^{1}

^{1}

^{2}

Changing coordinates using appropriate mathematical models from one reference system to another may be influenced if the operation requires the change of datum. A set of transformation parameters has been adopted for Nigeria. However, the critical concern usually associated with the problem of transformation of coordinates is the issue of recoverability of the original values of transformed coordinates. The recursive effect of variables associated with spatial problems can be aptly modelled with an appropriate algorithm that set out a process to achieve a definite output. Consequently, the main thrust of this paper is to highlight the critical elements of the mathematical algorithm associated with the National Transformation Version 2 (NTv2) model adapted for the Nigerian Datum Transformation process. The adapted NTv2 model adopts the bi-linear interpolation approach and the covariance function obtained were used to generate transformation elements in latitude (Δ
*φp*) and longitude (Δ
*λp*) and corresponding accuracies at the lattice nodes. The mathematical algorithm of this adapted NTv2 model underscores the likely attainment of better and significant values and statistical indicator of the improved accuracy as the average shift values for latitude and longitude for any transformed points in Nigeria. This capability makes the mathematical algorithm to be adaptable and fit for the purpose of the transformation process. The improvement in the positional accuracy is directly attributable to the application of the NTv2 model which provides a flexible and robust system of modelling any inherent systematic error in the national network.

In view of the propagation of systematic errors in measurement and computational operations terrestrial geodetic methods are intricately subject to errors and create distortion which spread into the reference coordinate datum. These apparent distortions in the local geodetic networks, affect mainly the scale and orientation of the network [

The accuracy and reliability of the associated estimated parameters determined depend not only on the choice of the transformation model, but can be greatly influenced by the location and spread of the common points, especially for the conformal models. For a firm parameter solution, it is important that the locations are spatially well distributed. Similarly, [

In our quest to achieve improved accuracy of the datum transformation process of the Nigerian coordinate system, this work will examine the mathematical requirements cased in the algorithmic framework for the deployment of the adapted National Transformation Version 2 Model (NTv2) for Nigeria. Its uniqueness as a grid-based technique demonstrates an improvement in the accuracy of the positional values of the transformed points on the earth or near-earth surfaces within the Nigerian geodetic network as [

The process of interpolation encompasses making approximations of the interpolated value of an attribute variable at points where there is no existing information by using data for a sample of the point where we do have information. Several interpolation techniques abound and applicable in other areas of science and engineering and they are fundamentally based on [

The Office of the Surveyor-General of the Federation (OSGoF) [

S/N | STATION NAME/ID | MINNA LATITUDE DECIMAL DEGREES | MINNA LONGITUDE DECIMAL DEGREE | MINNA ELLIPSOIDAL HEIGHT | WGS84 LATITUDE DECIMAL DEGREES | WGS84 LONGITUDE DECIMAL DEGREE | WGS 84 ELLIPSOIDAL HEIGHT (M) |
---|---|---|---|---|---|---|---|

1 | A10 | 9.319181 | 12.23059028 | 274.116 | 9.31946547 | 12.2299544 | 287.036 |

2 | A24 | 10.60364 | 11.34032833 | 610.163 | 10.6037971 | 11.3396676 | 624.625 |

3 | A16 | 10.12471 | 12.37651389 | 773.545 | 10.1249196 | 12.375878 | 786.604 |

4 | A39 | 11.28878 | 10.41809139 | 479.759 | 11.2888627 | 10.4173975 | 495.499 |

5 | C21 | 6.968268 | 9.247406944 | 329.453 | 6.96865594 | 9.24683417 | 344.608 |

6 | C16 | 6.136606 | 9.027243056 | 613.763 | 6.13706426 | 9.02665414 | 628.430 |

7 | C32 | 7.757921 | 10.12114583 | 377.183 | 7.75822143 | 10.1205124 | 391.825 |

8 | CFL56 | 11.85322 | 13.1170425 | 344.927 | 11.853237 | 13.1164306 | 357.405 |

9 | CFH66 | 6.172683 | 6.750818611 | 39.549 | 6.17314447 | 6.7501325 | 57.368 |

10 | CFA 33A | 6.626514 | 3.323891389 | 47.165 | 6.62690544 | 3.32312688 | 70.412 |

45 | L040 | 9.635833 | 6.516389722 | 281.108 | 9.63602803 | 6.51564808 | 301.648 |

46 | N032 | 9.105738 | 7.202294444 | 689.447 | 9.10598537 | 7.20157031 | 708.804 |

47 | A001 | 9.188534 | 12.49675972 | 205.402 | 9.18882886 | 12.4961279 | 217.938 |

48 | U72 | 7.453301 | 5.871579444 | 638.312 | 7.45366794 | 5.87108417 | 658.509 |

49 | C036 | 7.998495 | 10.99365694 | 529.928 | 7.99877698 | 10.9930426 | 543.625 |

50 | A21 | 10.45509 | 11.62860417 | 744.895 | 10.455269 | 11.6279517 | 758.963 |

51 | H2 | 7.33078 | 9.053321667 | 494.999 | 7.33115025 | 9.05274885 | 510.712 |

52 | L41 | 9.585434 | 6.507209444 | 235.874 | 9.58563389 | 6.50646808 | 256.406 |

53 | N107 | 9.492585 | 6.775244167 | 544.930 | 9.49279582 | 6.77451014 | 565.051 |

S/N | STATION NAME/ID | MINNA ORTHOMETRIC HEIGHT | WGS 84 ELLIPSOIDAL HEIGHT (M) | PRIME VERTICAL (MINNA DATUM) | DH | MINNA ELLIPSOIDAL HEIGHT H_{MINNA} |
---|---|---|---|---|---|---|

1 | A10 | 268.127 | 287.0361 | 6,378,818.17851 | 12.92059779 | 274.1155022 |

2 | A24 | 605.629 | 624.6253 | 6,378,983.96092 | 14.46199188 | 610.1633081 |

3 | A16 | 768.768 | 786.6037 | 6,378,919.74827 | 13.05853023 | 773.5451698 |

4 | A39 | 474.746 | 495.4991 | 6,379,080.73508 | 15.73962794 | 479.7594721 |

5 | C21 | 324.6 | 344.608 | 6,378,568.51743 | 15.15490663 | 329.4530934 |

6 | C16 | 607.473 | 628.4296 | 6,378,497.10286 | 14.66658108 | 613.7630189 |

7 | C32 | 373.029 | 391.8245 | 6,378,644.54154 | 14.64100334 | 377.1834967 |

8 | CFL56 | 340.68 | 357.4045 | 6,379,164.77329 | 12.47759485 | 344.9269052 |

9 | CFH66 | 37.214 | 57.368 | 6,378,500.01578 | 17.81909126 | 39.54890874 |

10 | CFA 33A | 47.093 | 70.412 | 6,378,538.09317 | 23.2469371 | 47.1650629 |

11 | D29 | 505.549 | 526.7924 | 6,379,095.21757 | 22.77818301 | 504.014217 |

12 | D17 | 325.526 | 350.1121 | 6,379,005.62972 | 23.64860407 | 326.4634959 |

13 | H4 | 341.778 | 362.903 | 6,378,615.15240 | 16.4177838 | 346.4852162 |

14 | H5 | 279.859 | 300.431 | 6,378,619.98762 | 15.96341579 | 284.4675842 |

15 | H11 | 239.394 | 259.5107 | 6,378,695.66251 | 16.70430421 | 242.8063958 |

16 | L10 | 172.987 | 197.892 | 6,378,590.38326 | 23.7087912 | 174.1832088 |

17 | L18 | 404.216 | 430.3807 | 6,378,727.29593 | 22.8215704 | 407.5591296 |

18 | L16 | 498.209 | 523.7774 | 6,378,659.46205 | 22.65178951 | 501.1256105 |

19 | MW606 | 90.472 | 105.326 | 6,378,422.05140 | 14.4719753 | 90.8540247 |

20 | N25 | 570.12 | 591.959 | 6,378,780.00431 | 18.09340511 | 573.8655949 |

21 | N102 | 449.616 | 471.778 | 6,378,857.51769 | 20.48091504 | 451.297085 |

22 | N127 | 754.644 | 776.0028 | 6,379,210.28307 | 20.5385254 | 755.4642746 |

49 | C036 | 524.471 | 543.625 | 6,378,669.28662 | 13.69658631 | 529.9284137 |

50 | A21 | 740.331 | 758.963 | 6,378,963.74417 | 14.06832624 | 744.8946738 |

51 | H2 | 489.83 | 510.712 | 6,378,602.43 | 15.71285083 | 494.9991492 |

52 | L41 | 234.356 | 256.406 | 6,378,850.857 | 20.53221747 | 235.8737825 |

53 | N107 | 543.706 | 565.051 | 6,378,839.361 | 20.12052499 | 544.930475 |

The Minna datum (Clarke 1880) ellipsoidal height (h_{CLK1880}) of a station is computed directly from the corresponding WGS84 ellipsoidal height (h_{WGS84}) in Equation (1). The 5-parameter generalized form of Molodensky Standard formula [_{WGS84})_{ }which is applied to the WGS84 ellipsoidal height (h_{WGS84}) to convert it to the corresponding Minna datum value (h_{CLK1880}) [

h CLK188 0 = h WGS84 + Δ h WGS84 (1)

where,

Δ h WGS84 = Δ χ cos φ cos λ + Δ Y cos φ sin λ + Δ z sin φ − ( Δ a a V ) + ( Δ f b a ) V sin 2 φ (2)

φ, λ = WGS84 coordinates of the station

Δx, Δy, Δz = Datum shifts to transform WGS84 datum to Minna datum

Δa, Δf = (Minna minus WGS84) semi-major and flattening respectively

a_{WGS84} = Semi-Major Axis Radius of WGS84 Ellipsoid = 6,378,137 m

f_{WGS84} = Flattening of WGS84 ellipsoid = 1/298.257223563

a_{MINNA} = 6,378,249.145 m

f_{MINNA} = 1/293.465

Δa = 112.145 m

Δf × 10^{4} = 0.54750714 (Δf = 0.000054750714)

Δx = 93.708 m

Δy = 92.626 m

Δz = −121.33 m

V = RadiusofcurvatureofthePrimeVertical = a ( 1 − e 2 sin 2 φ ) 1 / 2 (3)

b / a = 1 − f (4)

e 2 = 2 f − f 2 (5)

The National Transformation Version 2 (NTv2) is a mathematical format that is built on the components of grid shift file that houses the coordinate shift values as determined at the grid nodes of a regular lattice. This format was developed by the Geodetic Survey Division of Geomatics Canada [

The bi-linear interpolation was used in this research to determine transformation component in latitude (Δφ_{p}) and longitude (Δλ_{p}). The algorithm is given as [

δ φ P = a 0 + a 1 V + a 2 W + a 3 V W (6)

where

a 0 = δ φ 1 (7)

a 1 = δ φ 2 − δ φ 1 (8)

a 2 = δ φ 4 − δ φ 1 (9)

a 3 = δ φ 1 + δ φ 3 − δ φ 2 − δ φ 4 (10)

V = ( λ P − λ 1 λ 2 − λ 1 ) (11)

W = ( φ P − φ 1 ) / ( φ 3 − φ 1 ) (12)

where

a_{0}, a_{1}, a_{2}, a_{3} are interpolation parameters.

V, W is interpolation scale factors.

From the foregoing, the relevant distortion components in latitude and longitude are represented as (δφ_{1}, δφ_{2}, δφ_{3}, δφ_{4}; δλ_{1}, δλ_{2}, δλ_{3}, δλ_{4}). The latitude and longitude of the interpolated point is represented as (φ_{P} and λ_{P}). Similarly, the value of the longitude transformation component of the interpolated point can be determined using Equations (7)-(12). However, the δφ terms in the equations was substituted with the equivalent δλ terms from the grid. The rudimentary contents of the Nigerian grid shift file depict the upper and lower limit of the latitude and longitude frame. The interval for the grid in seconds of arc for latitude and longitude. Equations (13) and (14) demonstrate the determination of the total numbers of rows and column within the limit of the study area. The grid shift values in relation to the determined rows and columns are also generated in seconds (rows, columns) [

No. of Rows

1 + ( ( φ Upper − φ Lower ) / Δ φ ) (13)

M = 1 + integer ( 55800 − 10800 ) / 30

No. of Columns

1 + ( ( λ Upper − λ Lower ) / Δ λ ) (14)

N ( NoofColumns ) = 1 + ( 55800 − 7200 ) / 30

The process of generating the shift (distortions) values from the grid nodes for the purpose of interpolation is given is as shown Equations (15) and (16):

row i = 1 + Integer φ p − φ Lower Δ φ (15)

Column j = 1 + Integer λ p − λ Lower Δ λ (16)

The ith and jth values for the rows and columns are generated thus:

i = 1 + ( 33549.051 − 10800 ) / 30 = 1 + 22749.051 / 30

i = 759

j = 1 + ( 44030.250 − 7200 ) / 30 = 1 + 1228

j = 1229

where

φ_{p}, λ_{p} the latitude and longitude of the interpolated point

φ_{Lower}, λ_{Lower} lower latitude and longitude extent of the grid

Δφ, Δλ Latitude and longitude interval between grid nodes

This process is carried out for the remaining grid nodes encompassing the point of interest within the grid file. Again, the computation of coordinate of grid nodes are determined using Equations (17) and (18) in terms of latitude and longitude. The transformed coordinates of the points of interest was derived as shown in Equations (19) and (20).

φ 1 = φ Lower + ( i − 1 ) ∗ Δ φ (17)

λ 1 = λ Lower + ( j − 1 ) ∗ Δ λ (18)

φ WGS84 = φ p + δ φ p (19)

Also,

λ WGS84 = λ p + δ λ p (20)

In another vein, covariance function and transformation accuracy at grid nodes was derived using the algorithm of the haversine model. The associated distances between data points from the grid node in addition with the surrounding data points is given in Equation (21):

d = 2 r sin − 1 ( sin 2 ( φ 2 − φ 1 2 + cos ( φ 1 ) cos ( φ 2 ) sin 2 ( λ 2 − λ 1 2 ) ) ) (21)

where, d is the distance between the two points along the great circle of the sphere, r is the radius of the sphere (sometimes referred to as the mean radius of the earth), φ_{1}, φ_{2} Latitude of locations 1 and 2, λ_{1}, λ_{2} Longitude of locations 1 and 2.

Consequently, the transformation accuracy computation at each grid node is based on the model proposed by [

σ = Σ w i 2 Σ ( δ i − δ ¯ ) 2 ( Σ w i ) 2 ( n − 1 ) (22)

where, σ is transformation accuracy, δ ¯ is the computed shift component at the grid node (by collocation), δ i is the known shift component at each data point (i), w_{1} is the weight derived from the covariance function and based on the distance of point from the interpolation point, n is the number of data points._{ }

The specimen values of the prime vertical, the undulation correction and computed Minna ellipsoidal heights based on Minna Datum are shown in _{1} = 33540"; λ_{1} = 44040". The values of the interpolation scale factor V = −0.3291663333 and W = 0.3017 were obtained. Hence, the latitude and longitude interpolation parameters required for the seamless transformation process are:

a 0 = δ φ 1 = − 0.250228

a 1 = δ φ 2 − δ φ 1 = − 0.208187 − 0.250228 = 0.042041

a 2 = δ φ 3 − δ φ 1 = − 0.249185 − 0.250228 = 0.001043

a 3 = δ φ 1 + δ φ 4 − δ φ 2 − δ φ 3 = − 0.250228 − 0.173237 − 0.042041 − 0.001043 = − 0.466549

a 0 = δ λ 1 = 0.075285

a 1 = δ λ 2 − δ λ 1 = 0.099675 − 0.075285 = 0.02439

a 2 = δ λ 3 − δ λ 1 = 0.079317 − 0.075285 = 0.004032

a 3 = δ λ 1 + δ λ 4 − δ λ 2 − δ λ 3 = 0.075285 + 0.156117 − 0.099675 − 0.079317 = 0.05241

Equation (6) was used to compute the interpolated shift (distortion) value (latitude);

δ φ P = − 0.250228 + 0.042041 × ( − 0.3291663333 ) + 0.001043 × 0.3017 + ( − 0.466599 ) × ( − 0.3291663333 ) × 0.3017 = − 0.250228 + ( − 0.0138384818 ) + 0.0003146731 + 0.0463377053 = − 0. 2174141 0 34

This value δ φ P = − 0.2174141034 is then added to the latitude value of the point of interest in Minna Clarke 1880 reference ellipsoid to obtain the corresponding value of latitude in WGS84 ellipsoid. Similarly, the value of δ λ P is also determined by substituting the values of longitude interpolation parameters substituted in Equation (6). The average shift values for latitude and longitude of Nigeria are −0.252138" and 0.007462" respectively.

The summary of Nigerian grid shift file depicts the limits and the associated accuracy values that ensures that any point of interest can be determined in terms of the transformation of coordinates [

φ_{Lower} 10800.0000 (in seconds)

φ_{Upper} 55800.0000 (in seconds)

λ_{Lower} 7200.0000 (in seconds)

λ_{Upper}_{ } 55800.0000 (in seconds)

Δφ 30 (in seconds)

Δλ 30 (in seconds)

No. of Rows 1501

No. of Columns 1621

Number of grid shift values (rows × columns)

The transformed values in a grid shift file relate to a defined transformation path which is denoted as the forward transformation. For the Nigerian grid shift files, the forward transformation path is from Minna (Clarke1880) ellipsoid to WGS84 ellipsoid. The process is thus, create summary of grid shift file; get coordinates of interpolation point; determine sub grid that the point lies within; retrieve shift values of the nearest 4 nodes from grid shift file; use bi-linear interpolation to compute coordinate shift valid, this can be categorized as a minor grid. Add the computed shift to transformed coordinates and transform another point, by this, one cycle of the forward transformation process is completed. The transformation of coordinates in the opposite direction (WGS84 to Minna-Clarke 1880) in Nigeria is referred to as a back transformation [

1. Initialise Minna-Clarke1880 coordinates:

φ ′ Minna-Clarke1880 = φ WGS84

λ ′ Minna-Clarke1880 = λ WGS84

2. Repeat four times:

3. Interpolate shifts (δφ and δλ) at Minna-Clarke1880 point.

4. Compute new Minna-Clarke1880 coordinates:

φ ′ Minna-Clarke1880 = φ WGS84 − δ φ

λ ′ Minna-Clarke1880 = λ WGS84 − δ λ

The outcome of an efficient mathematical algorithm is the development of behind the scene code for the solving of the particular problem. Computer programming language was employed in adding functionality, in a code-behind-file model which has PC version [

The uniqueness of this adapted model is predicated on-grid shift files with corresponding grid shift parameters. [

the algorithmic propelled in the mathematical model provided the framework for the development of an efficient transformation procedure in line with the objective of this work. These parameters form the framework of the surface modelling of the two-reference datum in other to obtain the corresponding values (coordinate) in its transformed state. Furthermore, the interplay of these grid file data generates the accompanying basis of accuracy determination on point by point basis with no geometric bias. The associated improvement in the positional accuracy as against the use of geometric models of transformation is directly attributable to the application of the adapted NTv2 mathematical algorithm (model) for the Nigerian datum transformation process.

The authors declare no conflicts of interest regarding the publication of this paper.

Hart, L., Jackson, K.P. and Okeke, F.I. (2020) Modern Development for the Improvement of Accuracy of Nigerian Coordinate Transformation Process Using the Adapted NTv2 Model: The Critical Issues of the Mathematical Algorithm. International Journal of Geosciences, 11, 768-781. https://doi.org/10.4236/ijg.2020.1111039