PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
July 2019
477
ered at the time, too small to warrant the extra effort. For
instance, the Lambert Conformal Conic projection was used
only to the cubic term in the formulae for the tables of the de-
veloped meridional distances. This resulted in French Army
projection tables that have become part of the arcane lore of
computational cartography.
Furthermore, another idiosyncrasy of the French Army for-
mulae is that the Lambert (fully) Conformal Conic projection
normally utilizes one of the principal radii of the ellipsoid
called the Radius of Curvature in the Plane of the Meridian
(ρ
o
). The French Army instead substituted the Length of the
Ellipsoid Normal Terminated by the Semi-Minor Axis (υ
o
) at
the Latitude of Origin (
λ
o
). Although not strictly conformal,
this is the system that was commonly used by the French in
all colonies (before WWII) that utilized the Lambert Conic
projection (including Syria;
PE&RS
, September 2001).
Standard Lambert formulae will not work for Algeria un-
der certain conditions, and the improper use of the fully con-
formal projection will yield computational errors that can ex-
ceed 15 meters! As an example, consider a test point where
f
= 33° N and
l
= 3° E. For Nord Algerie on the French Army
Truncated Cubic Lambert Conic Grid, X = 528,064.182 m and
Y = –32,764.881 m; for the same test point on the
Nord Al-
gerie
Lambert fully Conformal Conic Grid, X = 528,074.691 m
and Y = –32,776.731 m. The computational difference of the
two formulae at the same test point is ∆X = –10.509 m and ∆Y
= +11.850 m, for a total error of 15.839 meters! Mathematical
elegance is not what matters in a country’s coordinate trans-
formations; what matters is computational conformity to lo-
cal legal standards. The certain condition when a fully con-
formal Lambert Conic will work in Algeria is based on
when
a particular Algerian map was compiled. That is, when the
Algerian triangulation was recomputed for the European Da-
tum of 1950, the French dropped usage of the Truncated Cu-
bic version on the old Voirol 75. In summary, for surveys and
maps before 1948, one must use the French Army Truncated
Cubic Lambert Conic. After 1948, one must use the Lambert
fully Conformal Conic. The parameters of the two Lambert
zones did not change for the Colonne Voirol Datum of 1875;
only the formulae changed. Things soon got more complex.
In 1953-1954 the first-order coastal parallel chain was reob-
served by the French. In 1959, the Institut Géographique Na-
tional (IGN), Paris, readjusted the entire first-order and first-or-
der complementary triangulation to the European Datum 1950
(ED50), incorporating the results of all previous surveys and ad-
justments. The rule of thumb for this Datum Shift is to increase
both Latitude and Longitude from the Colonne Voirol Datum
of 1875 to the European Datum 1950. The UTM Grid was used
for this purpose, as were all Datums that were transformed to
ED50. Like most countries, the ED50 UTM Grid was reserved
for military topographic mapping, and local native systems con-
tinued in use. That tradition has resulted in some convoluted
transformations being perpetuated in Algeria.
The North Sahara Datum of 1959 was obtained (in 1957-
1958) by recomputing the results of the first-order nets and
the first-order complimentary nets adjusted to the ED50, but
referenced to the Clarke 1880 (modified) ellipsoid where a =
6,378,249.145 m and
1
/
f
= 293.465. The adjustment on the
Clarke 1880 (modified) ellipsoid was performed such that it
optimized the fit of the shape of the geoid in North Africa,
i.e., by reducing to a minimum the sum of the squares of the
relative deflections of the vertical in the areas involved. This
principle was intended to minimize the mean discrepancies
between the geodetic net used in the northern part of Algeria
and the astronomic net used primarily in the southern part
of Algeria. Some maps were stereocompiled on the North Sa-
hara Datum of 1959 with the UTM Grid at 1:200,000 scale.
However, many maps were not cast on the UTM Grid.
The Lambert North Sahara Auxiliary Grid was directly
applied to the geodetic coordinates in accordance with the
definition of the
Nord Algerie
Zone with the fully conformal
formulae. However, it was never used in any publication or
in mapping because of the large discrepancies found between
the rectangular coordinates of any given point on the Colonne
Voirol Datum of 1875 (Voirol 75) or the North Sahara Datum
of 1959. This computational experiment is the reason for the
development and subsequent adoption of the Lambert Voirol
60 Grid System. This curious system adds 135 meters exactly
to the X coordinates and adds 90 meters exactly to the Y co-
ordinates of the original
Nord Algerie
Zone parameters. In
other words, the Lambert Voirol 60 Grid has a False Easting =
500,135 m and a False Northing = 300,090 m. According to the
French Army in June of 1970, “Under these conditions, when
we compare the LAMBERT – VOIROL 75 with the LAMBERT
VOIROL 60 coordinates, the shift between the two is always
less than 50 m in absolute value. This value does not represent
a mathematical relation, but rather the result of comparing
the two sets of coordinates. It shows up the inaccuracies in
the initial VOIROL 75 system. The maps made with the LAM-
BERT VOIROL 60 rectangular coordinates are all referenced
to the geographic coordinates of the NORTH SAHARA geodet-
ic system.” The current parlance for this in English is the “Voi-
rol Unified 1960 Grid” on the “North Sahara Datum of 1959.”
Note that there is no classical origin for this Datum due to the
fact that it is derived from the ED50.
In 1966, the Army Map Service (AMS) developed a series
of conversions on a Carton-by-Carton basis for transforming
from Voirol 75 to ED50 with UTM coordinates. As an example
of the transformation series for Algeria, the following is for
coordinates in UTM Zone 31 whose eastings are greater than
355,000 m: Carton 59: N = 0.9998873966 n – 0.l0000869984
e + 691.561 m and E = 0.9999391272 e + 0.0000869984 n –
416.633. The stated RMSE for this Carton is ±0.200 m. The
adjacent Carton 60, when used with the appropriate coeffi-
cients, has a stated RMSE of ±2.759 m!
In recent years, the IGN derived a seven-parameter trans-
formation from ED50 to WGS84 for North Africa. The quot-
ed accuracy is ±2 m in X, Y, and Z, and, when applying this
transformation, the resulting heights are approximately 30
m higher than expected for Algeria. The parameters are ∆X =
