Orthometric Height Determination using GPS to Fast Track Development

Orthometric Height Determination using GPS to Fast Track Development: a Case study of Nairobi County, Kenya Benson Kipkemboi Kenduiywo1, Patroba Achola Odera2 and Edward Hunja Waithaka3 1,2,3

Jomo Kenyatta University of Agriculture and Technology Department of Geomatic Engineering and Geospatial Information Systems, E-Mail: [email protected], [email protected] and [email protected] Abstract The Global Positioning System (GPS) is commonly considered a threedimensional system. However, the heights obtained from GPS are typically heights above an ellipsoidal model of the Earth. The heights are not consistent with levelled heights above mean sea level, often known as orthometric height. Orthometric heights reflect the nature of terrain and are useful in geodetic and surveying applications. Conversion from ellipsoidal heights to orthometric height requires a geoid undulation model. The objective of this study was to design a geoid undulation model of Nairobi province using geometric approach in order to facilitate transformation of ellipsoidal heights to orthometric heights. The approach is based on a simple relationship between ellipsoidal h, orthometric H and geoid undulation N heights. We used a second order polynomial to interpolate geoid undulation variation within the study area. Parameters of the polynomial were computed from a set of controls with predetermined ellipsoidal and orthometric heights using least squares indirect observation method. A designed user interface based on the approach transformed ellipsoidal heights to orthometric heights to an accuracy of cm. The accuracy level enables the use of GPS for applications like engineering feasibility studies, approximating earth works and topographic mapping. Exploiting such approach for determination of orthometric heights reduces time and cost taken in ordinary levelling surveying. Surveying takes a bigger percentage cost in most engineering projects like road construction and power line route surveys. Therefore, the designed approach will spur sustainable infrastructural development by providing fast and accurate alternative means of determining orthometric heights. Moreover, orthometric heights are significant in producing contours in topographic and thematic maps which are important sources of spatial baseline information. . Keywords: Geoid undulation, ellipsoidal height, orthometrci height, GPS

1

INTRODUCTION

Determination of the geoid has been one of the main research areas in geodesy for several decades. More and more accurate geo-potential models have been developed. A vertical reference frame or datum forms the basis for most development projects in which heights are used. In particular, most of the Geomatic Engineering, surveying, geodetic and geophysics applications are require orthometric height (H), height above the geoid, because it relates to the Earth’s physical surface (Heiskanen & Moritz, 1967). Orthometric heights are generally considered to refer to Mean Sea Level (MSL), and most vertical reference frames attempt to approximate MSL as the datum for heights. In principle the geoid (a level surface which globally best fits MSL) is the ideal datum. With the development of Global Positioning Systems (GPS) techniques, a great attention has been paid to the precise determination of local/regional geoids, aiming at replacing the classical levelling with GPS surveys (Engelis et al., 1985). Several methods have been developed, which can be classified into two basic approaches: The geometric approach and the gravimetric approach. The reference frame for GPS is the World Geodetic System (WGS84), where ellipsoidal heights (h) are referred. Orthometric heights on the contrary are referred to the MSL or geoid. In order to convert ellipsoidal heights into orthometric heights knowledge of the geoid and ellipsoid surfaces and how they relate is important (Heiskanen & Moritz, 1967; Veronez et al., 2011). Although conventional methods for establishing vertical control are precise, they are laborious, costly and time consuming (Erol et al., 2008). On the other hand, the advent of GPS, ellipsoidal heights can be established efficiently and relatively inexpensively. The main problem with GPS technique is that the heights refer to a reference ellipsoid approximating the true shape of the Earth but not the geoid or MSL. The aim of this study is to develop an approach for transforming GPS heights to orthometric heights using GPS in Nairobi County. Our approach is aimed at making efficient the process of determination of orthometric heights. Consequently, engineering projects like road construction, Geographic Information Science (GIS) projects and topographic mapping will be fast tracked. 2 2.1

MATERIALS AND METHODS STUDY AREA

This study was conducted using control points covering Nairobi province in Kenya. A total of 22 points with GPS and orthometric heights evenly distributed

in the province were used. 11 points were used for parameter estimation and 10 points for testing the geoid undulation surface. Figure 1 shows the spatial distribution of the points over the study area. Figure 1: Points with ellipsoidal and orthometric heights used in the study

2.2

Data

Two categories of data namely: orthometric and ellipsoidal data were used. GPS data was obtained from Survey of Kenya (SOK) based on a survey jointly done in March 2003 by SOK and Japan International Cooperation Agency (JICA) for the densification of photogrammetric control. The GPS survey data was supplemented with GPS satellite ephemeris and Continuous Operating Reference Stations (CORS) data downloaded from (SOPAC, 2013). The levelling data was also obtained from SOK. 2.3

Methodology

The GPS raw data was processed and cycle slips corrected using Leica Geo Office software. Cycle slips occur as a result of loss of lock by the GPS receiver or when satellite signals fail to reach the receiver antennae. Only one station (KJ17) had cycle slips. This was minimized by shifting all the observations taken after the cycle slip by a given integer value. However, it is important to note that there are no unique methods for eliminating cycle slips, that is, available procedures are not guaranteed to work in all cases (Leick, 1995). Coordinates of the first order controls 148ST2, 149S2 and 149S3 were fixed and used to adjust the new observations during processing. Output from processing was projected to WGS 1984 ellipsoid on Universal Transvers Mercator (UTM) projection zone 37 south. Ellipsoidal heights (h) and orthometric heights (H) can be linked using geoid undulation (N) as illustrated in 1 and

Figure 2. This relationship was used to determine geoid undulation surface over Nairobi province. Geoid undulation surface was then plotted using contours. Transformation parameters were derived from the computed geoid undulation using ( 2 )

. Orthometric heights were then interpolated from the parameters at any location within the area of study. (1)

Figure 2: An illustration of the relationship between H, N and h.

(2)

Where (x, y) are coordinates of a given reference point, and k0, k1, k2, k3, k4 and k5 are the parameters. At least six reference points are needed in order to solve a second degree equation. An adjustment is necessary to determine parameters when the number of reference points (control points) exceeds six. This study used least squares parametric adjustment or indirect observations method. The least squares observation equation used is expressed in ( 3 ). (3)

Here ν is the residual defined as: and (i, j) are positive integer values. The observation equation can be expressed in a matrix form as:

(4)

The control points were split into two sets; 11 points for parameter determination and 10 points for testing of the outcome of the model. Estimation of parameters was done by solving ( 4 ). The parameters were used to develop a user interface using visual basic application. The interface facilitated transformation of ellipsoidal heights to orthometric heights within the area in GPS mapping platforms with windows mobile. The remaining 10 points were then used to evaluate the accuracy of the model using standard error and root mean square (RMS). Standard error was used with the assumption that systematic errors were negligible. A comparison of interpolated orthometric heights (HCOMP) against orthometric heights obtained by levelling (H) was done during accuracy assessment. 3

RESULTS

Geoid undulation surface using a vertical interval of 0.05 m over Nairobi province is illustrated in Figure 3. The geoid surface varies from -13.90 m and to -14.40 m. Figure 3: Geoid Undulation model over Nairobi province

The computed parameters (k0, k1, k2, k3, k4 and k5) are illustrated in Table 1. These results were obtained after subtracting constant values, 9845000 and 239000, from Northing and Easting respectively. This was done in order to minimize scaling effect or singularity of the design matrix.

Table 1: Coefficients of transformation

Parameters K0 K1 K2 K3 K4 K5

Values (m) -14.022 -9.558 x10-006 -2.674 x10-005 3.917 x 10-010 1.327 x 10-011 5.785 x 10-010

Standard Errors ±0.123 ±1.179 x 10-005 ±7.406 x 10-006 ±3.718 x 10-010 ±3.210 x 10-010 ±1.394 x 10-010

Orthometric height transformation parameters in

Table 1 were used to design a user interface for use in GPS mapping platforms with windows mobile. The interface allows a user to compute orthometric heights for given points. More also, the parameters can be modified when new estimates are computed as illustrated in Figure 4. Figure 4: Designed graphic user interface for use in windows mapping platforms

The designed interface was used to interpolate orthometric heights using a set of 10 sampled points excluded from parameter estimation process. A measure of accuracy, standard error σ, was computed from the deviation of interpolated COMP orthometric heights from leveling heights ∆H (H - H) as shown in

Table 2. Table 2: Transformation model test and accuracy results

SITE ID KJ02 KJ04 KJ05 KJ08 KJ09

HCOMP 1539.59 1627.91 1517.22 1487.33 1944.01

H 1539.47 1627.74 1517.12 1487.42 1943.98

∆H 0.12 0.17 0.11 -0.09 0.04

(∆H)2 0.013 0.029 0.011 0.008 0.001

KJ10 1812.83 1812.80 KJ14 1496.04 1496.11 KJ17 1862.12 1862.17 KJ19 1793.84 1793.78 KJ22 1605.42 1605.35 2 Σ(∆H) = 0.079 Standard Error (σ) = 0.090 Standard error was computed as:

0.03 -0.07 -0.05 0.06 0.08

0.001 0.005 0.002 0.004 0.006

(5)

Where, n (10) is the number of points used in testing the model. 4

DISCUSSION AND CONCLUSION

We determined the geoid undulation surface using the geometric approach. The geometrically derived geoid-ellipsoid separation is limited to the combined accuracy of the GPS data, MSL heights and the interpolation approach used (Featherstone et al., 1998). In this study, the GPS survey was done using dual frequency receivers which have higher accuracy and the MSL heights were part of national control with acceptable accuracy. Therefore, the accuracy constraints with respect to GPS and MSL heights are deemed negligible. Estimated model parameters are acceptable because normally the first coefficient of any polynomial, in this case K0, is expected to be near or within the range of the expected results. The geoid undulation ranges between -14.398 m and 13.899 m which the first coefficient falls in. In addition the validation results indicate that orthometric heights can be interpolated with an accuracy of cm. (Pikridas et al., 2011) notes that, in many geodetic and surveying applications, geoid undulation must be determined with an accuracy of a few centimeters. According to (Rapp, 1993) most applications require geoid undulation accuracy require geoid undulation information to an accuracy of cm with controls points in a resolution of 50 km. The accuracy achieved in this study is deemed to suffice for applications like engineering feasibility studies (i.e. earth work determination), GIS topographic mapping, remote sensing land-cover mapping projects and oceanographic applications which need height information. Moreover, errors between control points in such GPS survey cancel out. The user interface developed in this study works on windows mobile platform used by receivers like Trimble Juno SB. Our interface provides real time transformation of heights of points of interest to a user. It is aimed at fast tracking

mapping activities that need orthometric heights. This interface has eliminated the labour intensive, time consuming and costly levelling process especially in applications that require a few centimetres of accuracy. Our next study will implement interpolation using Support Vector Machines (SVM) with a novel regression technique as noted by (Zaletnyik et al., 2008). The efforst will also evaluate possibilities of including gravimetric data into the interpolation process. 5

ACKNOWLEDGEMENT

We are highly indebted to SOK Geodetic department for availing us the data sets that were used in this study. 6

REFERENCES

Engelis, T., Rapp, R., & Bock, Y. (1985). Measuring orthometric height differences with GPS and gravity data. Manuscripta Geodaetica, 10(3), 187-194. Erol, B., Erol, S., & Çelik, R. N. (2008). Height transformation using regional geoids and GPS/levelling in Turkey. Survey Review, 40(307), 2-18. Featherstone, W., Dentith, M., & Kirby, J. (1998). Strategies for the accurate determination of orthometric heights from GPS. Survey Review, 34(267), 278-296. Heiskanen, W. A., & Moritz, H. (1967). Physical Geodesy. San Francisco and London: WH Freeman and Company. Leick, A. (1995). GPS satellite surveying. New York: Wiley. Pikridas, C., Fotiou, A., Katsougiannopoulos, S., & Rossikopoulos, D. (2011). Estimation and evaluation of GPS geoid heights using an artificial neural network model. Applied Geomatics, 3(3), 183-187. Rapp, R. H. (1993). Geoid undulation accuracy. Geoscience and Remote Sensing, IEEE Transactions on, 31(2), 365-370. SOPAC. (2013). Scripps Orbit and Permanent Array Center (SOPAC). Retrieved 21st August, 2013, from http://sopac.ucsd.edu/

Veronez, M. R., Florêncio de Souza, S., Matsuoka, M. T., Reinhardt, A., & Macedônio da Silva, R. (2011). Regional Mapping of the Geoid Using GNSS (GPS) Measurements and an Artificial Neural Network. Remote Sensing, 3(4), 668-683. Zaletnyik, P., Völgyesi, L., & Paláncz, B. (2008). Modelling local GPS/levelling geoid undulations using Support Vector Machines. Civil Engineering, 52(1), 39-43. This work is licensed under the Creative Commons Attribution 3.0 License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/legalcode Proceedings of Global Geospatial Conference 2013 Addis Ababa, Ethiopia, 4-8 November 2013