land survey - Convert Cassini-Soldner Projection coordinates from Kenya to UTM

I have a list of coordinates from Kenya. The coordinates were recorded in old survey maps using Cassini-Soldner Projection, with Clarke 1858 Datum. The approximate geographic coordinates (WGS84) of the area where the coordinates are apparently located are lat: -1.46 and lon : 36.9.

As an example, one set of the coordinates is (X: 150052.09 , Y: 4690.62). My interest is to convert these set of coordinates to UTM projection.

Answer

You can find details on the transformation from Cassini-Soldner to ARC1960 UTM in this article:

http://cae.uonbi.ac.ke/sites/default/files/cae/cae/Gordon%20Okumu%20Wayumba%20F80-81401-2009.pdf

In Table 4.8, a set of four reference points is given, with an equation to transform from Cassini to UTM:

E= b * n + a * e + ∆E                                                

N= a * n - b * e + ∆N 

Where a= Scos (θ),    b= S sin(θ) 
N and E are the local UTM coordinates
(n,e) are local Cassini  coordinates 

Computed values are given in Table 5.8:

Parameter  Value  Accuracy  Units 
S  1.0000169  ±0.000002  - 
θ  -0.000886  ±0.000002  rad 

∆N  10000167.51  ±0.35  m 
∆E  277419.49  ±0.35  m 

I get a standard deviation of the four points of 10 meters with those values.

Alternatively, you may use:

E = d * n + f * e + ∆E
N = a * n - b * e + ∆N

with:

a    1.00023

b   -0.00077
∆N  10000201.67
d   -0.00086
f    1.00033
∆E    277420.75

Quality of the reference points is now within 2 meters.

These transformations are valid for the Cassini Soldner projection centered on the 37° meridian, against UTM 37S based on ARC1960 datum (not WGS84!).

For the 39° meridian, transformation parameters are given in table 5.3:

PARAMETER  VALUE 

a  1.002248791 
b  0.0007371372 
∆N  10001507.607 
∆E  500339.6901 

Again, only valid for transformation to UTM zone 37S. Appendix 11 and 12 of the document have coordinates for both systems, and here the standard deviation is under 1 meter.