Simplified Flat Coordinate Model for Northern Parts of Arabian Gulf

The method of coordinate conversion is still considered important and laborious due to the shift from the spatial ellipsoidal (geographic) to the flat planned system. The most common method uses a contiguous UTM system as one of the most reliable systems in the conversion process; however, this system faces a problem in large areas that contain more than one zone. The goal of this research is to create a simple and low computational cost model to represent a non-contiguous semi-UTM geographic coordinates for confined regions of the globe. The considered region taken in this study is the northern parts of Arabian Gulf (including parts of Iraq, Kuwait, Iran, and Saudi Arabia). The determined mathematical model was based on using two dimensional Taylor sequence. The most accurate representation met in this study was based on the 6 th two dimensional polynomial. The estimation of equations’ coefficients was done using the least square criterion for the overall error of estimating coordinate values of either (latitude, longitude) or (Easting, Northing). The two basic determinations were applied for the forward and the backward; in the first step, the conversion of coordinates was calculated from the ellipsoidal coordinates (i.e., Longitude, Latitude) to UTM (WGS84) coordinates (i.e., Easting, Northing) and vice versa. The attained results indicated that the mathematical model used is successful for achieving the conversion process. With the use of the 6 th order 2D-polynomial equations, a very small error of less than 1 m was achieved in the Easting and Northing coordinates.


Introduction
Universal Transverse Mercator (UTM) is an international plane (rectangular) coordinate system developed by the U.S. Army [1]. In this system, the world is divided into 60 zones, each covering 6 degrees of longitude. In latitude, it extends from 80° S to 84° N. The origin of each zone is the intersection of the central meridian at the equator. High degree of accuracy is possible due to separate projections for each UTM zone. UTM values are calculated in meters. To eliminate negative coordinates, the projection alters the coordinate values at the origin. The value given to the central meridian is the false easting and the value assigned to the Equator is the false northing. For locations in the Northern hemisphere, the origin is assigned a false easting of 500,000 meters and a false northing of 0. For locations in the southern hemisphere, the origin is assigned a false easting of 500,000 meters and a false northing of 10,000,000 meters (10,000 km) [2,3,4]. The surface of the Earth cannot be flatten unless it is converted to a plane form .The UTM can provide a method to represent every point on earth by using a list of flat (X, Y or Easting, Northing) coordinates [5]. The benefits of using the flat coordinates for representing the location on earth is for easy planar mapping and easy derivation of spatial information from locations coordinates (e.g., distances, angles, areas, etc.), added to that the capability to determine the location of a point is dependent on spatial information. The measurement of distances, directions, and areas can be performed more efficiently than the geographic coordinate system, due the shifting of the actual location in the geographic coordinate [6]. There is a problem when converting coordinates in UTM system, but it is a small problem when compared with other systems. The principle of UTM system is to divide the world into zones, where each area within the zone has its own special attributes. Most of the areas are within more than one zone; therefore, moving from one zone to another makes the converting result inaccurate, which in fact is the root of the problem [7,8]. In Iraq, the same problem exists in the UTM Extended Zones, as the territory of Iraq falls within the UTM system in three regions (37, 38 and 39). Zone 37 includes parts of western Iraq, zone 38 covers the central region and includes most of Iraq, and zone 39 covers a small part of the city of Basra in eastern Iraq [9]. The same problem happens with the north parts of the Arabian Gulf, where the area of the gulf lies in three different zones, which are (38, 39, 40); zone 38 covers parts of Basra, Kuwait and Saudi Arabia, zone 39 covers Bahrain, Qatar and United Arab Emirates, and zone 40 covers parts of Iran. Figure-1 illustrates the coverage area of zones 38, 39 and 40 in the northern parts of Arabian Gulf.

Theoretical Framework
In this section, the mathematical framework of the transfer is explained. Four models of polynomial equations were used to perform the transformation.

Relative Normalized Polynomial Representation
where λ k , φ k , X k , and Y k are the latitude and longitude (East, North) coordinates, respectively, of k th point, and the corresponding mean values are: where {a L }, {b L }, {c L }, {d L } are the sequence of polynomial coefficients. After testing the four modes, the attained results show that the Relative Normalized Polynomial is the best to use.

Forward Representation
The coordinate values of East and North (X and Y) can be represented mathematically, with changing the symbols of latitude and longitude from {φ,

Backward Representation
Also, the inverse mapping equations can be approximately written in the form: Since the above equation is continuous in the region, then the Taylor series [11,12] for both equations (4c) and (4d) can be written as in equations (8a) and (8b):

Results and discussion
The dataset used in this study represents the grid of points' coordinates (Longitude, Latitude) covering the north gulf region with some surrounding areas. The separation t of difference between the grid points is taken as 0.25 degrees. These coordinates were then converted to (X, Y) coordinates using UTM system conversion software but we will determine this program on Zones 38 and calculate (X, Y) coordinate values, as shown in Figure-2. The tables below show error rates and explain why we selected the relative normalized polynomial over the absolute polynomial, relative polynomial, and normalized polynomial.

Forward
This step presents the calculation rate of error for test two at X east and Y north using different orders polynomial equations (up to 5 th order), depending on least square error criteria. The tables below list the error rates for these orders, depending on the relative normalized polynomial equation. 269368 According to the error rate values of the orders above, we concluded that the 6 th order gives the minimum error value. Tables-9 and 10 show the values of polynomial coefficients and values of attained error rates in X east and Y north using the 6 th order polynomial (equations 6a and 6b). Also, the error distributions as shown in Figures-(3

Backward
Tables-11 and 12 present the values of coefficients and values of error rates at the studied longitudes and latitudes, using the 6 th order polynomials equations (9a and 9b) to test the accuracy of the converting process. Also, the error distributions are shown in Figures-5 and 6.    This study showed that the application of the first test of the normalization method led to results that maintained a very high error rate, which resulted in the failure of the conversion process. However, when using the second test of the normalization method, the error rate was gradually reduced to the lowest ratio, when the sixth-degree polynomial equation was applied, which contributed to the success of the conversion and provided similar results to ideal conditions. Using the sixth-order polynomial equation, the value of the average error was approximately 0.17 meters at X east and about 0.008 meters at Y north. These results show a very efficient model, on which we can rely for the transmission process of X east and Y north from (Longitude and Latitude) for the northern part of the Arabian Gulf, due to the high precision found in this model.