Estimation of Regional Geoid Model using combined

PILOT PRPJECT ON

Estimation of Regional Geoid Model using combined method and Implementation in GNSS Receivers, for Improved Vertical Accuracy

P.G.C.C.Fonseka GNSS-2 Sabaragamuwa University of Sri Lanka Sri Lanka

Under the Guidance of

Mr Girish Khare SNTD/SNGG SCIENTIST/ENGINEER-"SD" Space Application Centre ISRO, Ahmedabad, India

Space Application Centre (SAC), Ahmedabad Indian Space Research organization (ISRO)

April, 2018

DECLARATION I do hereby declare that the work reported in this report was exclusively carried out by me under the supervision of Mr Girish Khare. It describes the results of my own independent research except where due reference has been made in the text. No part of this dissertation has been submitted earlier or concurrently for the same or any other post graduate diploma

.................................. P.G.C.C.Fonseka GNSS-2 Sabaragamuwa University of Sri Lanka Sri Lanka E mail: [email protected] Phone No: +94702062517

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CERTIFICATE This is to certify that the Pilot Project titled Estimation of Regional Geoid Model using combined method and Implementation in GNSS Receivers, for Improved Vertical Accuracy.which is being carried by Mr. P.G.C.C.Fonseka, student of GNSS-2, during the academic year 2017-2018 in partial fulfilment of the requirement for the award of “Post Graduate Diploma in Global Navigation Satellite Systems” as a record of students work carried out at Space Application Centre (SAC), ISRO, Ahmedabad, India under my supervision and guidance.

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Guide Mr Girish Khare SCIENTIST/ENGINEER-"SD" SNTD/SNGG GNSS Receiver Lab Space Application Centre ISRO, Ahmedabad, India E mail : [email protected] Phone No : 2691 2489 Mobile : 079-26915142

Course Director Dr. Raghunadh K Bhattar Sci/Engr 'SG' PPG/SAC Course Director, SATCOM & GNSS, CSSTEAP Space Application Centre ISRO E mail : [email protected] Phone No : 079-2691-2427 / 6005 / 6015 Mobile : 9409308359

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CONTENTS DECLARATION ........................................................................................................................ i CERTIFICATE ..........................................................................................................................ii LIST OF FIGURES ................................................................................................................... v LIST OF ACRONYMS AND ABBREVIATIONS ................................................................. vi ABSTRACT .............................................................................................................................vii ACKNOWLEDGEMENT ..................................................................................................... viii 1.0

INRTODUCTION .......................................................................................................... 1

1.1 Background ...................................................................................................................... 1 1.2 Research Problem ............................................................................................................ 3 1.3 Objective .......................................................................................................................... 3 2.0

LITERATURE REVIEW ............................................................................................... 4

2.1The Figure of Earth ........................................................................................................... 4 2.1.1 The Plane .................................................................................................................. 4 2.1.2 Ellipsoid .................................................................................................................... 4 2.1.3 Geoids ....................................................................................................................... 5 2.2 Height Systems and Vertical Datum ................................................................................ 6 2.2.1 Orthometric Height ................................................................................................... 6 2.2.2 Ellipsoidal Height ..................................................................................................... 6 2.2.3 Mean Sea Level......................................................................................................... 7 2.3 Geoid Models ................................................................................................................... 7 2.3.1 Applications of Geoid ............................................................................................... 8 2.3.2 Geoid Modeling Methods ......................................................................................... 9 3.0

METHODOLOGY ....................................................................................................... 13

3.1

Data Used .................................................................................................................. 13

3.1.1 RTM Geoid Heights ................................................................................................ 13 3.1.2 EGM 2008 Data ...................................................................................................... 16 3.2 Software and Programs Used ......................................................................................... 19 3.3 Study Area ..................................................................................................................... 19 3.4 Geoid Modeling Procedure ............................................................................................ 21 3.4 Flow Chart ..................................................................................................................... 23 .............................................................................................................................................. 23 iii

4.0 RESULTS AND DISCUSSION ........................................................................................ 24 4.1 Polynomial for Local Geoid Model ............................................................................... 27 5.0 CONCLUSION AND FUTURE SCOPE .......................................................................... 30 5.1 Future Scope for Mtech. ................................................................................................ 31 5.2 Proposed Methodology ............................................................................................... 32 6.0 BIBLIOGRAPHY .............................................................................................................. 33 7.0 LIST OF APPENDICES .................................................................................................... 36 7.1 MATLAB codes............................................................................................................. 36 7.1.1 Terrain Correction ................................................................................................... 36 7.1.2 Coordinate Projection ............................................................................................. 39 7.1.3 Final Polynomial ..................................................................................................... 40 7.2 ArcGIS steps .................................................................................................................. 41 7.2.1 SRTM for Rathnapura district ................................................................................ 41 7.2.2 Datum convertion of shape file ............................................................................... 42 7.2.3 Clipped file.............................................................................................................. 42 7.2.4 Masking SRTM ....................................................................................................... 43 7.2.5 Extracted SRTM ..................................................................................................... 43 7.2.6 Raster to point conversion ...................................................................................... 44 7.2.7 Latitude calculation ................................................................................................. 44 7.2.8 Extracting points to WordPad ................................................................................. 45 7.2.9 GPS locations .......................................................................................................... 45 7.2.10 Intersection of GPS points and SRTM heights ..................................................... 46 7.2.11 Trend ..................................................................................................................... 46 7.2.12 Kriging .................................................................................................................. 47 7.3 FORTAN77 Harmonic Synthesis Interface ................................................................... 47

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LIST OF FIGURES Figure 1 Relationship between Ellipsoidal and Orthometric Heights (Yilmaz, 2010) .............. 1 Figure 2 Geocentric & Geodetic (Britannica, 1994) .................................................................. 4 Figure 3 Relationships between Ellipsoids and Geoids (Babu, 2017) ....................................... 6 Figure 4 Relationships between Ellipsoid, Geoid and Orthometric Heights. (Canada, 2017) .. 7 Figure 5 Principle of SRTM (Simon, 2015) ............................................................................ 14 Figure 6 Notation Used for the definition of a prism (D.Nagy, 2000) .................................... 15 Figure 7 Global 2.5 Minute Geoid Undulations (NGA, n.d.) .................................................. 18 Figure 8 Ratnapura District (Lanka, 2014) .............................................................................. 19 Figure 9 Morphological Regions in Ratnapura District ........................................................... 21 Figure 10 Trend Surface .......................................................................................................... 24 Figure 11 Kriging Surface ....................................................................................................... 25 Figure 12 RTM effect .............................................................................................................. 26 Figure 13 Prediction Bounds ................................................................................................... 27 Figure 14 Geoid Undulations over Rathnapura District .......................................................... 28 Figure 15 Error per location point ........................................................................................... 29

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LIST OF ACRONYMS AND ABBREVIATIONS DEM

Digital Elevation Model

DTM

Digital Terrain Model

EGM2008

Earth Gravitational Model

FBM

Fundemental Bench Mark

FFT

Fast Fourier Tranform

GGM

Global Geopotential Model

GIS

Geographical Information System

GNSS

Global Navigation Satellite System

GOCE

Gravity field and steadt-state Ocean Circulation Eplorer

GPS

Global Positioning Sysytem

GRACE

Gravity Recovery and Climate Experiment

MSL

Mean Sea Level

NGA

U.S National Geospatial-Intelligence Agency

RMSE

Root Mean Square Error

RTM

Residual Terrain Model

SLCG2012

Sri Lankan Combined Geoid 2012

SRTM

Shuttle Radar Topographic Mission

STD

Standard Deviation

USGS

United States Geoilogical Survey

UTM

Universal Transverse Mercator

WGS84

World Geodetic System 1984

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ABSTRACT Land Surveying techniques are rapidly changed from conventional methods towards modern technological methods during the past two decades. The invasions of Global Navigation Satellite Systems (GNSS) in surveying applications have made it more efficient and cost effective. The uses of GNSS have substantially increased the requirement of a high resolution geoid model in converting the ellipsoidal height to orthometric height which has a physical meaning, as GNSS systems provides height and location information with respect to World Geodetic System 1984 (WGS84). Conventional geoid determination techniques require dense and homogeneous gravity measurements. In most of the regions gravity measurements are confined to more accessible low elevation terrain and also to establish such dense gravity network requires much cost and effort which is a highly concerned matter for most of the developing countries. Hence the observed gravity anomalies do not represent the true anomalies over the topography. This study concerns geoid modeling using a high resolution geopotential model and terrain data termed combined geoid determination through Residual Terrain Model (RTM) reduction method, where no dense land gravity data is required. This method was applied for Ratnapura district in Sri Lanka which covers an area of 3,275 km2 and located to the southwest and south of the Central Highlands and lies between 6° 15' N~ 6°55'N latitude and 80° 10'.E~ 80°57'E longitude. In estimation of the local geoid model, the high resolution global geopotential model Earth Gravitational Model (EGM2008) is used as the reference field. It is capable of representing long to medium wavelength gravity field whilst truncation at its maximum degree and order leads to omission error where short wavelength features are not represented. Global Positioning System (GPS) leveling data is used to constrain long wavelength geoid errors. RTM reduction method accounts for the omission errors. Shuttle Radar Topography Mission (SRTM3) data and Digital Terrain Model (DTM2006.0) spherical harmonic model elevation data are used to calculate the RTM effect. The topographic corrections were applied through rectangular prism forward modeling methods and Bruns formula. Hence the correction was applied to the reference global geoid model for better vertical accuracies in deriving orthometric heights. A polynomial was fitted to the final correction model and estimated coefficients are presented in results with 95% confidence. Accuracy assessment of the estimated model through cross validation and real time filed data is yet to be done. Key Words: Geoid, Orthometric Height, Global Geopotential Model, GNSS, RTM, SRTM,

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ACKNOWLEDGEMENT This research work would not have been success in this short without the timely and valuable guidance, inspiration, technical discussion, support and supervision of the guide Mr. Girish Khare, Dr. Sreejith K.M, Dr. H.M.I.Prasanna & Dr. D.R.W.Welikanna. Their in-depth knowledge, vast experience, noble intellectual, continuous enthusiasm for research has helped me immensely and inspired to work further. They have also always been accessible, approachable, and willing to help besides their own tight schedules. The SAC-ISRO & CSSTEAP administration has been kind and helpful enough in providing necessary infrastructure free Wi-Fi facilities in guesthouse, and in particular, library facilities of SAC for providing access to their bought journal papers of various acclaimed journals like IEEE with access to authentic and reliable resources. I am very much grateful for this opportunity. I would also like to thank CSSTEAP course director, Dr. Raghunadh K. Bhattar, course coordinator Mr Vishal Agarwal, Mr Pariek and Mr B.N Panchal for their kind support and motivation. I am also thankful to all my friends who have given their thoughtful feedback and suggestion in course of doing this research work and making this a successful project.

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1.0 INRTODUCTION 1.1 Background In modern era, Global Navigation Satellite Systems (GNSS) is widely used for Geodesy, Geophysics and surveying applications. The GNSS observations are referred to geodetic ellipsoid World Geodetic System 1984(WGS84) and characterized by rectangular or geodetic coordinates. A problem arises while taking orthometric heights which are more meaningful and physical. The orthometric height of a point is the distance along a plumb line from the point to the geoid and it is generally known as "height above Mean Sea Level ". It is used for all construction and leveling works in surveying field. GNSS receivers give height (Z) coordinates with respect to World Geodetic System 1984(WGS84) which has only a mathematical meaning. If the geoid undulation is known orthometric height can be derived directly which has a physical meaning.

Figure 1 Relationship between Ellipsoidal and Orthometric Heights (Yilmaz, 2010)

Ellipsoidal height of point P on the earth is distance between points P and ellipsoid surface at ellipsoidal center direction. Orthometric height of points P is distance between point P and geoid surface at gravity vector direction. Practically GPS (Global Positioning System) observations are done at point P using DGPS (Differential Global Positioning System) techniques. After processing location coordinates can be derived either Cartesian (X, Y, Z) or geographic (φ, λ, h). (RAO, 2010). Estimation of orthometric height H of point P from ellipsoidal height h depends on precise determination of geoid undulation N. In practice, for 1

determination of geoid undulations, especially for local applications, requires control points which both ellipsoidal and orthometric heights are known (Soler, 2015). Thus geoid undulation N of control points can be calculated from difference between ellipsoidal and ortometric heights. For calculation of geoid undulation of a point, in order to derive the orthometric height with geographic (φ, λ) or cartesian (x, y) coordinates, it is mandatory to determine geoid surface model of application area by using control points geoid undulations. (Soler, 2015) The geoid plays a very important role in geodesy. It can not only be seen as the most natural shape of the earth, but it also serves as the reference surface for most of the height system. Geoid is the equipotential surface of the Earth gravity field that best approximates the mean sea level. Such a reference surface is needed for a number of modern mapping, oceanographic and geophysical applications. (RAO, 2010) Geoid model determination techniques can be divided into two methods. The geoid Model could be obtained by GPS/leveling undulation measurements and it is called as geometric method. In the geometric method geoid model is determined by subtracting the orthometric height which is measured by using leveling techniques, from ellipsoidal height which is measured by using GPS techniques (Yong-qi, 1999). The other method is gravimetric method .In the gravimetric method geoid model is determined by using the difference of gravity data of actual gravity field and normal gravity field which is mathematically converted towards a model. (Sideris, 2003).Most common approach is the remove-restore approach. The geoid undulations due to long wavelength gravity field variations are estimated by a global geopotential model (GGM), and the high frequency components are obtained through a regional topographic model. The remaining medium to short wavelength features of geoid undulations are estimates by stokes or Molodensky based approaches using gravity measurements. (H.M.I.Prasanna, 2013). (N SRINIVAS, 2012) In this study the requirement of dense and homogeneous gravity measurements is omitted by using the method RTM.(Residual Terrain Modeling).In brief, Local Geoid determination for the area of interest by using Earth Gravitational Model 2008 (EGM 2008) and Shutter Radar Topographic Mission (SRTM3) data along with Digital Terrain Model (DTM2006.0) using residual terrain & rectangular prism forward modeling methods. Hence the correction will be applied to the reference geoid model for better vertical accuracies in deriving orthometric heights. Existing Global Positioning System (GPS) leveling points will be used for numerical validation.

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1.2 Research Problem

Due to the efficiency and cost effectiveness, Global Navigation Satellite Systems (GNSS) techniques have invaded the survey industry of most countries. But have been unable to resolve the height information for geophysical, surveying and construction requirements as it provides coordinates with respect to Word Geodetic System 1984 (WGS84) which has a mathematical value. Hence most common approach for obtaining orthometric heights is leveling techniques which are highly time consuming and very less cost effective when compared to GNSS techniques. The determination of orthometric heights directly from GNSS observations, which is meaningful for above purposes, requires the knowledge of geoid undulations. This study will focus on the following facts.  Geoid undulation in south India region and Sri Lanka express an extreme difference of hundred meters. (Geoid Surface is roughly 100m below the reference ellipsoid).  In most widely used standard high resolution Earth Gravitational Model (EGM2008) unable to resolve all gravity fields due to truncation of its maximum degree and order which leads to omission errors. More evident in mountainous regions where the topography is the dominant source. But surveying and many other applications requires orthometric height to the sub centimeter level accuracy.  Commonly used gravimetric geoid determination methods require a dense gravity network which necessitates cost and effort. Available gravity measurements are confined to more accessible low elevation terrain. Conducting gravity surveys is a matter of concern for developing countries such as Sri Lanka.

1.3 Objective



Assessing and validating the applicability of combined geoid determination using heterogeneous data of Global Positioning System (GPS) leveling, Earth Gravitational Model (EGM2008) and Digital Elevation Models (DEM) along with Residual Terrain Modeling (RTM) method where gravity and GPS leveling data coverage is limited.



Fitting the final model to polynomial in order to directly obtain orthometric heights from GPS-Leveling data. 3

2.0 LITERATURE REVIEW 2.1The Figure of Earth The figure of the earth in geodesy refers to those surfaces used as an approximation to the physical shape/size of the earth for the purpose of computational convenience. Surfaces of computations are the referenced surfaces used in surveying, to which observations/ measurements are referred. Basically, there are 3 types of these surfaces which are used for the representation of the earth in surveying. They are the plane, ellipsoid and geoid. (Johns, 1959) 2.1.1 The Plane The surface that corresponds to the plane is the physical terrain of the earth surface. The terrain is a topographical surface characterized by hills and valleys. It is an irregular surface, thus it cannot be used for exact mathematical computations. In Geodesy, the plane is not used as a computation surface due to the curvature of the earth. 2.1.2 Ellipsoid The shape of the Earth is not a perfect sphere, the effects of rotation and variations in gravity combine to produce an irregular, non-uniform shape. It was Isaac Newton (1643 – 1727) who first propose dosing a rotational ellipsoid as a representative figure for the Earth (H.M.I.Prasanna, 2013) .Ellipsoids are commonly used to model the Earth because they approximate the shape of the Earth on a global scale and are relatively simple mathematically to describe. Just two variables are required to model an ellipsoid; which is the length of the semi-major axis, and b which is the semi-minor axis. (RAO, 2010) Figure 2

Figure 2 Geocentric & Geodetic (Britannica, 1994)

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The semi-major and semi-minor axis can be used to calculate f, which is the flattening ratio as shown in equation 2.1.2 as:

(Equation2.1.2)

Several ellipsoid models have been developed through time. Two commonly used recent models include WGS84 (World Geodetic System 1984) and GRS80 (Geodetic Reference System 1980).WGS84 has major axis of 6,378,137.0m and a flattening of 1/298.257223563 (Leick, 1990), andGRS80 has major axis of 6,378,137.0m and a flattening of 1/298.257222101.Although ellipsoids are useful for representing a position in the horizontal plane, being a purely mathematical representation, they fail to accurately represent a meaningful vertical datum, such as mean sea level. (RAO, 2010)

2.1.3 Geoids A geoid is typically defined as an equipotential surface of the Earth’s gravity field coinciding with the mean sea level of the oceans surfaces as regarded as extending under the continents. It is a complex geometrical figure dependent on the gravity of the Earth and its motion. A geoid is of importance to engineering and geosciences as a physically defined surface for determining orthometirc heights (Torge, 2001). Orthometric heights are of primary importance because they predict the flow of fluids. This is because the flow of fluids and the level of bodies of fluids such as the oceans are determined by the Earth’s gravity field and are reflected in the gradient between orthometric heights. In comparison, ellipsoid heights do not represent the gravity field and hence the ellipsoid height gradient between two points could be opposite to the orthometic height gradient. This suggests that water could run uphill, which of course it cannot. Therefore it is orthometric heights which are of greatest concern. (Torge, 2001).Figure 3 illustrates the variation between terrains, ellipsoid, mean sea level and geoid over a certain cross section where the curvature is taken into consideration.

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Figure 3 Relationships between Ellipsoids and Geoids (Babu, 2017)

2.2 Height Systems and Vertical Datum The term in which a height system is defined is dependent on the way that the Earth’s gravity field is observed or modeled 2.2.1 Orthometric Height The orthometric height is defined as the distance along the curved plumb line between a surface point and the ellipsoid (Torge, 2001). This definition corresponds to the common understanding of height above sea level. However, because of the curved nature of the plumb line and unknown variations in gravity down the plumb line, it is not possible to physically observe or compute a true orthometic height (Bernhard Hofmann-Wellenhof, 2005). The use of gravity field models does enable its relative determination. 2.2.2 Ellipsoidal Height The ellipsoid height is the distance along the normal to the ellipsoid to the Earth’s surface. Unlike the previous definitions, it is independent of gravity and therefore does not correctly determine the flow of fluids. In some cases the gradient of ellipsoid heights can be in the opposite direction to orthometric heights. (Soler, 2015)

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2.2.3 Mean Sea Level Because of the assumption that MSL and the geoid coincide in the open oceans, it is possible to relate MSL to the geoid using tide gauge measurements and therefore define a vertical datum. In order to accurately define MSL, it is necessary to take regular measurements of MSL over a period sufficient to cancel out the effects of features such as long term tidal cycles and sea surface topography variations. A continuous record of at least 19 years is recommended for this purpose (Bernhard Hofmann-Wellenhof, 2005).

2.3 Geoid Models A geoid model is a model of the separation values between an ellipsoid and the geoid for a given area, be it global, regional or local. It is used to convert ellipsoid heights to an orthometirc height, such as mean sea level. The relationship between the different height systems is described by the following equation: H = h – N (Equation2.3)

Where: H = Orthometric height h = Height above Ellipsoid N = Geoid – Ellipsoid separation value

Therefore the orthometric height (H) can be determined by subtracting the geoid - ellipsoid separation value from the ellipsoid height. Note that if the geoid is above the ellipsoid, the N value is positive, if it is below the ellipsoid, the N value is negative. This relationship is shown in Figure 4 below.

Figure 4 Relationships between Ellipsoid, Geoid and Orthometric Heights. (Canada, 2017)

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As gravity is a function of the density of the Earth’s mass, which is not evenly distributed, the shape of the geoid is not regular, nor can it be represented by a regular mathematical expression. As a result, geoid models are often tabulated files containing grid parameters and corresponding geoid values, within terminate values being interpolated. Alternatively, spherical harmonic equations can be used to determine geoid heights (Bernhard HofmannWellenhof, 2005).Geoid models have particular application to GPS (Global Positioning System) surveying as GPS instruments measure their position relative to a chosen ellipsoid. Therefore the height measured will be the ellipsoid height. With the increased use of GPS equipment in surveying, the need for an accurate geoid model has become evident. This is particularly the case when GPS equipment is being used in preference to traditional spirit leveling, which in mountainous terrain is labor intensive and slow. (H.M.I.Prasanna, 2013)

2.3.1 Applications of Geoid

 Vertical datum for orthometric heights.  Determination of orthometric heights from ellipsoidal heights. Global Navigation Satellite Systems (GNSS) provide us with heights referring to geocentric ellipsoids; the ellipsoidal height is of little use for day-to-day requirements of height. Usual your need for height is in the form of orthometric or normal height, and to obtain these, geoid height is needed.  Understanding of ocean circulation patterns and dynamics  Description of the positions of satellites and ground stations in suitable reference frames. The fact that the geoid reflects gravity field irregularities means that, a better understanding of the geoid enables refinement of satellite orbits.  Oceanography, hydrographic surveying and maritime. The geoid is valuable for  Oceanographers, hydrographic surveyors and maritime industries in general as vessels navigate vast bodies of water. The knowledge of the geoid is essential to better model ocean currents and undersea mapping especially in soundings. Seagoing vessels can take advantage of the currents and characteristics of the ocean to plan faster and safer routes, which in turn will use less fuel (conservation) and cost less.  Knowledge of the geoid is important to model geodynamical phenomena (e.g. polar motion, Earth rotation, crustal deformation). Geoid is useful in interpretation 8

of precursors to geo-hazards researches such as the study of post-glacial rebound, earthquakes, volcanoes, landslides, tsunamis etc. and mitigation.  Vertical and horizontal control networks definition, establishment, transformation and adjustment.  Vertical datum unification: 

When direct land access is a problem: Datum transfer and unification is possible with good geoid model even when the land parcels cannot directly be accessed by land.



When there is datum inconsistencies brought about say by differences in MSL/tide gauges; geoid and ellipsoidal heights (e.g. GPS heights on benchmarks) can minimize by far the problem.

2.3.2 Geoid Modeling Methods Several methods exist for modeling geoid, either local or global models derived as part of a global or regional geodetic infrastructure. Globally, the determination of the geoid has been carried out by various researchers. In all the studies that have been carried out, it has been concluded that the combination of Geometrical interpolation method with other convectional geoid modelling methods gives better accuracy, especially for local geoid determination. (H.M.I.Prasanna, 2013) However some studies observed that interpolation of the geometrically derived geoid can prove superior to a gravimetric geoid for some survey areas that are smaller than the resolution of the gravimetric geoid, therefore caution must be taken in drawing judgment. (Bernhard Hofmann-Wellenhof, 2005) (N SRINIVAS, 2012) (Torge, 2001) (Sideris, 2003) (H.M.I.Prasanna, 2013) (Soler, 2015) Basically, there are six major techniques by which the geoid can be determined. They include: a. The GPS/leveling technique (Geometric approach) b. The Gravimetric technique c. The Astro -geodetic technique d. The Satellite technique e. The Astro-gravimetric technique f. The combination of the geometric approach and gravimetric technique Few are explained below. 9

2.3.2.1 Geometric This approach is the one most favored for GPS surveys of small extent (area covered less than 10 - 20 km2).GPS measurements are taken at benchmarks with known orthometric height (H). The difference between the GPS-derived ellipsoidal height (h) and the orthometric height (H) provides the geoidal height N at that point. In the simplest case a single control point would provide a constant height shift. However, the most common case involves three benchmarks at which GPS measurement are made. This allows two tilts (North-South and East-West) and a shift to be determined. In effect, the geoid is modeled by a tilted plane. (Yong-qi, 1999) These tilts and shift can then be applied to the heights determined from GPS at any new points in the area. The "calibration" of RTK GPS prior to the start of a survey incorporates this model, as well as a model for horizontal shifts and a rotation in azimuth. It is possible to take GPS measurement sat more than three benchmarks. In this case the commercial software will use these data to get a more reliable model of the tilted plane (i.e. the geoid is still assumed to take this shape over the region of interest). It is possible to use the extra data to generate a more complex model for the geoid, using for example polynomials, splines or Kriging, but this will require some expertise on the part of the user (Yong-qi, 1999). The chief limitations of the geometric case are: 

The model is only valid over the area encompassed by the known benchmarks (extrapolation beyond this area is highly inadvisable).



The simple tilted plane model can only be safely used over a small area (the geoid is much more complex in shape).



It is not always possible (or convenient) to find sufficient benchmarks with known orthometric heights in the area of interest.

2.3.2.2 Gravimetric This approach provides a uniform grid of geoidal heights over a large area, in contrast to the scattered point values over a limited area provided by the geometric method. However, it requires a complex numerical integration of gravity anomalies to determine a geoidal height. The key formula is that due to Stokes: ∬

( )

(Equation 2.3.2.2)

(Bernhard Hofmann-Wellenhof, 2005)

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Without going into the details of the other terms, the core "observation" is the gravity anomaly Δg, which represents the difference between observed gravity and a theoretical gravity value. The integration takes place over the entire surface of the earth, so that to get a single geoidal height, tens of thousands of gravity observations are needed. The gravity anomaly is also requires a correction for the effect of terrain variations, so a detailed digital elevation model is also needed (N SRINIVAS, 2012). An alternative representation of the geoid is to use a spherical harmonic expansion of the Earth's geopotential (a spherical harmonic model is basically two-dimensional Fourier series). Typically the coefficients of this model are determined from the analysis of the perturbations in the orbits of low-orbiting satellites. (Sideris, 2003) Nowadays it is common to use these two representations together in one of two ways: 

Use the satellite-based spherical harmonic expansion to determine the long wavelength (low frequency) component of the geoid, and a modified version of Stokes' formulator determine the high frequency component, integrating over a spherical "cap", not the entire Earth. (N SRINIVAS, 2012)



Use both satellite data and terrestrial gravity anomalies in a single solution for a highorder spherical harmonic expansion of the Earth's gravity field. This requires significant computer resources to compute the coefficients, and is beyond the capability of all but the largest agencies. Although the gravimetric approach provides good spatial coverage overlarge areas and can provide the detailed structure of the geoid, it suffers from some drawbacks. (N SRINIVAS, 2012)



It is computationally intensive and mathematically complex.



The results are only as good as the underlying gravity measurements. If there are errors in the gravity data, there will be errors in the geoid. More seriously, there are many parts of the world where gravity data are sparse or non-existent, leading to gaps or smoothing in the geoid model. (H.M.I.Prasanna, 2013)



The gravimetric geoid is susceptible to biases and tilts, due to errors in satellite orbit modeling and to gaps in terrestrial gravity data sets. (H.M.I.Prasanna, 2013)

2.3.2.3 Gravimetric and Geometric It is possible to combine the gravimetric and geometric approaches. The gravimetric geoid is computed first, and then "calibrated" by using GPS-determined geoidal heights at discrete points over the entire region. This calibration would, in the simplest case, consist of a bias 11

and two tilts. However, much more complex correction surfaces are possible, and in the case of continental geoid models desirable. This approach also compensates for any tilts or biases that may exist in the precise leveling networks and enables the user to transform his GPSderived heights directly to the national leveling datum. (Yong-qi, 1999) 2.3.2.4 Remove-Compute-Restore (RCR) method In the RCR technique, the anomalous potential T is split into three parts: T = TEGM + TRTM + TRES (Equation 2.3.2.4.1) Where TEGM is the contribution of an Earth Geopotential Model (EGM).TRTM are the terrain affects from Residual Terrain Modeling (RTM), and TRES the residual gravity field. T is treated as a spatial function. One reason for subtracting an EGM is to represent the gravity field outside the area covered with data. For the terrain-reduction, the terrain effects are reduced relative to a mean elevation surface. The terrain potential is subtracted from the observations using prism integration, i.e. representing the mass between the actual topography and the mean elevation surface as mass prisms of either positive or negative density, nominally 2670 kg/m3. The prism implementation of the RTM method has an inherent problem: the method leaves a point above the mean elevation surface in the massfree domain, whereas a point below the mean elevation surface after the reduction would correspond to the value inside the reference topography mass. As all geodetic gravity field modeling methods require observations derived from a harmonic function, i.e. in a mass-free environment above the geoid, the harmonic correction is applied to the gravity anomaly points below the mean elevation surface. (N SRINIVAS, 2012). The quasi gravimetric geoid undulation (N) is expressed as: ∫ ( )

(Equation 2.3.2.4.2)

Where NGGM is geoid undulation computed from Global Geopotential Model (GGM), which presents long wavelength contribution, NRTM is the indirect effect of the terrain on the geoid undulation estimated through a regional topographic model, S (ψ) is Stokes’ function, Δgres is residual gravity anomaly effect and γ is mean normal gravity. (N SRINIVAS, 2012). Dense and homogeneous gravity measurements are required to omit the aliasing effect. (H.M.I.Prasanna, 2013).

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3.0 METHODOLOGY 3.1 Data Used 1. Residual terrain model (RTM) data 

Shuttle Radar Topographic Mission (SRTM) 90m resolution



Digital Terrain Model (DTM2006.0) spherical harmonic model of global earth’s topography.

2. Height Data 

Differential GPS (DGPS) height values of control points within the area of interest.



Orthometric height values of the control points within the area of interest.

3. Global geopotential model (GGM) data for geoid undulations 

Earth gravitational model (EGM2008).

3.1.1 RTM Geoid Heights EGM2008 inherits omission errors due to truncation. The RTM method is capable of computing EGM2008 omission errors. DTM2006.0 and SRTM elevation data are two freely available data sources necessary to obtain RTM data. (C. Hirt, 2010) 3.1.1.1 SRTM Digital Elevation Model (DEM) SRTM is an international project, a joint endeavor of the National Aeronautics and Space Administration (NASA), the National Geospatial-Intelligence Agency (NGA) of USA government, and the German and Italian Space Agencies. On February 11, 2000, the Shuttle Radar Topography Mission (SRTM) payload onboard Space Shuttle Endeavour was launched into space. To acquire topographic (elevation) data, the SRTM payload was outfitted with two radar antennas. One located in the shuttle's payload bay, and the other on the end of a 60metre (200-feet) mast that extended from the payload once the Shuttle was in space. With its radars sweeping most of the Earth's surfaces, during its ten days of operation, the Shuttle Radar Topography Mission collected topographic data over nearly 80 percent of Earth's land surfaces, creating the first-ever most complete near-global high-resolution database of the Earth's topography, further reference is found at: (Technology, 2018).SRTM data is available at the USA Geological Survey's (USGS) EROS Data Centre for download via File Transfer Protocol (ftp); thus ftp to ( (USGS, 2018). 13

Several new web sites have posted SRTM data in different formats than available at the USGS ftp site or the Seamless Server and some have improved it for deficiencies like data voids and spikes, for example one may want to check the Global Land Cover Facility at (Anon., 2017) or the CGIAR Consortium for Spatial Information (CGIAR, 2017) and Jonathan de Ferranti site (Ferranti, 2014). Although the USGS SRTM data is available in two resolutions of 1”and3”, the 1-arc second data is available only for the USA government and its allies. In this research, the 3”-SRTM (SRTM3) was downloaded from the CGIAR site given above; it is in 50×50tiles.The chosen AOI for our research task is the area bounded by 50≤ϕ ≤ 100and 800≤ λ ≤ 850 (latitude is given by ϕ and longitude by λ)as a Geo Tiff format file.

Figure 5 Principle of SRTM (Simon, 2015)

3.1.1.2 DTM2006.0 Spherical Harmonic Model DTM2006.0 is a spherical harmonic model of global Earths topography which comprises 2.4 million pairs of fully normalized height coefficients

(C. Hirt, 2010).Data can be

retrieved through the following link (NGA, 2014) in the file of Coef_Height_and_depth _to2190_DTM2006.0 (in units of meters). Heights {H} above Mean Sea Level (MSL) are reckoned positive (+), while Depths are reckoned negative (-). The elevation model is given by:

14

(

)

∑ ∑(

)

(

)

(Equation 3.1.1.2.1) (C. Hirt, 2010) This model is complete to degree and order 2190. The file contains 2401336 ASCII formatted records, each record containing: {n, m, HC, HS nm}

{2i5, 2d 25.15} this file can also be

read with free format. This file can be used to compute (among other things) the Height_Anomaly_to_Geoid_Undulation

conversion

term.

For

geoid

undulation

computations, where the full resolution of EGM2008 is sought, it is recommend the use of the EGM2008 gravitational model to degree 2190, with the parallel use of this elevation expansion to degree 2160. (Nikolaos K. Pavlis, 2012) DTM2006.0 has the same spatial resolution as EGM2008.So it can be considered as a low pass filter which removes long wavelength topographic components associated to EGM2008 from height resolution (90m) SRTM data. (C. Hirt, 2010) RTM equations are computed by

(Equation 3.1.1.2.2) (H.M.I.Prasanna, 2013) RTM elevations are converted to the corresponding geoid heights prism forward modeling method. (D.Nagy, 2000)

using the rectangular

Figure 6 Notation Used for the definition of a prism (D.Nagy, 2000)

Each grid point represents a rectangular prism of constant density ρ0 for which the gravitational potential V is computed. With the corner coordinates (x1, y1, z1) and (x2, y2, z2) of a single prism. The rectangular prism potential is given by 15

(

)

(

)

(

)

(Equation 3.1.1.2.3) (D.Nagy, 2000) x = Xp y = Yp z = Zp x1 = X1-Xp y1 = Y1-Yp z1 = Z1-Zp x2 = X2-Xp y2 = Y2-Yp z2 = Z2-Zp √ Where r is the distance between the point (x, y, z) and the origin of the coordinate system, and G is the gravitational constant. The standard topographic mass-density of ρo= 2, 670 kgm-3.

The SRTM/DTM2006.0RTMdata is transformed to RTM height anomalies using the prismintegration forward-modeling method. For the conversion of the prism’s potential V to its height anomaly contribution N prism, a variant of Bruns’s equation is applied. (Equation 3.1.1.2.4) (C. Hirt, 2010) Where

is normal gravity on the quasigeoid. The height anomaly contribution

prisms forming the RTM is then obtained as the sum of the height anomalies

of all prism

implied by all the single prisms: ∑

( ) (Equation 3.1.1.2.5) (C. Hirt, 2010)

RTM data is estimated through the above process.

3.1.2 EGM 2008 Data EGM2008 is a spherical harmonic model of the Earth’s gravitational potential, developed by a least squares combination of the ITG-GRACE03S gravitational model and its associated error covariance matrix, with the gravitational information obtained from a global set of areamean free-air gravity anomalies defined on a 5 arc-minute equiangular grid. This grid was formed by merging terrestrial, altimetry-derived, and airborne gravity data. Over areas where 16

only lower resolution gravity data were available, the inspectoral content was supplemented with gravitational information implied by the topography. EGM2008 is complete to degree and order 2159, and contains additional coefficients up to degree 2190 and order 2159. Over areas covered with high quality gravity data, the discrepancies between EGM2008 geoid undulations and independent GPS/Leveling values are on the order of _5 to _10 cm. EGM2008 vertical deflections over USA and Australia are within _1.1 to _1.3 arc-seconds of independent astrogeodetic values. These results indicate that EGM2008 performs comparably with contemporary detailed regional geoid models. EGM2008 performs equally well with other GRACE-based gravitational models in orbit computations. Over EGM96, EGM2008 represents improvement by a factor of six in resolution, and by factors of three to six in accuracy, depending on gravitational quantity and geographic area. EGM2008 represents a milestone and a new paradigm in global gravity field modeling, by demonstrating for the first time ever, that given accurate and detailed gravimetric data, a single global model may satisfy the requirements of a very wide range of applications. (Nikolaos K. Pavlis, 2012) In order to compute height anomalies NEGM2008 from the set of EGM2008 fully normalized spherical harmonic coefficients Cnm, Snm the standard series expansion of spherical harmonic synthesis is used. (C. Hirt, 2010)

(

)

[

∑ ( ) ∑( ̅

̅

)̅

(

)]

(Equation 3.1.2.1) (Nikolaos K. Pavlis, 2012) with degree n and order m of the harmonic coefficients and nEGM max indicating the maximum degree of the series expansion (e.g., 2,190), GM (geocentric gravitational constant) and a (semi major axis) are the EGM2008 scaling parameters, γ is normal gravity on the surface of the reference ellipsoid, Pnm(cosθ) are the fully normalized associated Legendre functions (Bernhard Hofmann-Wellenhof, 2005). The coordinate triplet (r, θ, λ) denotes the geocentric polar coordinates of radius, geocentric co-latitude and longitude, which are computed from the geodetic coordinates (ϕ, λ, h) (geodetic latitude, longitude and ellipsoidal height) for each computation point (Torge, 2001) Following website provides with the necessary spherical harmonic models ((a), (b)) and harmonic synthesis software. (NGA, 2014) 17

a) EGM2008_to2190_TideFree.gz This file contains the fully-normalized, unit-less, spherical harmonic coefficients of the Earth’s gravitational potential { ̅ standard deviations {sigma ̅ { ̅

, ̅

, sigma ̅

, ̅

} and their associated (calibrated) error } as implied by the EGM2008 model. The

} coefficients are consistent with the expression: (Equation 3.1.2.1). Where

GM= 3986004.415*108m3s-2, a= 6378136.3m.The file contains 2401333 ASCII formatted records, each record containing: {n, m, ̅

, ̅

} , sigma ̅

, sigma ̅

}to {2i5, 2d25.15, 2d20.10}. Missing and non-existent coefficients are written as zeros. The file can also be read with free format. (Nikolaos K. Pavlis, 2012) b) Zeta-to-N_to2160_egm2008.gz This file contains fully-normalized spherical harmonic coefficients of Height Anomaly to Geoid Undulation conversion term {

,

} in units of meters.

These conversions are applied to EGM2008 height anomalies computed on the WGS 84 ellipsoid, to yield EGM2008 geoid undulations with respect to WGS 84. The {

,

} coefficients are consistent with the series: (

)

∑ ∑(

)

(

)

(Equation 3.1.2.1) (Nikolaos K. Pavlis, 2012) This model is complete to degree and order 2160. The file contains 2336041 ASCII formatted records, each record containing: {n, m,

,

}

{2i5, 2d25.15}

This file can also be read with free format. For geoid undulation computations, where the full resolution of EGM2008 is sought, it is recommend the use of the EGM2008 gravitational model to degree 2190, with the parallel use of this Height Anomaly to Geoid Undulation conversion expansion to degree 2160. (Nikolaos K. Pavlis, 2012)

Figure 7 Global 2.5 Minute Geoid Undulations (NGA, n.d.)

18

3.2 Software and Programs Used     

ArcGIS 10.3 MATLAB 14a Microsoft excel 2010 Fortran77 program for very-high-degree harmonic synthesis Python Language

Arc GIS software was used to deal with the SRTM image and EGM 2008 point data for extracting height data relevant to the Ratnapura district. Trend and Interpolation (kriging) were also done using Arc GIS. Matlab software was used to do mathematical computations regarding prism integral formula theories, terrain corrections, converting coordinate systems (WGS 84 to UTM 44N) and also used for producing surfaces and polynomials. Microsoft Excel was used to store and manage height data, coordinate data, trend data, kriging data and the geoid undulation data etc. FORTRAN was used for spherical harmonic synthesis in obtaining the geoid undulations for EGM 2008 and corresponding heights from DTM2006.0. In the preceding sections the process which was done in the research task using these programs will be discussed in detail.

3.3 Study Area The study area is Ratnapura district in Sri Lanka. It covers an area of 3,275 km2 and located to the southwest and south of the Central Highlands and lies between 6° 15' N ~ 6°55'N latitude and 80° 10'E.~80°57'E longitude. The general elevation of the district ranges from 30 m to 2,135 m. Mountain ranges; high peaks, dissected plateaus, escarpments etc. cover a greater part of the district.

Figure 8 Ratnapura District (Lanka, 2014)

19

From its height and slope characteristics the district can be divided into three main morphological regions (Figure 8)

(a) The lowlands which include mainly the basins of the Kalu Ganga and the Walawe Ganga. On the basis of elevation the lowlands may be further subdivided into two distinctive units: the first with an elevation up to 30 m and the second from 30 m to 270 m.

(b) The uplands with an elevation of 270 m to 1,060 m consist of a ridge and valley topography. Furthermore, the uplands are also characterized by highly dissected plateaus of the Sabaragamuwa ridges, the Rakwana massif (hills) and the Southern Platform of the Central Highlands. The Sabaragamuwa ridges and the Rakwana hills are detached from the Central Highlands and extend from northwest to southeast within the region, the Rakwana mass is much smaller in area extent and lower in height than the Sabaragamuwa ridges. Several peaks in the area are over 1,060 m. The highest part of the massif consists of a series of high plains such as Handapan Ella and Tangamale at a general elevation of 1,060 m to 1,220 m. They are surrounded by several peaks and escarpments such as Beralagala (1,385 m), Gongala (1,346 m), Suriyakanda (1,310 m) and the Abbey Rock (1,300 m). On the average the slopes vary from 10° to 35° in the upland ridges, (depending on their litho logy and structure). Well-developed steep scarps are common in this area.

(c) The highlands which lie at the elevation of over 1,060 m consist of plains and plateaus, mountain peaks and ridges, rock-knob plains, erosion remnants, steep rock lands and lithosols. These features characterize the highest elevations of the southern rim of the Central Highlands, Rakwana massif and the Southern platform. It may be noted that the southern rim of the Central Highlands extends along the northern part of the Ratnapura district is at elevations ranging from about 1,250 m to over 2,000 m.

20

Figure 9 Morphological Regions in Ratnapura District

3.4 Geoid Modeling Procedure 1. The high frequency component of the gravitation field which stems from the topographic masses was removed by taking the differences between GPSleveling undulations and the global geoid model heights. Rectangular Prism Forward modeling method was used in removing the topographic contribution on geoid undulations from both GPS leveling data and EGM2008 geoid model using Matlab. (Appendix 7.3, Section 3.1.1.2 DTM2006.0 Spherical Harmonic Model, Appendix 7.1.1 for rectangular prism forward modeling Equation 3.1.1.2.3 and below procedure was followed. EGM2008 geoid undulations were estimated as described in section 3.1.2 EGM 2008 Data equation 3.1.2.1 with fortan77 Spherical Harmonic Synthesis Program). ArcGIS was used in extracting cell values from raster image according to the Rathnapura district shape file (Appendix 7.2.1 to 7.2.10). These raster values were converted into point file and were used in prism integral formulas. GPS leveling undulations were calculated using orthometric heights obtained in relevant coordinates from spirit leveling techniques. RTM effects were estimated as mentioned in section 3.1.1 RTM Geoid Heights. Geodetic coordinates we converted from WGS84 to UTM(44N) using Matlab (Appendix 7.1.2)

21

GPS levelling data were reduced by high resolution DTM SRTM3. The EGM2008 geoid model was reduced by DTM2006.0 model as it is used for EGM2008 topographic correction. (Nikolaos K. Pavlis, 2012). The reduced geoid difference

at a GPS leveling point is given by

( ( ( Where

(

)

(

) )

) )

(Equation 3.4.1) (H.M.I.Prasanna, 2013)

- GPS leveling geoid undulation - EGM2008 Geoid Undulation - SRTM3 topographic effect on geoid undulations - DTM2006.0 topographic effect on geoid undulation - RTM effect on geoid undulation.

The RTM corrections were estimated as the difference between high resolution SRTM3 and the low resolution DTM2006.0 models. As DTM2006.0 is the topography model incorporated in EGM2008, the RTM corrections represent the fine geoid features which are not included in the EGM2008 model.

2. The trend of the RTM reduced geoid undulation differences was modeled through a low order trend function. The undulation difference can be represented as (Equation

3.4.2)

(H.M.I.Prasanna, 2013) Is the RTM reduced geoid undulation differences using spherical harmonic degree 2190 of EGM2008 and 2160 of DTM2006.0 reference surfaces. parameters(x1, x2, x3,); ai is a n x 1 vector of coefficient and

Is an n x 1 vector of unknown denotes the residual vector.

The simplest parametric form which is a linear model was used in this study by the platform is ArcGIS. (Appendix 7.2.1.1) (Equation 3.4.3) 3. The remaining residual were modeled using a continuous corrective surface.

22

The detrended residuals

were further modeled using interpolation techniques. Since a

systematic bias was found kriging interpolation technique was used by the platform of Arc GIS. If not weighted average must be used. (Appendix 7.2.12) 4. The combined geoid for the region was obtained by adding corrective surface, trend surface and RTM effect to the global geoid model. The model was fitted to a polynomial and the coefficients were calculated using MATLAB. (Appendix 7.1.3)

3.4 Flow Chart GPS Leveling

EGM 2008

DTM 2006.0

SRTM

ΔN = (𝑁 𝐺𝑃𝑆 𝐿𝐸𝑉𝐸𝐿𝐼𝑁𝐺

𝑁 𝐸𝑀𝐺

= (𝑁 𝐺𝑃𝑆 𝐿𝐸𝑉𝐸𝐿𝐼𝑁𝐺

) - (𝑁 𝑆𝑅𝑇𝑀 𝑁 𝐸𝑀𝐺

)

𝑁 𝐷𝑇𝑀

)

𝑁 𝑅𝑇𝑀

Trend Modeling (T) Δ N = 𝑎𝑖𝑇 xi + Ɛ𝑖 = T + Ɛ𝑖

Corrective surface (C) Obtained by Kriging Of residuals Ɛ𝑖

Final Geoid Height (N) N=𝑁 𝐸𝐺𝑀 + 𝑁 𝑅𝑇𝑀 + T + C

Fit to polynomial Estimation of Polynomial efficient

Minimum RMSE

23

4.0 RESULTS AND DISCUSSION

The main concern of this study was determining precise geoid using combined method where both GPS leveling data and gravity measurement data are inadequate. In between the process there were several surfaces created.

Figure 10 Trend Surface

24

The

(Equation 3.4.2) was analyzed through geostatistical wizard of ArcGIS and detected

trend was modeled through the low order trend function. (Equation 3.4.3). It indicates a minimum of -1250.969482421875m maximum of 791.0538940429688m undulation values. Mean value is 25.15534172867088m and standard deviation is 169.156161199107m. The extent of data is as follows in decimal degrees. Top 6.812081112, bottom 6.333618888, left 80.273862888 and right 80.849542536. (Figure10). The detrended values were interpolated using kriging method. Data was not extrapolated as it reduces the truthfulness of surface data.

Figure 11 Kriging Surface

25

Residuals were calculated by predicting the sample values from the estimated trend surface. The difference between true values and the trend values for the same latitude and longitude was interpolated to obtain the kriging surface.(Figure 11). The extent is same as the trend surface.

It

shows

a

minimum

of

-75.49160003662109m

and

a

maximum

of

106.0877990722656m. Mean value is 0.09379979071719463m and standard deviation is 10.12754546251276m. The EGM2008 global model is unable to resolve some of the short wavelength gravitational fields due to truncation and DTM model being used for topographic correction. In this study the topographic correction is replaced by high resolution SRTM values. Hence the RTM reduction method is applied in account for the topographic irregularities relative to smooth reference surface. The advantage of RTM method is that it only accounts for the topographic effect that has not been included in the spherical harmonic model of the gravity field.

Figure 12 RTM effect

Figure 12 obtained as mentioned in section 3.1.1.2. Linear interpolation was used. Piecewise linear surface computed where X(UTM) is normalized by mean 4.515e+05 and standard deviation 1.837e+04 and where Y(UTM) is normalized by mean 7.266e+05 and standard deviation 1.527e+04. Sum of squared errors of prediction is 3.981e-12 and R-squared value is 1 which indicates the goodness of fit. Figure 12 represents the relationship between earth 26

masses and the terrain correction. In comparison with SRTM values the figure shows a smooth proportional relationship between the terrain correction and the terrain mass volume. As the terrain mass increases, the correction value increases. The effect of terrain to the geoid undulation is in centimeters. That result matches with what can be expected from the terrain. Final surface model was obtained by adding trend surface, kriging surface and RTM effect to the global geoid model.

4.1 Polynomial for Local Geoid Model The final refined geoid model was fitted to polynomial using MATLAB in order to predict geoid undulations within the area of extent. Prediction bound was set to 99%. (Figure13)

Figure 13 Prediction Bounds

Polynomial model is as f(x,y) in order to derive user required geoid undulations at any desire location x, y (Projected coordinates from WGS84 to UTM 44N latitude and longitude respectively) within the boundary limits. f(x,y) = p00 + p10*x + p01*y + p20*x^2 + p11*x*y + p02*y^2 + p30*x^3 + p21*x^2*y + p12*x*y^2 + p03*y^3 + p40*x^4 + p31*x^3*y + p22*x^2*y^2 + p13*x*y^3 + p04*y^4 + p50*x^5 + p41*x^4*y + p32*x^3*y^2 + p23*x^2*y^3 + p14*x*y^4 + p05*y^5 Where x is normalized by mean 4.515e+05 and standard deviation 1.837e+04and where y is normalized by mean 7.266e+05 and standard deviation 1.527e+04.Coefficients (with 95% confidence bounds): p00 =

11.48

p10 =

-77.52

p01 =

-271.9 27

p20 =

-26.93

p11 =

-41.72

p02 =

-81.02

p30 =

23.67

p21 =

120.3

p12 =

92.33

p03 =

95.45

p40 =

0.7288

p31 =

-1.954

p22 =

3.777

p13 =

5.56

p04 =

13.51

p50 =

-2.699

p41 =

-5.534

p32 =

-1.562

p23 =

-9.144

p14 =

6.434

p05 =

8.242

Figure 14 Geoid Undulations over Rathnapura District

Goodness of fit for the final geoid model is as follows.

28

Sum of squared errors of prediction is 1.879e+09. The R-squired value is 0.5513 as well as the adjusted R-squired value is 0.5512. The root mean square error was found out to be 157.5. In order to evaluate the polynomial the GSS8000 GNSS multi constellation simulator made by Spirent, which is capable of providing GPS IRNSS and SBAS simulations, was tested with orthometric heights obtained through leveling technques. It was found out that a sub meter level deviation was there between the true values and the simulated values for the same location set of coordinates.

Figure 15 Error per location point

Figure 15 represents the variation in error between the two reference frames considering the field observed orthometric heights are as the more accurate value. Sample data was not sufficient to model or to find out a systematic bias in the error. The mean error was found out to be 0.4172m and the standard deviation is 0.2730m. The Global geoid model is also capable of estimating orthometric height values in a sub meter level accuracy. Hence the aim of this study is to refine the global geoid model and provide more accurate results it was decided not to validate the locally derived precise geoid model using the GNSS simulator as it also inherits a sub meter level error in estimating orthometric heights.

29

5.0 CONCLUSION AND FUTURE SCOPE Gravity related quantities are vital for geodetic and geophysical applications. (Ex: - gravity anomaly, gravity gradient, vertical deflection, geoid undulation etc.) The density and the accuracy of these data is a matter of concern for related studies and precise interpretation. In most of the regions gravity measurements are confined to more accessible low elevation terrain. And also there are many regions where high quality and dense gravity related measurements due to financial (most of the developing countries) and accessible difficulties. The main concern of this study was to avoid the requirement of such land gravity data in precise geoid determination for direct estimation of orthometric heights with improved vertical accuracy from GNSS observation surveying techniques. This proposed precise geoid determination method is comprised with global gravity model EGM2008, sparse GPS leveling data along with global digital terrain model SRTM. RTM geoid undulations were initially removed from the differences between GPS-leveling and the global model geoid heights. Afterwards, the remaining residuals were interpolated as trend and continuous corrector surfaces. Finally, the combined geoid was obtained by adding corrective surface, trend and RTM effect to the global model geoid heights. This method can be beneficial for many developing countries and regions where there are no dense gravity and GPS-leveling networks. In this study it was assumed that the spirit leveling heights were the most accurate. But there can be more blunders and random errors in spirit leveling data. It is advisable to use a geoid based model such as SLCG2012 in estimating orthometric heights for the desired control points. The method followed in this study solely depends on the geometric distribution and the accuracy of the GPS leveling points. The size of the sample set which are used as control points (GPS leveling points) must be tested with the level of accuracy achieved by estimated geoid model for various sample sizes. At present more accurate and high resolution and gravitational models derived from GRACE, CHAMP, GOCE satellite missions, and SRTM improved versions in combined with ASTER global digital elevation elevations maps which provides an accuracy of 1 arc second are available. These data sources can be used in deriving the local geoid model. For the rectangular prism forward modeling method standard topographic mass density was taken as 2, 673 kgm−3. If prism integral theories can be applied pieces wise according to the density change more accurate RTM effects could be derived. 30

5.1 Future Scope for Mtech. 1) In order to investigate the most appropriate RTM corrections for the test area, the computation is to be repeated using different EGM2008/RTM combinations by changing the maximum spherical harmonic degree of EGM2008 and DTM 2006.0. [

]

Various combinations will be trialed out in order to get the minimum

.

2) In geodetic literatures various parametric models have been used in different adjustment schemes depending on the amount of systematic errors in triplet height data sets (h, H, and N) for trend modeling. Many studies have used three or four parameter models.

Also five parameter extension

And seven parameter models also have been used

Where, (

)

e is the first eccentricity of the reference ellipsoid and

,

are the latitude and longitude

vectors. Various trend models will be trialed out in order to get the minimum

.

3) Evaluation of the performance of the estimated geoid model accuracy by iterative cross validation and real time field data for accuracy assessment.

31

5.2 Proposed Methodology GPS Leveling

DTM 2006.0

EGM 2008

SRTM

ΔN = (𝑁 𝐺𝑃𝑆 𝐿𝐸𝑉𝐸𝐿𝐼𝑁𝐺

𝑁 𝐸𝑀𝐺

= (𝑁 𝐺𝑃𝑆 𝐿𝐸𝑉𝐸𝐿𝐼𝑁𝐺

) - (𝑁 𝑆𝑅𝑇𝑀 𝑁 𝐸𝑀𝐺

)

𝑁 𝐷𝑇𝑀

)

𝑁 𝑅𝑇𝑀

Different spectral combinations of EGM 2008 and DTM2006.0

Minimum ΔN

𝐸𝐺𝑀 𝐷𝑇𝑀 𝑁𝑚𝑎𝑥 = 𝑁𝑚𝑎𝑥 ∈ [

]

Trend Modeling (T) Δ N = 𝑎𝑖𝑇 x + Ɛ𝑖

Testing with various trend models

= T + Ɛ𝑖

Minimum Ɛ𝑖 Corrective surface (C) Obtained by Kriging Of residuals Ɛ𝑖

Final Geoid Height (N) N=𝑁 𝐸𝐺𝑀

+ 𝑁 𝑅𝑇𝑀 + T + C

Fit to Polynomial Estimation of Polynomial coefficients

Minimum RMSE

Accuracy assessment

Cross Validation

Real time field data

32

6.0 BIBLIOGRAPHY Anon., 2017. Global Land Cover Facility. [Online] Available at: http://glcf.umd.edu/data/srtm/ [Accessed 14 April 2018]. Babu, S., 2017. GEOPHYSICS. [Online] Available at: http://gpsurya.blogspot.in/2017/11/ [Accessed 12 April 2018]. Bernhard Hofmann-Wellenhof, H. M., 2005. Physical Geodesy. 1st ed. Graz: SpringerVerlag Wien. Bradford W Parkinson, J. J. S. J., 1996. Global Positioning System: Theory and Applications, Volume II. 2nd ed. Washington,DC: American Institute of Aeronautics and Astronautics,Inc. Britannica, E., 1994. Reference Ellipsoid. [Online] Available at: https://www.britannica.com/science/reference-ellipsoid [Accessed 12 April 2018]. C. Hirt, W. E. F. ,. U. M., 2010. Combining EGM2008 and SRTM/DTM2006.0 residual terrain model data to improve quasigeoid computations in mountainous areas devoid of gravity data. Journal of Geodesy, 84(9), pp. 1-11. Canada, N. R., 2017. Height Reference System Modernization. [Online] Available at: http://www.nrcan.gc.ca/earth-sciences/geomatics/geodetic-referencesystems/9054 [Accessed 13 April 2018]. CGIAR, 2017. CGIAR-CSI. [Online] Available at: http://srtm.csi.cgiar.org/ [Accessed 14 April 2018]. D.Nagy, G. J., 2000. The Gravitational Potential and its Derivatives for the Prism. Journal of Geodesy, 74(8), pp. 552-560. Ferranti, J. d., 2014. DIGITAL ELEVATION DATA. [Online] Available at: http://viewfinderpanoramas.org/dem3.html [Accessed 14 April 2018]. H.M.I.Prasanna, 2013. Precise Geoid Determination and its Geophysical Implication in Sri Lanka (Phd Thesis). s.l.:The Hong Knog Polytechnic University. Johns, R. K. C., 1959. The Figure of the Earth. Journal of the Royal Astronomical Society of Canada, Volume 53, pp. 257-263. Lanka, S. D. o. S., 2014. Topo Map/Data Indormation. [Online] Available at: http://www.survey.gov.lk/surveyweb/Home%20English/Map%20Index.php [Accessed 15 April 2018]. 33

N SRINIVAS, V. M. T. S. T. P. E. M. S. N., 2012. Gravimetric geoid of a part of south India and its comparison with global geopotential models and GPS-levelling data. Journal of Earth System Science, 121(4), pp. 1025-1032. NGA, 2014. Office of Geomatics. [Online] Available at: http://earth-info.nga.mil/GandG/update/index.php?action=home [Accessed 28 April 2018]. NGA, n.d. EGM2008 GIS Data. [Online] Available at: http://earth-info.nga.mil/GandG/wgs84/gravitymod/egm2008/egm08_gis.html [Accessed 14 April 2018]. Nikolaos K. Pavlis, S. A. H. C. K. K. F., 2012. The development and evaluation of the Earth GravitationalModel 2008 (EGM2008). JOURNAL OF GEOPHYSICAL RESEARCH, Volume 117. RAO, G. S., 2010. Global Navigation Satellite Systems with Essentials of Satellite Communications. 1st ed. Chennai: Mc Graw Hill Education. Sideris, F. A. B. &. M. G., 2003. Two different methodologies for geoid determination from ground and airborne gravity data. Geophysical Journal International, 155(3), pp. 914-922. Simon, 2015. Digital Geography. [Online] Available at: http://www.digital-geography.com/srtm-1-1-arc-second-now-available-largeglobal-coverage/ [Accessed 14 April 2018]. Soler, G. W. a. T., 2015. Measuring Land Subsidence Using GPS: Ellipsoid Height versus Orthometric Height. Journal of Surveying Engineering, 141(2). Technology, C. I. o., 2018. Shuttle Radar Topography Mission. [Online] Available at: https://www2.jpl.nasa.gov/srtm/cbanddataproducts.html [Accessed 12 April 2018]. Torge, W., 2001. Geodesy. 2rd ed. Berlin ,New york: Walter de Gruyter . USGS, 2018. EarthExplorer. [Online] Available at: https://earthexplorer.usgs.gov/ [Accessed 14 April 2018]. Yilmaz, M., 2010. ResearchGate. [Online] Available at: https://www.researchgate.net/publication/229038967_Evaluation_of_recent_global_geopoten tial_models_based_on_GPSlevelling_data_over_Afyonkarahisar_Turkey/figures?lo=1 [Accessed 10 04 2018]. Yong-qi, a. Z.-j., 1999. Research Gate. [Online] Available at: 34

https://www.researchgate.net/publication/245291696_Determination_of_Local_Geoid_with_ Geometric_Method_Case_Study [Accessed 11 April 2018]. Yong-qi, Y. Z.-j., 1999. Determination of Local Geoid with Geometric Method: Case Study. Journal of Surveying Engineering, 3(125).

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7.0 LIST OF APPENDICES 7.1 MATLAB codes 7.1.1 Terrain Correction for k=1: 429453 longitude=ts(k,1); latitude=ts(k,2); %Asking for user inputs %longitude=input('longitude: '); %latitude=input('latitude : '); % converting entered longitudes and latitudes into UTM x,UTM y lon=longitude*(pi)/180; lat=latitude*(pi)/180; lam=(81)*(pi)/180; f=1/298.257223563; No=0; Eo=500000; ko=0.9996; a=6378137; n=f/(2-f); A=(a/(1+n))*(1+((n^2)/4)+((n^4)/64)); alpha1=n/2-(2*(n^2)/3)+(5*(n^3)/16); alpha2=(13*(n^2)/48)-(3*(n^3)/5); alpha3=61*(n^3)/240; beata1=n/2-(2*(n^2)/3)+(37*(n^3)/96); beata2=(1*(n^2)/48)+(1*(n^3)/15); beata3=17*(n^3)/480; gama1=(n*2)-(2*(n^2)/3)-(2*(n^3)); gama2=(3*(n^2)/7)-(8*(n^3)/5); gama3=56*(n^3)/15; t1=(2*n^0.5/(1+n)); t=sinh(atanh(sin(lat))-(t1*(atanh(t1*sin(lat))))); ep=atan((t/(cos(lon-lam)))); eata=atanh((sin(lon-lam))/((1+(t^2))^0.5)); e1=(alpha1*cos(2*ep))*(sinh(2*eata)); e2=(alpha2*cos(4*ep))*(sinh(4*eata)); e3=(alpha3*cos(6*ep))*(sinh(6*eata)); E=Eo+ko*A*(eata+e1+e2+e3);

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n1=(alpha1*sin(2.*ep))*(cosh(2.*eata)); n2=(alpha2*sin(4.*ep))*(cosh(4.*eata)); n3=(alpha3*sin(6.*ep))*(cosh(6.*eata)); N=No+ko.*A.*(ep+n1+n2+n3); % creating m1 matrix(UTM x,UTM y, SRTM height) m1=xlsread('srtm.xlsx','F3:H429453'); % creating M matrix(x1,x2,y1,y2,z1,z2 in prism intergral formula) M(429453,6)=0; % calculating values for M matrix for e=1:429451 M(e,1)=m1(e,2)-45-E; M(e,2)=m1(e,2)+45-E; M(e,3)=m1(e,3)-45-N; M(e,4)=m1(e,3)+45-N; M(e,5)=m1(e,1); M(e,6)=0; end sum=0; % applying prism intergral formula for calculting gravitational potential for x=1:429451 r1=((M(x,1).^2)+(M(x,3).^2)+(M(x,5).^2)).^0.5; r2=((M(x,2).^2)+(M(x,4).^2)+(M(x,6).^2)).^0.5; ra=((M(x,2).^2)+(M(x,4).^2)+(M(x,5).^2)).^0.5; rb=((M(x,1).^2)+(M(x,4).^2)+(M(x,5).^2)).^0.5; rc=((M(x,1).^2)+(M(x,4).^2)+(M(x,6).^2)).^0.5; rd=((M(x,2).^2)+(M(x,3).^2)+(M(x,6).^2)).^0.5; re=((M(x,1).^2)+(M(x,3).^2)+(M(x,6).^2)).^0.5; rf=((M(x,2).^2)+(M(x,3).^2)+(M(x,5).^2)).^0.5; A1=M(x,2)*M(x,4)*log(M(x,6)+r2); A2=M(x,4)*M(x,6)*log(M(x,2)+r2); A3=M(x,6)*M(x,2)*log(M(x,4)+r2); A4=-1*(0.5*M(x,2).^2*atand((M(x,4)*M(x,6)/(M(x,2)*r2)))); A5=-1*(1*0.5*M(x,4).^2*atand((M(x,2)*M(x,6)/(M(x,4)*r2)))); A6=-1*(0.5*M(x,6).^2*atand((M(x,2)*M(x,4)/(M(x,6)*r2)))); B1=-1*(M(x,1)*M(x,4)*log(M(x,6)+rc)); B2=-1*M(x,6)*M(x,4)*log(M(x,1)+rc); B3=-1*(1*M(x,6)*M(x,1)*log(M(x,4)+rc)); B4=(0.5*M(x,1).^2*atand((M(x,4)*M(x,6)/(M(x,1)*rc)))); B5=+(0.5*M(x,4).^2*atand((M(x,1)*M(x,6)/(M(x,4)*rc)))); B6=+(0.5*M(x,6).^2*atand((M(x,1)*M(x,4)/(M(x,6)*rc)))); C1=-1*((M(x,2)*(M(x,3))*log(M(x,6)+rd))); C2=-1*((M(x,6))*(M(x,3))*(log(M(x,2)+rd))); C3=-1*((M(x,6))*M(x,2)*log(M(x,3)+rd)); 37

C4=(0.5*M(x,2).^2*atand((M(x,3)*M(x,6)/(M(x,2)*rd)))); C5=(0.5*M(x,3).^2*atand((M(x,2)*M(x,6)/(M(x,3)*rd)))); C6=(0.5*M(x,6).^2*atand((M(x,2)*M(x,3)/(M(x,6)*rd)))); D1=((M(x,1)*(M(x,3))*log(M(x,6)+re))); D2=((M(x,6))*(M(x,3))*(log(M(x,1)+re))); D3=((M(x,6))*M(x,1)*log(M(x,3)+re)); D4=-1*(0.5*M(x,1).^2*atand((M(x,3)*M(x,6)/(M(x,1)*re)))); D5=-1*(0.5*M(x,3).^2*atand((M(x,1)*M(x,6)/(M(x,3)*re)))); D6=-1*(0.5*M(x,6).^2*atand((M(x,1)*M(x,3)/(M(x,6)*re)))); E1=((M(x,2)*(M(x,4))*log(M(x,5)+ra))); E2=((M(x,5))*(M(x,4)*log(M(x,2)+ra))); E3=((M(x,5))*M(x,2)*log(M(x,4)+ra)); E4=-1*(0.5*M(x,2).^2*atand((M(x,4)*M(x,5)/(M(x,2)*ra)))); E5=-1*(0.5*M(x,4).^2*atand((M(x,2)*M(x,5)/(M(x,4)*ra)))); E6=-1*(0.5*M(x,5).^2*atand((M(x,2)*M(x,4)/(M(x,5)*ra)))); F1=-1*((M(x,1)*(M(x,4))*log(M(x,5)+rb))); F2=-1*((M(x,5))*(M(x,4))*(log(M(x,1)+rb))); F3=-1*((M(x,5))*M(x,1)*log(M(x,4)+rb)); F4=(0.5*M(x,1).^2*atand((M(x,4)*M(x,5)/(M(x,1)*rb)))); F5=(0.5*M(x,4).^2*atand((M(x,1)*M(x,5)/(M(x,4)*rb)))); F6=(0.5*M(x,5).^2*atand((M(x,1)*M(x,4)/(M(x,5)*rb)))); G1=-1*((M(x,2)*(M(x,3))*log(M(x,5)+rf))); G2=-1*((M(x,5))*(M(x,3))*(log(M(x,2)+rf))); G3=-1*((M(x,5))*M(x,2)*log(M(x,3)+rf)); G4=(0.5*M(x,2).^2*atand((M(x,3)*M(x,5)/(M(x,2)*rf)))); G5=(0.5*M(x,3).^2*atand((M(x,2)*M(x,5)/(M(x,3)*rf)))); G6=(0.5*M(x,5).^2*atand((M(x,2)*M(x,3)/(M(x,5)*rf)))); H1=((M(x,1)*(M(x,3))*log(M(x,5)+r1))); H2=((M(x,5))*(M(x,3))*(log(M(x,1)+r1))); H3=((M(x,5))*M(x,1)*log(M(x,3)+r1)); H4=-1*(0.5*M(x,1).^2*atand((M(x,3)*M(x,5)/(M(x,1)*r1)))); H5=-1*(0.5*M(x,3).^2*atand((M(x,1)*M(x,5)/(M(x,3)*r1)))); H6=-1*(0.5*M(x,5).^2*atand((M(x,1)*M(x,3)/(M(x,5)*r1)))); sum=sum+6.674*10.^(11)*267*(A1+A2+A3+A4+A5+A6+B1+B2+B3+B4+B5+B6+C1+C2+C3+C4+C5+C6 +D1+D2+D3+D4+D5+D6-E1-E2-E3-E4-E5-E6-F1-F2-F3-F4-F5-F6-G1-G2G3-G4-G5-G6-H1-H2-H3-H4-H5-H6); end %caculting the normal gravity at the user input point B=9.7803185*(1+0.005278895*((sin(latitude*pi/180)).^2)0.000023462*((sin(latitude*pi/180).^4))); %calculating the terrain correction C=sum/B; disp('Terrain Correction : '); disp(C); end 38

7.1.2 Coordinate Projection lon(156,1)=0; lat(156,1)=0; longitude(156,1)=0; latitude(156,1)=0; t(156,1)=0; e1(156,1)=0; e2(156,1)=0; e3(156,1)=0; n1(156,1)=0; n2(156,1)=0; n3(156,1)=0; eata(156,1)=0; ep(156,1)=0; E(156,1)=0; N(156,1)=0; for x=1:156 longitude(x,1)=m(x,1); latitude(x,1)=m(x,2); lon(x,1)=longitude(x,1).*(pi)/180; lat(x,1)=latitude(x,1).*(pi)/180; lam=(81)*(pi)/180; f=1/298.257223563; No=0; Eo=500000; ko=0.9996; a=6378137; n=f/(2-f); A=(a/(1+n))*(1+((n^2)/4)+((n^4)/64)); alpha1=n/2-(2*(n^2)/3)+(5*(n^3)/16); alpha2=(13*(n^2)/48)-(3*(n^3)/5); alpha3=61*(n^3)/240; beata1=n/2-(2*(n^2)/3)+(37*(n^3)/96); beata2=(1*(n^2)/48)+(1*(n^3)/15); beata3=17*(n^3)/480; gama1=(n*2)-(2*(n^2)/3)-(2*(n^3)); gama2=(3*(n^2)/7)-(8*(n^3)/5); gama3=56*(n^3)/15; t1=(2*n^0.5/(1+n)); t(x,1)=sinh(atanh(sin(lat(x,1)))(t1*(atanh(t1*sin(lat(x,1)))))); ep(x,1)=atan((t(x,1)./(cos(lon(x,1)-lam)))); 39

eata(x,1)=atanh((sin(lon(x,1)-lam))/((1+(t(x,1).^2))^0.5)); e1(x,1)=(alpha1*cos(2.*ep(x,1)))*(sinh(2.*eata(x,1))); e2(x,1)=(alpha2*cos(4.*ep(x,1)))*(sinh(4.*eata(x,1))); e3(x,1)=(alpha3*cos(6.*ep(x,1)))*(sinh(6.*eata(x,1))); E(x,1)=Eo+ko.*A.*(eata(x,1)+e1(x,1)+e2(x,1)+e3(x,1)); n1(x,1)=(alpha1*sin(2.*ep(x,1)))*(cosh(2.*eata(x,1))); n2(x,1)=(alpha2*sin(4.*ep(x,1)))*(cosh(4.*eata(x,1))); n3(x,1)=(alpha3*sin(6.*ep(x,1)))*(cosh(6.*eata(x,1))); N(x,1)=No+ko.*A.*(ep(x,1)+n1(x,1)+n2(x,1)+n3(x,1)); 7.1.3 Final Polynomial function [fitresult, gof] = createFit3(x, y, z) %CREATEFIT3(X,Y,Z) % Create a fit. % % Data for 'Gravity model' fit: % X Input : x % Y Input : y % Z Output: z % Output: % fitresult : a fit object representing the fit. % gof : structure with goodness-of fit info. % % See also FIT, CFIT, SFIT. %

Auto-generated by MATLAB on 16-Apr-2018 12:18:24

%% Fit: 'Gravity model'. x=xlsread('Trend_Kriging.xlsx','L2:L75803'); y=xlsread('Trend_Kriging.xlsx','M2:M75803'); z=xlsread('Trend_Kriging.xlsx','K2:K75803'); [xData, yData, zData] = prepareSurfaceData( x, y, z ); % Set up fittype and options. ft = fittype( 'poly55' ); % Fit model to data. [fitresult, gof] = fit( [xData, yData], zData, ft, 'Normalize', 'on' ); % Plot fit with data. figure( 'Name', 'Gravity model' ); h = plot( fitresult, [xData, yData], zData, 'Style', 'PredObs', 'Level', 0.99 ); 40

legend( h, 'Gravity model', 'z vs. x, y', 'Location', 'NorthEast' ); % Label axes xlabel( 'X (UTM)' ); ylabel( 'y (UTM)' ); zlabel( 'Geoid Undulation' ); grid on; surf(peaks); colorbar %shading interp;

7.2 ArcGIS steps 7.2.1 SRTM for Rathnapura district

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7.2.2 Datum convertion of shape file

7.2.3 Clipped file

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7.2.4 Masking SRTM

7.2.5 Extracted SRTM

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7.2.6 Raster to point conversion

7.2.7 Latitude calculation

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7.2.8 Extracting points to WordPad

7.2.9 GPS locations

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7.2.10 Intersection of GPS points and SRTM heights

7.2.11 Trend

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7.2.12 Kriging

7.3 FORTAN77 Harmonic Synthesis Interface

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