum extent possible, the HPA should be based on statistical results generated from your survey. The HPA calculation needs to include an allowance for the HPA of the source geodetic control or the RTN/RTK base you are using. The accuracy must be expressed to a 95% confidence level. At this point, a brief review of the difference between the standard deviation and the 95% confidence level is in order. In statistics, the standard deviation S is a measure of confidence. A fact of the normal distribution is that 68% of the measured values in a data set will fall within the upper and lower limit of the standard deviation. For example, if a distance is measured as m, and the standard deviation S= 0.01 m, then the upper limit is calculated as m m= m, and the lower limit is

2 calculated as m 0.01 m= m, and 68% of any observations made will fall between m and m. Most surveying standards reference the 95% confidence level. The 95% confidence level is achieved by multiplying the standard deviation by Using the numbers from the previous example, if S= 0.01 m, then the 95% confidence levels would be calculated as 0.01 m X 1.96 = 0.02 m, and 95% of the observations would fall between m +/ m; that is 95% of the values would fall between m and m. 1. Static GNSS surveys using the Precise Point Positioning (PPP) service The PPP service provides a report which includes the 95% confidence levels for latitude and longitude (the PPP report calls these confidence levels Sigmas 95% ), and the HPA is readily calculated using these statistics. Example: Calculating the HPA from data provided by a PPP report: Estimated Position for TH 043 Latitude (+n) Longitude (+e) Ell. Height NAD83(CSRS) (2002) 49º º m Sigmas(95%) m m m The HPA for the PPP-derived base station is calculated as the square root of the sum of the squares of the 95% confidence levels for the latitude and longitude; HPA= (95% confidence level Latitude)² + (95% confidence level Longitude)² = (0.013² ²)= m= 21 mm 2. Static GNSS surveys using the manufacturer s software to process a baseline vector Often, a GNSS survey will derive coordinates for a base station using the PPP service, and process a vector to a second GNSS point. In these cases, the GNSS processing software will provide statistical analysis for the second GNSS point, which will consider errors in the baseline vector. It is important to be mindful of the following: The software should be reporting at the 95% confidence level. If the software is reporting at the 68% confidence level, then the multiplier of 1.96 should be applied to achieve the 95% confidence level. The software should consider the positional uncertainties of the base station in the calculation. Be wary of situations where the GNSS software reports an HPA at the rover which is more accurate than that of the base station. Example: A GNSS survey is conducted where a receiver is set up at TH 043 and is designated the base station. A second receiver is set up at TH 044 and several hours of GNSS observations are collected. Use the HPA for the base station calculated in the previous example and the following vector calculation report (provided by the GNSS processing software) to calculate the HPA for TH 044.

3 Vector processing report From Pt. TH 043 To Pt. Start Time End Time Duration Horizontal Precision 95% (m) TH 044 9:10 05/06/ :10 05/06/2015 Vertical Precision 95% (m) dn (m) de (m) 3: The HPA calculation for TH 044 is straight forward- we simply add the positional uncertainty at the base station obtained from PPP to the Horizontal Precision of the vector from the processing report: HPA= error base station² + error vector² = 0.021² ²= m= 22 mm 3. Real Time Kinematic (RTK) GNSS surveys There are several approaches for calculating the HPA for RTK surveys: i. Most GNSS manufacturers provide some sort of post-mission statistical reporting, often in terms of coordinate quality for RTK surveys, which is available through a desk-top interface. For example, one GNSS manufacturer s software offers a Statistics window where the Horizontal Precision (95%) is displayed. Using this figure for the HPA on a survey plan is a reasonable approach. It is important to be aware that these software applications may or may not consider the errors at the base station. ii. The positional uncertainty related to the vector between the base station and the rover can be calculated by referencing the manufacturer s specifications. The vector error can then be added to the HPA at the base station to calculate an HPA for the rover. The HPA for the base station could come from the published standard deviations from a MASCOT monument (which would require applying the multiplier to achieve the 95% confidence level), from the 95% confidence levels in a PPP report, or from the HPA of a real time network base station. This approach is not as rigorous as other methods, as it will not consider the conditions under which the survey was completed. However, if no other options are available, this method can be used. One manufacturer, for example, specifies an accuracy of 8 mm + 1 ppm at the 68% confidence level (in normal to favorable conditions ) for RTK survey measurements. Example: Using the manufacturer s accuracy specifications, what is the HPA of a RTK point, where the baseline is 15 km long and the base station is MASCOT monument ? MASCOT GCM No: Latitude Longitude D M S SD D M S SD /-0.003m /-0.003m a. MASCOT lists standard deviations (68% confidence level); it follows that we first multiply the standard deviations by 1.96 to achieve the 95% confidence levels at the base station: 95% Confidence level Latitude= m X 1.96= m

4 95% Confidence Level Longitude= m X 1.96= m b. Second, the HPA for the Base Station is calculated: HPA Base station= (95% confidence level Latitude)²+ (95% confidence level Longitude)² = ( )= m c. Next, the uncertainty associated with the RTK baseline is calculated. The baseline is 15 KM long. The manufacturer s specifications relate an accuracy of 8 mm + 1 ppm for RTK surveys. Uncertainty baseline vector (68%)= 8 mm + (15,000,000 mm X 1/1,000,000)= 8 mm + 15 mm= 23 mm = m To calculate the 95% confidence level we apply multiplier 1.96: 95% confidence level= m X 1.96= m= 46 mm Remember that the manufacturer states these specifications for normal to favorable conditions. If conditions for GNSS surveying are inclement, then this method may not be sufficient. d. Finally, we add the uncertainty in the base station to the uncertainty in the RTK vector to calculate the HPA of the RTK point: HPA= (8 mm)² + (46 mm)²= 47 mm (For normal to favorable conditions) This approach can also be applied to the situation where corrections come from a Real Time Network. iii. The final approach for calculating HPA can be used when several RTK measurements have been taken for a single point, and the measurements are sufficiently time separated (e.g. 1-2 hours between points). In this method, a statistical approach is taken. The statistical theory behind this technique is nicely explained in the article The Normal Distribution by Dr. Charles Ghilani in the June 2015 edition of xyht. Given several time separated RTK coordinate values for a single point, the following working example demonstrates how the HPA can be calculated. This method is well suited to a spreadsheet application. Example: The following RTK coordinates for TH 045 were taken with a 1 hour time separation between observations. What is the HPA for TH 045, given that the baseline is 15 km long, and that the base station is MASCOT GCM No: Measurement 1: TH 045 N: E: Measurement 2: TH 045 N: E: Measurement 3: TH 045 N: E: a. First, we calculate the mean coordinate values: Mean N= ( )/3= m Mean E= ( )/3= m

5 b. Second, we calculate the residuals v for each measurement (where v= mean value individual observation): Measurement 1 vn= = m ve= = m Measurement 2 vn= = m ve= = m Measurement 3 vn= = m ve = = m c. Next, we calculate the standard deviation S (68% confidence level) for the Northings and Eastings S = v²/ n-1, where n = the number of observations S Northing= (-0.010² ² ²)/(3-1)= /2 = m S Easting= (-0.008² ² ²)/(3-1)= /2 = m d. To achieve the 95% confidence level, we apply the multiplier 1.96 to the standard deviations: 95% confidence level N= m X 1.96= m 95% confidence level E = m X 1.96= m e. Now we can calculate the horizontal confidence level of the point using the 95% confidence levels for the Northing and Easting: Horizontal confidence level (95%) = (95% confidence level N)² + (95% confidence level E)² = (0.061² ²)= m =73 mm f. The HPA we have just calculated does not account for the error associated with the distance from the base station or the positional error at the base station. So, we now apply these errors, both of which were calculated in the previous example: HPA= error baseline vector² + error base station² + confidence level calculated at point² = (47 mm) ² + (8 mm)² + (73 mm)² = 87 mm Additional Considerations for Georeferencing with RTK. If a survey is completed entirely by RTK, then every point in the survey will have associated coordinate values. It follows that the georeferencing of the survey plan of an RTK survey essentially becomes a matter of choosing two RTK points for which good surveying practices have been employed. What are good practices for RTK surveying? This topic is discussed in Guidelines for RTK/ RTN GNSS Surveying in Canada, which is published by Natural Resources Canada and is available at the NRCan website. This manual is suggested reading, and it recommends the following techniques for precision work with RTK equipment (pages 19-21):

6 The computation of a mean position over a specified time period (time window averaging) is recommended, however this technique on its own is not enough to guarantee positional quality. Re-occupation - To optimize the benefits from changes to the satellite geometry and atmospheric conditions, re-occupying the RTK point after a time lapse of 1-2 hours is recommended. Checks to known survey control will verify that there are no systematic errors degrading the survey accuracy. Surveying each RTK point using corrections from a second base station will provide checks on all point determination factors. Unlike static GNSS surveys, RTK point solutions are usually based on relatively few observations- It follows that redundancy is imperative when using RTK techniques for georeferencing. Static surveys also require redundancy. One member has shared that he has processed baselines where the processing software has reported excellent positional quality- however, due to multipath, the positional solution had a 1-2 metre error. Similarly, base stations have been known to move, either by tectonic shifts or other reasons (See the article Black Boxes in the December 2013 issue of The Link). 4. Georeferencing using conventional surveying equipment If a survey is georeferenced conventionally by ties to passive geodetic control (e.g. it is in an Integrated Survey Area), then PAD recommends that the coordinates should be shown at the geodetic control monuments, and the coordinates will be the MASCOT published coordinates. This technique has the benefits of simplicity (there are no calculations required) and accuracy (the horizontal positional accuracy will be calculated from the standard deviations published on the MASCOT coordinate listing). It is worth repeating that the standard deviations included in the MASCOT listings represent the 68% confidence level. These figures must be multiplied by 1.96 to achieve the 95% confidence level prior to calculating the Horizontal Positional Accuracy. Conclusion While this article emphasizes an approach based on statistical analysis, it is important to remember that it is incumbent on the surveyor to ensure that appropriate measurement techniques and redundancies are employed before relying on statistical data. In other words, don t rely on the statistical output that your system displays unless you have incorporated appropriate redundancy and observational methods into your survey. Above all else, the HPA should be a meaningful and realistic figure. This article has offered a handful of options to use for calculating HPA- some of the methods discussed are more rigorous than others. Regardless of the method used, the HPA you show on your plan needs to be a considered figure which makes sense. Looking ahead to 2016, ParcelMap BC will use the HPA value to weight the coordinates of the georeferenced points in order to guide the adjustment process. Assigning an HPA of 20 cm to georeferenced points when the survey has been done more accurately would negatively skew the adjustment of the cadastral fabric being inserted into ParcelMap BC. Alternatively, overstating the HPA will cause PMBC to put too much weight on a specific plan s information.