the concept of scale factors has been fully dealt with and it only remains to deal with their application it should be clearly understood that scale factors transform distance on the eliposid to distance on the scale on the project
plane of project it can be seen that a horizontal distance at ground level AB must first be reduced to its equivalent at msl ( geoid) A,B , using the altitude correction , thence to the ellipsoid A,B using the geoid - ellipsoid value (N) and then multiplied by the scale factor to produce the projection distance A2B2.
whilst this is theoretically the correct approach , lack of knowledge of N may result in this step being ignored . thus the practical approach is to reduce to MSL and then to the projection plane , i.e from D to S to G
the basic equation for scale factor is given in equation . where the size of the ellipsoid and the value of the scale factor on the central meridian (f) are considered . specific to the os GB system , the following formula may be developed , which is sufficiently accurate for most purpose .
scale difference (SD) is the difference between the scale factor at any point (F) and that at the central meridian (F0) and varies as the square of the distance from the central meridian , I,e
scale on the project
national reference system of great britian showing 1000km squares the figures used to designate them in the former system , and the letters which have replaced the figures - courtesy ordnance survey , crown copyright reserved
As already intimated in equation the tratment for highly accurate work is to compute F for each end of the line and in the middle , and then obtain the mean value from Simpson,s rule however , for most practical purpose on short lines ,
it is sufficient to compute F at the mid - point of a line . in OSGB (36) the scale factor varies , at the most , by only 6 ppm per km , and hence a single value for f at the centre of small site can be regarded as constant throughout the area , on long motor way or projects , however one need to use different scale factor for different section .
distance reduction
the following example will serve to illustrate the classical application of scale factors .
Worked examples
Grid to ground distance any distance calculated from NG coordinates will be a grid distance if this to be set out on the ground it must :
(1) Be divided by the LSF to give the ellipsoidal distance at MSL , I.e.S = G/F
(2) Have the altitude correction applied in reverse to give the horizontal ground distance .
Consider two points , A and B , whose coordinate are :
A:E 638824.074 N 307 911.843
B:E 644 601 N 313 000.421
_______________________________
∴ ⧍ E = 5776.935 ∴ ⧍ N = 5088.578
2 2 1/2
Grid distance = ( ⧍ E + ⧍ N )
= 7698.481 m = G
Mid-easting of AB = E 641 712 m
∴ F= 1.0003188 (from equation (8.49)
∴ Ellipsoidal distance at MSL = S = G/F = 7696.027 m
Now assuming AB at a mean height (H) of 250 m above MSL . the altitude correction Cm is
SH 7696 ×250
Cm = __= __________ = + 0.301 m
R 6384 100
∴ Horizontal distance at ground level = 7696.328 m
This situation could arise where the survey and design coordinates of a project are in OSGB (36)/OSTN02 .
Distances calculated from the grid coordinates would need to be transformed to their equivalent on the ground for setting -out purposes .
Example 8.2 Ground to grid distance When connecting surveys to the national grid, horizontal distance measured on the ground must be :
(a) Reduced to their equivalent on the ellipsoid ,
(b) Multiplied by the LSF to PRoduce the equivalent grid distance , I.e, G= S × F .
Consider now the previous problem worked in reverse
Horizontal ground distance = 7696.328 m
Altitude correction Cm = - 0.301 m
________
∴ Ellipsoidal distance S at MSL = 7696.027 m
F = 1.0003188
∴ Grid distance G = S × F = 7698.481 m
This situation could arise in the case of a link traverse computed in OSTN02 . the length of each leg of the traverse would need to be reduced from its horizontal distance at ground level to its equivalent distance on the NG .
There is on application of grid convergence as the traverse commences from a grid bearing and connects into another grid bearing . the application of the (t-T) correction to the angles would generally be negligible ,
being a maximum of 7 ؛؛ for a 10-km line and much less than the observational errors of the traverse angles . it would only be necessary to consider both the above effects if the angular error was being controlled by taking gyro- theodolite observations on intermediate lines in the traverse .
The two applications of first reducing to MSL and then to the planr of the projection (NG) , illustrated in the examples , can be combined to give :
Fa = F (1 - H/R)
where H is the ground height relative to MSL andis positive when above and negative when below MSL then from Example 8.1 :
Fa = 1.0003188 ( 1-250/6 384 100 ) = 1.0002797
Fa is then scale factor adjusted and can be used directly to transform from ground to grid and vice versa .
From Example 8.2
7696.328 × 1.0002797 = 7698.481 m
