This is the accepted manuscript, posted on academic social networks on 8 July
2017, 12 months after Taylor & Francis had published the paper online.
ABSTRACT For three-parameter datum transformations to be applied rigorously, geodetic coordinates on the first ellipsoid need to be converted to Cartesian coordinates before application of the shifts, then converted to geodetic coordinates on the second ellipsoid. The Standard Molodensky method of datum transformation is more direct but is inexact. It also fails to reproduce the original coordinates when applied forward and back. However, this paper shows a pattern of proportionality between the misclosures and the errors in the forward approximations. This gives rise to a new method of computing the transformations, best described as “Standard Molodensky in two stages with applied misclosures” (SMITSWAM). The method is shown to be more than 1600 times more accurate than Standard Molodensky, coming close to the accuracy of the rigorous approach. SMITSWAM is also shown to be around 48% faster than the traditional form of the rigorous method which uses iteration. Keywords: Molodensky, datum transformations, geodetic datums, SMITSWAM, applied misclosures
Introduction Datum transformations are mathematical models for converting coordinates of points in one geodetic datum
to coordinates in another. Usually the coordinates being transformed are geodetic (, , h). Typically the
models are used to convert coordinates between a local datum and a worldwide datum such as the World
Geodetic System 1984 (WGS 84).
The accuracy of the mathematical model is primarily the extent to which the mathematical model fits the
observed coordinate differences at the control points, although the model must also provide a smooth fit to
ensure credible results in the area of interest as a whole. Multiple regression equations are particularly accurate
models used for datum transformations, but they use a large number of statistically-derived parameters.
Generic formulae and examples can be found in NIMA (2004).
For many purposes, however, accuracy can be traded for simplicity in the choice of mathematical model.
One way is to define a relationship between the respective ellipsoids involving up to 7 parameters. In the
general case they cater for a shift of origin, changes in orientation and scale.
This paper concentrates on the transformation model which has just 3 shift parameters, namely X, Y, Z.
Derived sets of 3 parameters for transforming local geodetic datums to WGS 84 can be found in NIMA (2004).
The differences between the respective ellipsoids in semi-major axes & flattening, a & f, are not normally
counted as parameters but they need to be included in the model.
Page 2 of 12
This paper considers the accuracy with which the mathematical model is computed as opposed to the
accuracy of the model itself.
The rigorous 3-parameter datum transformation consists of 3 stages:
• conversion of geodetic coordinates to Cartesian coordinates in the first datum;
• application of the shifts X, Y and Z to the Cartesian coordinates;
• conversion of Cartesian coordinates to geodetic coordinates in the second datum.
The final stage is more difficult than the first, the problem being the computation of latitude. The traditional
method - and the method most easily understood - requires iteration.
The rigorous method is the most precise implementation of the mathematical model. In this paper, it is the
benchmark against which approximate methods are assessed for computational accuracy.
The most direct methods for computing the 3-parameter datum transformation are those associated with
Mikhail Sergeevich Molodensky. Changes in latitude, longitude and ellipsoidal height are computed as linear
combinations of X, Y, Z, a and f. There are the Standard Molodensky formulae and the simpler Abridged
Molodensky formulae.
The rigorous approach The procedure for rigorous computation of the three-parameter transformation model is well known. Firstly the
coordinates, , , h in Datum 1 are converted to Cartesian coordinates. In this context, , , h, a and e2 refer to
the ellipsoid of Datum 1.
.sin1 22
e
a
−= (1)
.coscos)(1 hX += (2)
.sincos)(1 hY += (3)
.sin])1([ 2
1 heZ +−= (4)
The derivation of these equations can be found, for example, in Section 5-3 of Heiskanen and Moritz (1967).
Secondly, the Cartesian coordinates are converted from Datum 1 to Datum 2.
.12 XXX += (5)
.12 YYY += (6)
.12 ZZZ += (7)
Thirdly, the Cartesian Coordinates in Datum 2 are converted to geodetic coordinates. The process used is
essentially that recommended in Section 5-3 of Heiskanen and Moritz (1967). For equations (8)-(21), , , h, a
and e2 refer to the ellipsoid of Datum 2.
2
2
2
2 YXr += (8)
which is the distance from the polar axis.
22 ,2arctan YX= (9)
unless 1210/ar , in which case should be set to 0. (Longitude is actually indeterminate along the polar
axis.) See “Note on arctan2”.
Unless 12
2 10/aZ (in which case should be set to 0) the following iterative approach is used to
compute .
−=
re
Z
)1(arctan
2
2 (10)
as a first estimate.
. = (11)
.sin1 22
−
=e
a (12)
.sin
arctan2
2
+=
r
eZ (13)
While |−|>10-12, repeat (11)-(13).
If ||</4,
,cos
−=
rh (14)
otherwise
Page 3 of 12
).1(sin
22 eZ
h −−=
(15)
For the height computation, the test ||</4 is one of many that can be used to avoid a small denominator.
Between latitudes 84, |cos| will exceed 0.1 and (14) could be used every time.
It is worth stating that for Cartesians-to-latitude computation, the iterative approach is not the only method.
A near-exact alternative has been proposed by Bowring (1966) and is noted in Bomford (1980). This uses the
semi-minor axis b and the quantity = e2/(1-e2). The calculation-of-latitude stage is replaced by
,)/()/(arctan 2 barZu = (16)
where r is defined by (8), and
.cos
sinarctan
32
3
2
−
+=
uaer
ubZ (17)
Fukushima (1999) recommended a variation in the implementation of Bowring’s method. Based on
trigonometric identities, it is designed to optimise the computation of latitude from Cartesians. In his own
tests, using Microsoft Fortran Powerstation 4.00 on the Intel Pentium II processor, it produced a saving in
processing time. The calculation-of-latitude stage is replaced by
),/()/( 2 barZT = (18)
where r is defined by (8),
𝐶 = 1/√1 + 𝑇2, (19)
𝑆 = 𝐶𝑇, (20)
and
𝜙 = arctan [𝑍2+𝜀𝑏𝑆3
𝑟−𝑒2𝑎𝐶3]. (21)
For the purposes of this paper, the rigorous approach using Cartesians and iteration is called RAUCAI while
the rigorous approach using Cartesians and Bowring is called RAUCAB.
Since 1970, analytical methods that avoid iteration have been proposed by, for example, Paul (1973),
Borkowski (1989), and Vermeille (2002, 2004, 2011). They are, of course, only exact theoretically because
rounding-off always occurs in their implementation. An extensive list of studies on (, , h)(X, Y, Z)
coordinate transformations can be found in Featherstone and Claessens (2008).
Note on arctan2 This paper adopts the convention that arctan2[x,y] is the solution of tan = y/x in the range − to which has
the same sign as y. Programming languages usually have a two-argument arctangent function. A notable
exception is VBA where the only built-in arctangent function is the single-argument Atn(x). The user-defined
function in Listing 1 is suggested; it caters for all possibilities except x = y = 0.
Listing 1: User-defined arctan2 for VBA Function Atn2(x As Double, y As Double)
Dim i As Integer
If y < 0# Then
i = -1
Else
i = 1
End If
If Abs(y) > Abs(x) Then
Atn2 = 1.5707963267949 * i - Atn(x / y)
Else
Atn2 = Atn(y / x)
If x < 0# Then
Atn2 = Atn2 + 3.14159265358979 * i
End If
End If
End Function
Standard Molodensky Datum transformations based on three parameters are often computed by Standard Molodensky formulae,
documented in Molodensky et al (1962). They are explicitly recommended by NIMA (2004) for most
applications involving local geodetic datums and WGS 84.
In general, there are three Standard Molodensky formulae, (22)-(24) below. Those for and are in radians.
The h formula would only be needed if the requirement includes ellipsoidal heights, because heights referenced
to the geoid are not affected by datum changes. The superscript “SM” distinguishes the Standard Molodensky
approximations from the rigorously-computed shifts.
Page 4 of 12
)./(}cossin)]/()/([
/)cossin(cos
sinsincossin{
2
habbaf
aeaZ
YXSM
+++
++
−−=
(22)
.cos)(
cossin
h
YXSM
+
+−= (23)
.sin)/()/(sin
sincoscoscos
2
abfaaZ
YXhSM
+−+
+= (24)
Of the radii of curvature, is given by (1) and is given by
𝜌 = 𝑎(1 − 𝑒2)/(1 − 𝑒2 sin2 𝜙)3/2. (25)
The research for this paper included computation of errors from the Standard Molodensky formulae relative to
the rigorous approach. The regions chosen were Great Britain, Australia, Congo (Zaire) and India.
For Great Britain, 53 points were transformed from Ordnance Survey of Great Britain 1936 to WGS 84 using
shift parameters recommended by NIMA (2004). Figure 1 shows the errors for points on the Airy ellipsoid. As
Table 1 shows, there is no significant difference for the equivalent points 500m, 2km & 10km above the ellipsoid.
Table 1 OSGB 36 → WGS 84 by Standard Molodensky: Worst-Case Errors
h in latitude in longitude in height
0 -0.00026 -0.00066 -0.005m
500m -0.00026 -0.00066 -0.006m
2km -0.00026 -0.00066 -0.006m
10km -0.00026 -0.00066 -0.006m
1 Errors in Standard Molodensky for OSGB 36 to WGS 84
For Australia, 63 points were transformed from Australian Geodetic Datum 1984 to WGS 84 using shift
parameters recommended by NIMA (2004). Figure 2 shows the errors for points on the Australian National
Spheroid. As Table 2 shows, there is no significant difference for the equivalent points 500m, 2km & 10km
above the ellipsoid.
Table 2 AGD 84 → WGS 84 by Standard Molodensky: Worst-Case Errors
h in latitude in longitude in height
0 -0.00008 0.00007 -0.003m
500m -0.00008 0.00007 -0.003m
2km -0.00008 0.00007 -0.003m
10km -0.00008 0.00007 -0.003m
Page 5 of 12
2 Errors in Standard Molodensky for Australian Geodetic Datum 1984 to WGS 84 For Congo, 55 points were transformed from ARC 1950 to WGS 84 using shift parameters recommended by
NIMA (2004).. Figure 3 shows the errors for points on the Clarke 1880 ellipsoid. As Table 3 shows, there is no
significant difference for the equivalent points 500m, 2km & 10km above the ellipsoid.
Table 3 ARC 1950 → WGS 84 by Standard Molodensky: Worst-Case Errors
h in latitude in longitude in height
0 -0.00082 -0.00005 -0.008m
500m -0.00082 -0.00005 -0.008m
2km -0.00082 -0.00005 -0.008m
10km -0.00081 -0.00005 -0.008m
3 Errors in Standard Molodensky for ARC 1950 (Zaire) to WGS 84
For India & Nepal, 50 points were transformed from the Indian Datum to WGS 84 using shift parameters
recommended by NIMA (2004). Figure 4 shows the errors for points on the Everest (India) ellipsoid. As Table
4 shows, there is no significant difference for the equivalent points 500m, 2km & 10km above the ellipsoid.
Page 6 of 12
Table 4 Indian → WGS 84 by Standard Molodensky: Worst-Case Errors
h in latitude in longitude in height
0 -0.00081 -0.00117 -0.009m
500m -0.00081 -0.00117 -0.009m
2km -0.00081 -0.00117 -0.009m
10km -0.00081 -0.00117 -0.009m
4 Errors in Standard Molodensky for Indian Datum to WGS 84
Misclosures from Standard Molodensky forward and back In the examples above, the Standard Molodensky formulae were applied to the reverse transformation from
Datum 2 to Datum 1. X, Y, Z, a & f were reversed in sign and their coefficients were this time in terms of
Datum 2. The results were “misclosed” coordinates , , h in Datum 1 that differed slightly from the original
coordinates in Datum 1. A significant aspect of the misclosures is that they were consistently around twice the
errors in the first transformation. To put it another way, the errors measured by “Standard Molodensky minus
rigorous equivalent” were around half the differences “transformed-both-ways minus original”.
These results pointed to a way of using misclosures to eliminate the errors in Standard Molodensky.
Appplying the misclosures (SMITSWAM) SMITSWAM stands for Standard Molodensky in two stages with applied misclosures, in the sense that half the
misclosure is subtracted from the result obtained by Standard Molodensky.
The first application of Standard Molodensky produces the transformation
).,,(),,( 222111
SMSMSM hh → (26)
The second application of Standard Molodensky can be regarded as “Reverse Standard Molodensky” as it
is from Datum 2 to Datum 1, and the original X, Y, Z, a & f are replaced by their negatives. It produces
the transformation
).,,(),,( 111222
RSMRSMRSMSMSMSM hh → (27)
The new approximation to the transformed position is obtained by subtracting half the misclosure from the
first approximation:
.2/)( 1122 −−= RSMSMSMITSWAM (28)
.2/)( 1122 −−= RSMSMSMITSWAM (29)
.2/)( 1122 hhhh RSMSMSMITSWAM −−= (30)
For the data sets covering Great Britain, Australia, Congo and India/Nepal, the errors relative to the rigorous
approach are all at sub-millimetre level. The worst-case errors are:
5.0510-7 for latitude,
3.6810-9 for longitude, and
−4.86m10-6 for ellipsoidal height.
In particular, the errors in Figures 1 to 4 are reduced to zero at every point.
Page 7 of 12
Computation analysis SMITSWAM retains the simplicity of Standard Molodensky although the computation is roughly doubled. Given
the accuracy of SMITSWAM, it is more relevant to compare its computational speed with that of the rigorous
approach. Accordingly, SMITSWAM, RAUCAI and RAUCAB were programmed in VBA within Excel 2007,
with Sub procedures measuring the respective transformation times per point. The computer used was an HP
p6032uk with RAM = 3GB and processor = AMD Phenom (tm) 8550 TripleCore Processor 2.20GHz. The
software used was Windows Vista (Service Pack 2, 32-bit) and Microsoft Office 2007. In each case, the methods were programmed to minimise the number of multiplications, divisions and series-
based functions, avoiding repeat computations as far as possible. Timings for the transformations excluded one-
off calculations (like a/b and b/a) and conversions between degrees and radians. As Table 5 shows, SMITSWAM
has more products and quotients than the rigorous approach but makes less use of series-based functions.
Table 5 Computation statistics (per point) for SMITSWAM and the rigorous approach
Initial results showed SMITSWAM was transforming points at least twice as fast as RAUCAI and RAUCAB.
However, it quickly became clear that the times for RAUCAI and RAUCAB were inflated by the use of the
Excel worksheet function Atan2 in the absence of a VBA version. To ensure a fair comparison, the
transformations were recomputed with the user-defined Atn2 function given in Listing 1. The results are shown
in Table 6.
Table 6 Average timings per point for the respective transformations
On this evidence, SMITSWAM is 39% to 56% faster than RAUCAI depending on whether the latter requires
3 or 4 iterations. If the convergence criteria for RAUCAI was weakened to make 2 iterations the norm (thereby
reducing its accuracy), SMITSWAM would still be 23% faster.
SMITSWAM has very much the same speed as RAUCAB, provided the latter is programmed with an efficient
arctan2 function for the longitude. Surprisingly, the Fukushima optimisation of RAUCAB made negligible
difference to processing time (in VBA implementation, at any rate). The practical choice between SMITSWAM
and RAUCAB comes down to whether one wants (or needs) the superior accuracy of the latter or whether one
prefers the simplicity of the former (which uses nothing more complicated than the Standard Molodensky
formulae).
Reference has been made to variations on the rigorous approach for obtaining latitude from Cartesian
coordinates analytically without iteration. Those proposed by Paul (1973), Borkowski (1989), and Vermeille
(2002, 2004, 2011) make several calls of Arctangent & Square Root, and require at least one cube root. They
involve more computation than Bowring’s near-exact method, so the associated transformations are bound to
be slower than RAUCAB. That in turn means they are bound to be slower than SMITSWAM.
Mathematical reasoning behind the applied misclosure technique An introductory way to understand the applied-misclosure technique is to consider an analogous problem
involving a scalar function of a single variable. A smooth continuous function y = f(x) is known only at a single
point x0 but its first derivative can be evaluated anywhere. The objective is to find an approximation to f(x0+h).
The first approximation is obtained from f(x0) + hf(x0). In Figure 5, it is the y value at the point C. The error is
the unknown distance between B and C.
Page 8 of 12
5 Forward approximation of f(x0+h) from starting point A
The reverse process is then applied from where x = x0+h. The first derivative at B can be evaluated but the
tangent at B is applied to point C (since the position of B is unknown). The new approximation to f(x0) is
yC−hf(x0+h). In Figure 6, it is the y value of the point D. The distance AD is the misclosure of the forward-and-
back process.
6 Reverse approximation of f(x0) from starting point C
Unlike BC, the distance AD is known, and equates to hf(x0)−hf(x0+h). Using half the distance AD as a
correction to the approximation yC we should get much closer to yB, the value of f(x0+h). In other words,
),(21
ADCB yyyy −− (31)
or
)].()([
)()()(
0021
000
hxfxfh
xfhxfhxf
+−
−++ (32)
This is equivalent to
)].()([)()( 0021
00 hxfxfhxfhxf ++++ (33)
The accuracy of this approximation can be analysed using Taylor’s Theorem. The approximation used in
the forward process,
),()()( 000 xfhxfhxf ++ (34)
comes from the truncation of the Taylor series
),()()()( 2
21
000 fhxfhxfhxf ++=+ (35)
where is between x0 and x0+h.
For the reverse process, noting that the square of −h is h2,
)()()()( 2
21
000 fhhxfhhxfxf ++−+= (36)
where is between x0 and x0+h.
).()()()( 2
21
000 fhhxfhxfhxf −++=+ (37)
Adding (35) to (37) and dividing by 2,
)].()([
)]()([)()(
2
41
0021
00
ffh
xfhxfhxfhxf
−
++++=+ (38)
Applying Taylor’s Theorem to the final term,
Page 9 of 12
)()(
)]()([)()(
2
41
0021
00
fh
xfhxfhxfhxf
−
++++=+ (39)
where is between and .
The final term in (39) is effectively an h3 term since − is closer to zero (perhaps considerably) than h. It
provides the error in (33) and hence the error in (31). Double use of a first-order approximation, with applied
misclosure, has achieved a second-order fit.
There is an alternative way of demonstrating that (33) is superior to (34) as an approximation. Both take the
form f(x0) + gh where g is a gradient. The closer g is to the gradient of the chord AB, the better the approximation.
In (34) the starting point is used to obtain a forward gradient, that of the tangent AC. In (33) the gradient is a
mean between DC and AC, the gradient MC in fact, where M is the midpoint of DA. As Figure 7 shows, the
gradient of MC is much closer than the gradient AC to the gradient of chord AB.
7 Mean gradient MC that approximates the gradient of chord AB
This can be substantiated mathematically, because by Taylor’s Theorem,
),()()( 00 fhxfhxf +=+ (40)
where is between x0 and x0+h, although not necessarily the same as the used in (35). is likely to be close to
halfway between x0 and x0+h, so )]()([ 0021 xfhxf ++ is an obvious approximation to ).(f
Applying half the misclosure from a forward and reverse approximation can therefore be seen as replacing
a forward gradient by an estimated mean gradient in a Taylor-derived approximation.
Similar reasoning can be used to show why SMITSWAM is more accurate than Standard Molodensky. In this
case a “forward Jacobian” is replaced by an estimated “mean Jacobian” in a Taylor-derived approximation.
Using underline notation for column vectors, v and p can be defined as follows:
;][ Thv = (41)
.][ TfaZYXp = (42)
Treating Datum 1 as fixed, then for any given point on Datum 1, v is a non-linear function of p. The Standard
Molodensky approximation of v, however, is a linear function of p. This can be written as
pSvSM
= (43)
where S is a 35 Jacobian matrix (of partial derivatives). The elements of S can be found from the coefficients
in equations (22)-(24). For this analysis, what matters is that S is a function of , , h, a & f.
The first application of Standard Molodensky gives us
pSvSM
C 112 = (44)
where S1 is computed from 1, 1, h1, a1 & f1. This produces approximations 2C, 2C, h2C to the corresponding
coordinates 2, 2, h2 in Datum 2. The notation used here differs from that used in (26).
The reverse application of Standard Molodensky gives us
)(212 pSv C
SM
CC −= (45)
where S2C is computed from 2C, 2C, h2C, a2 & f2. This produces approximations 1C, 1C, h1C to the
corresponding coordinates 1, 1, h1 in Datum 2. The notation used in (44) and (45) can be related to the
notation used in (26) and (27) by the following description of the process:
Page 10 of 12
.
1
1
1
12
2
2
2
12
1
1
1
⎯⎯⎯ →⎯
⎯⎯ →⎯
RSM
RSM
RSM
SM
CC
SM
SM
SM
SM
C
h
v
h
v
h
(46)
At this point we note the implementation of Taylor’s theorem that relates to the true value of v12.
pSv m=12 (47)
where m indicates intermediate values of , , h, a & f (meaning that the used in Sm is between 1 and 2,
etc). The matrix ][ 2121
CSS + is virtually the average of the Jacobians at both ends, so a new approximation
(NA) for v12 can be written as follows.
pSSv C
NA][ 212
112 += (48)
Making the reasonable assumption that Sm is much closer to the average of S1 and S2 than to S1, the
approximation to (47) by (48) is better than (44). That is in spite of the fact that (48) uses S2C rather than the
unknown S2.
But substituting (44) and (45) into (48) gives us
].[ 121221
12
SM
CC
SMNAvvv −= (49)
Lastly, it needs to be shown that the new approximation (NA) is the same as SMITSWAM. Rearranging
(49),
].[ 121221
1212
SM
CC
SMSMNAvvvv +−= (50)
Adding the start coordinates to both sides,
].[ 121221
12
1
1
1
12
1
1
1
SM
CC
SMSMNAvvv
h
v
h
+−
+
=+
(51)
Applying (46), this means that
,2
1
11
11
11
2
2
2
2
2
2
−
−
−
−
=
hhhh RSM
RSM
RSM
SM
SM
SM
NA
NA
NA
(52)
which is precisely the SMITSWAM approximation. So the superiority of SMITSWAM over Standard
Molodensky can be attributed to a near-average Jacobian replacing the forward Jacobian computed at the starting
point.
SMITSWAM without heights Since Standard Molodensky is often used without ellipsoidal heights, it is natural to ask whether the
SMITSWAM approach works when heights are disregarded. Setting h to zero in the forward application of (22)
and (23) is easy enough. However, it won’t stop the transformed point having a non-zero height with respect to
the second datum. Setting h to zero for the backward application of (22) and (23) will introduce errors into the
latitude & longitude shifts.
For the data sets covering Great Britain, Australia, Congo and India/Nepal, the errors relative to the rigorous
approach are smaller than for Standard Molodensky but are no longer at sub-millimetre level. The worst-case
errors are −6.0610-5 for latitude and 4.4910-5 for longitude. It is recommended that if SMITSWAM is
employed when heights are not required, equation (24) should be used to transform zero heights to the second
datum so that they can be fed into (22) and (23) for the reverse transformation.
Abridged Molodensky Another method of approximating three-parameter datum transformations consists of the so-called Abridged
Molodensky formulae. They are given by (53)-(55) below. Those for and are in radians. The h formula
would only be needed if the requirement includes ellipsoidal heights, because heights referenced to the geoid are
not affected by datum changes. The superscript “AM” distinguishes the Abridged Molodensky approximations
from the rigorously-computed shifts.
/}2sin)(cos
sinsincossin{
affaZ
YXAM
+++
−−= (53)
where is given by (25).
Page 11 of 12
cos
cossin YXAM +−= (54)
where is given by (1) .
.sin)(sin
sincoscoscos
2 aaffaZ
YXh AM
−+++
+=
(55)
According to Moore and Smith (1998), on the surface of the Earth the difference between the coordinates of
any point transformed using the two versions of Molodensky is less than 0.6m, and at an altitude of 12km the
difference is of the order of 1m.
For comparison, the examples considered earlier were revisited, this time for the computation of errors from
the Abridged Molodensky formulae relative to the rigorous approach. The results are summarised in Tables 7 to
