## Where on Earth is it?

Millions of archaeological air photos have been made in the past, and many more will be made in the future. Each photo is a unique record which contains spatial information of historical value. It's worth as a record depends on being able to find features on the ground and to protect them from destruction or for scientific study. Thus, the spatial data in the photography needs to be related to its position on a large scale map. The map in turn, is an abstraction and generalisation of the true appearance of the terrain, and it is based on a transformation of that appearance relative to the three dimensional shape of the Earth. The techniques for making such transformations, which are erroneously called 'projections' for historical reasons, have long been based on higher mathematics and hundreds have been invented during the long history of map making.

Positions on the Earth are described in terms of a grid system appropriate to a sphere, designated as latitude and longitude. For reasons to be discussed below, large scale maps use other grid systems which have certain more desirable features and which take into account the true shape of the Earth to some extent. Usually, the positions of the sites visible in aerial photographs are recorded in one of these grid systems, appropriate to the maps and measurement techniques available.

### Latitude and Longitude:

Latitude and longitude express the co-ordinates of a point any place on the Earth. Historically, since it was thought that the Earth is a sphere, these quantities are expressed as parts of a circle in degrees, minutes and seconds or in decimal form. In a few countries like France, the circle is divided into 400 parts so that a decimal division of each quadrant is possible. In reality, the Earth's cross section is not a circle, but rather to a first approximation. it is an ellipse with the horizontal radius roughly 1/300th longer than the vertical radius. This pole flattening is due to the Earth's rotation and its consequent affect on it's shape. In reality, the shape of the Earth is extremely complex, and for the most precise measurements, this has to be taken into account.

Latitudes and longitudes have historically been measured relative to an initial reference point in each area to make a datum, the name given in geodesy to define an ideal mathematical surface equal to average sea level in the area of interest. The initial point of the datum was assigned a latitude and longitude based on astronomical measurements, as well as a height above an ideal chosen ellipsoid. Hence, the datum consists of both the parameters for the ellipse and the chosen initial point. The real surface of the earth is anything but smooth, and the terrain, the ellipsoid and the ideal surface, the geoid, depart considerably from each other. Satellite position measurements which are valid over the whole surface take this into account and the ellipsoid used is usually different from that used in a national map system. Hence latitudes and longitudes obtained from them are not the same as those obtained from classical measurements as recorded on large scale maps.

From this it can be seen that latitudes and longitudes of the same point depend on the choice of the initial point. Even with the same ellipse, a different datum will shift a point's position for the same latitude and longitude, or put differently, the same point will have slightly different latitudes and longitudes when using the maps of adjacent countries.

Failure to take a change of datum into account can result in very significant errors as shown in this diagram for the position of a single point using different datums given in the on-line reference by Dana (see below). Nearly every national mapping system has its own datum, or one limited to a small region. The only system which is truly universal is that used in the Global Positioning System (GPS). Unhappily, the GPS positions themselves are deliberately made less accurate for military reasons, but simultaneous (differential) GPS measurements made relative to a known point can remove this intentional error, offering highly accurate results. Perhaps, one day, the deliberately induced errors will be removed, and the high accuracy of which these measurements are capable will be available to all of us, not only the military.

Summarising, for high accuracy, the datum must be specified when using latitudes and longitudes. One can convert between one datum and another using several different transformation techniques, providing that a number of constants are known. These constants must be determined through a long series of satellite measurements by a national mapping service. They are made at a number of points throughout the area of responsibility and a least squares fit is taken to relate classical latitudes and longitudes to those determined by the satellite data. Depending on the number of points used and the time taken for the observations, different sets of constants may be offered. These are tabulated in the on-line references given at the end of this paper.

There are three widely used methods for transforming from one set of latitudes and longitudes to another depending on which constants are available and what accuracy is required. The simplest technique, very well illustrated in the on-line reference by Dana, is to transform to a geocentric Coordinate system and then back again to the ellipsoidal coordinates of the second system. This is sometimes called a Bursa-Wolff transformation. It can be done in one operation when three constants are available using the Molodensky transformation, very well described in the on-line documentation offered by the US Defence Mapping Agency, now called NIMA. For the highest accuracy, 7 constants are required, and a Helmert transformation is used. This is described in the paper by Reit (1997). A very clear mathematical explanation for those interested is given in the on-line Australian documentation (see below).

In the author's AirPhoto program, for the following grid systems, a 7 parameter high precision (sub-centimetre error) Helmert 3D transformation between national and GPS datums is are available:

Australia, Austria, Finland KKJ, France, Gauss-Krueger 3 (Germany), Gauss-Krueger 6 (Czech Rep., Former DDR, Poland, Slovakia, Hungary), Netherlands, Sweden RT90, Switzerland.

A 3 parameter lower precision (ca. 2-3 meters) transformation between national and GPS datums is offered for all other supported systems. The North American Datum, NAD27 is supported with an average value for the whole of the USA and Canada, giving an error of the order of 5-6 meters. These low precision conversions use the constants from the Molodensky transformation offered in the US Defence Mapping Agency's MADTRAN and its successor for Windows from the National Imagery and Mapping Agency (NIMA) DTCC (details below). Whenever more accurate 7 parameter Helmert 3D constants become available, these will be replaced.

In AirPhoto, Norway-Sweden-Finland GPS (WGS84 Datum) compatible grids are supported directly with sub-millimetre precision and require no conversion. Special transformation techniques worked out by Reit and Ollikainen (see references) are used. Norway is currently the only country using GPS compatible coordinates directly 1993 to construct the grid system used on its large scale maps, but others may change over during the coming years.

For highest accuracy, map calibration points should also be given a height above the Geoid. As a first approximation, height above mean sea level obtained from local system contours may be used, since geoid heights are rarely known. If zero height is given for a national system, conversion errors to GPS values will be somewhat higher than those described above. Height above the Geoid can be obtained for many areas using the DTCC program from NIMA (see below).

### Maps as transformations:

Maps must represent the ellipsoidal shape of the earth on a plane surface. This cannot be done without some distortion. Small scale maps of large regions were historically 'projected' from the sphere on to the plane before it was realised in the mid-18th century that the Earth is not spherical. Thus, the term, "Map Projection" is widely used.

However, the large scale maps used for precise plotting of archaeological sites or finds are usually designed so that angles and distances are distorted as little as possible. This is not possible using most "projections". Since measurement of angles was and to some extent still is essential in survey operations using theodelites, cartographers from the 18th century onward were concerned with creating mappings which are mathematical transformations of points on the ellipsoid to points on the plane constructed such that angles were preserved correctly. This was also of extreme importance to the military after the development of long range artillery in the 19th century. Although latitudes and longitudes are usually shown on the map borders, it is customary to superimpose a rectangular grid on the map whose values can be read as co-ordinates. This grid is usually designed to preserve angles. Thus, in nearly every, an angle preserving or 'conformal' grid system was developed for the national map grid.

These grid systems are not 'map projections' , because they are purely mathematical transformations of the latitude and longitude of points to grid co-ordinates and the inverse. Each country developed one or more grid systems during the last two centuries with little regard to what was being done in neighbouring countries. They are widely entrenched and are not readily changed, since they are incorporated in all land survey data for property ownership and appear directly or indirectly in millions of documents. They are also the primary references for the positions of archaeological sites found through aerial photography or conventional means.

A rectangular grid system cannot cover a wide area without severe distortion. Therefore the surface must be broken up into segments or strips which can be treated in a planar fashion without too much error. The numbering in such grid systems used in the large scale maps of all large countries undergoes sudden changes. At the borders of countries using differing map projection schemes, co-ordinate systems are not the same and transformations are required from one system to another. In North America, completely different systems were used in the countries concerned and even within most of the 50 US states in the past. In Africa, Asia and South America, the colonial heritage is preserved in many different incompatible mapping schemes. In Europe, there are still as many different systems as there are countries.

These non-conformities make it impossible to conduct a geographic search in a database across national boundaries or to apply spatial statistical methods to the grid co-ordinates for sites taken from large scale maps from a wide area. Map numbering and grid reference schemes are alphanumeric in many countries which also complicates searching in a database when distance information is required.

One solution lies in appropriate transformations for the various national co-ordinates from the national ellipsoids to a common reference. In many areas, the GPS (WGS84) datum is the most reasonable one to use. In Europe, a reference point, the European Datum of 1950 (ED50) was decided upon by international convention. It is located at the geodetic observatory at Potsdam near Berlin and uses the International Ellipsoid of 1924. However, most European national mapping systems do not use this datum. In North America, a common reference point was chosen in 1927 (NAD27) at Mead's Ranch in Kansas with the Clarke Ellipsoid of 1866. Now, with satellite measurements of precise position available, many advanced countries are beginning to refer their national systems to the WGS Ellipsoid of 1984 under various local names whose parameters, will not be subject to significant further change. Small adjustments which account for movements of tectonic plates and crustal uplift in deglaciated regions are needed in high precision geodesy, but these are not very significant for archaeological sites.

Peter Dana at the University of Texas estimates that almost 1000 different datums have been in use at one time or another. Parameters for conversion of over 200 of these to WGS 84 are readily available. The WGS84 system references the gravitational centre of the Earth. Almost all western countries have readily available maps at scales of 1:25000 or larger, and the sites are usually referenced to the grid systems used on these maps. In less well mapped areas, GPS measurements are now widely used to obtain positions.

### Ellipsoids:

Although the real shape of the Earth is obtained through accurate satellite measurements today, older ellipsoid approximations obtained astronomically are still widely used in cartography. One of the early models was due to Bessel whose name is known more by the functions named after him, than for his activity as director of the Prussian state mapping service in the early 19th century. The Bessel ellipsoid of 1841 despite its errors survives to this day in the German, Austrian, Swiss, Dutch, and Swedish mapping systems. The British physicist Airy made major contributions to the theory of optics, but he too was concerned with cartography. The Airy ellipsoid of 1822, one of the very first, is still used in the UK and in a modified form in the Irish Republic. In 1909, J. F. Hayford of the US Coast and Geodetic survey made measurements within the continental USA which were adopted by the International Union of Geodesy and Geophysics as the International Ellipse of 1924. This was used until very recently in Scandinavia, Italy, the USA and is still used in Luxembourg, Belgium and until 1993 in Norway and Finland. The British geodesist Clarke made several sets of measurements in widely different parts of the world in the mid-19th century and the Clarke Ellipsoid of 1866 is still used in the USA while that of 1880 is used in France. In 1942, Krassovsky published further measurements based on the Pulkovo observatory near Leningrad and the Krassovsky ellipsoid of 1940 became the standard of the former Warsaw Pact. All these ellipsoids differ slightly from each other in the values for the Earth's major and minor axis radii. And all have one or more reference points from which latitudes and longitudes have been assigned in the past.

Almost every country uses a different ellipsoid and a different datum for its mapping system. Although differences are small, they are significant enough so that latitudes and longitudes for the same point will differ somewhat when the ellipsoid and the datum is changed. National maps usually use an ellipsoid model which provides a best fit for the local system.

### Transformations for large scale maps:

Several major types of transformation or projection are used commonly in large scale mapping, Snyder (1982). The common transformations are to a cylinder wrapped transversally around the Earth either tangentially or cutting through two parallels of longitude (Transverse Mercator), to a cone either tangent to or cutting through two parallels of latitude (Lambert Conic Conformal), and more rarely to a cylinder lying obliquely along a meridian (Oblique Mercator), and still more rarely to a plane just touching the surface with co-ordinates projected from the opposite point on the other side of the Earth (Stereographic).

Sometimes the chosen ellipsoid model is first transformed to a sphere, and then another geometric figure which can be unrolled is mapped from the sphere. Usually, this two stage process is avoided through direct analytical calculation based on the work of Gauss in the early 19th century. However there are almost as many methods as there are and were cartographers! J.H. Lambert, the Alsatian mathematician and cartographer invented or analysed a number of these projections including the two major ones and published in 1772.
The major mappings are:
1. To a cone with a transformation, the cone being then unrolled. These belong to the family of Lambert mappings and are in use in countries which have a greater east-west than north-south extent.

2. Mapping to a cylinder which is then unrolled by means of a transformation formula which is not a geometric projection. These are the Gauss-Krüger mappings invented by Lambert and their derivative in the UTM or UK national grid system.

3. Transformation mapping to a tangent plane from the opposite side of the ellipsoid. These are Stereographic mappings as used in the Netherlands and a few other countries and known since antiquity Snyder (1982) states "that it was probably known in its polar form to the Egyptians, while Hipparchus was apparently the first Greek to use it." Until the early 16th century it was only used for star maps.

4. Transformation of the ellipsoid to a sphere, then taking a transverse cylinder mapping rotated so that the meridian at a point near the centre of the country becomes the zero meridian, as used in Switzerland.

5. Purely mathematical transformations which cannot be described by a geometric figure. These offer lower distortion than the four classical methods above. An example is the grid system used in New Zealand.

Most countries which have a greater North-South than East-West extent use a transverse cylinder tangent to a meridian with axis east-west. This is called a transverse Mercator mapping in English-speaking countries, and a Gauss - Krüger (GK) mapping elsewhere. It is the most popular for large scale maps.

C.F.C. Gauss may have earned his fame in many other ways in statistics and mathematics, but he earned his living as director of the mapping service of the Hannoverian Kingdom. Krüger (1912) systematised and revised the original Gauss (1822) design. It is used in the German speaking and the former Warsaw Pact countries, though with different ellipsoids. Meridian strips of either 3° or 6° are used as discussed below.

When the transverse cylinder is made slightly smaller than the ellipsoid, it cuts two meridians. This is the basis for the Universal Transverse Mercator (UTM) system which is the NATO standard for military maps and is used for recent maps in Denmark, Norway, Italy and the USA. The UTM system has an advantage over the GK method in that only half the number of map strips are required to cover the globe with lower distortion than a GK system would produce for strips of the same width. In the UK and the Irish Republic, a modified form of UTM is in use.

Both the GK and the UTM systems have the major virtue that angles are preserved locally, distance distortion is minimised, and the co-ordinate system is Cartesian on a given map sheet so that only a straight line drawn from border to border in each direction is needed to obtain the co-ordinates of any point on the sheet. UTM co-ordinates are ambiguous north and south of the Equator, because 10,000,000 is added to them in the southern hemisphere to avoid negative quantities.

The truncated cone is used for mapping in France and Belgium and in some of the older US. maps among others. This leads to the Lambert Conformal mapping. It had the advantage of being expressible in closed analytic form as opposed to the GK and UTM systems which require series expansions of functions of a complex variable. Back in the 18th century when it was invented, this was important, since the more sophisticated conformal techniques which use the series expansions were invented several generations later by Gauss. In the Lambert Conformal system, angles are preserved, and distance distortion is not excessive. France uses four truncated cones with slightly different scale factors for the northern, central, southern parts of the country plus Corsica for its four zones. These are separate Lambert Conformal mappings.

Belgium uses a modified Lambert Conformal system which has its origin at the old Royal Observatory near Brussels. It uses the International (Hayford) ellipsoid of 1924 and has but one zone. To prevent negative values, 15000 is added to the east-west and 5400000 to the north-south axis. Belgian Lambert co-ordinates are incompatible with the French Lambert system described above.

The Netherlands is one of the few countries in the world to use the Stereographic projection, in theory a true projection to a plane from a point opposite The Hague on the other side of the ellipsoid. Stereographic projections are also standard for polar latitudes above 80° north and south. Practically, however, a series expansion form is always used. This system should not be used at even minor distances outside the Netherlands, or significant errors will occur.

The Swiss system is one of a very few countries to use an oblique transverse Mercator scheme. This offers high accuracy in one single co-ordinate system for the whole country, but it too should not be used outside the Swiss borders.

All of the commonly used transformations preserve angles well enough and are of sufficiently low distortion so that linear distance measurements can be made on a map sheet of scales 1:50000 and larger. However, on Lambert Conformal maps, the horizontal lines of the grid are in fact arcs with such large radii that the departure from a straight line is hardly noticeable.

### Grid reference letter and number combinations:

Grid references are a convenient shortcut to refer to map positions. They are usually two numbers which are in the Cartesian co-ordinates of the map transformation used. However, In Britain and Ireland, they are preceded by a two letter code which designates a 100 km side length square in a modified transverse Mercator system. The first two digits of the 100 km reference grid are usually dropped. There is also a Military Grid Reference System (MGRS) for the whole globe with alpha-numeric prefixes. MGRS is used in the MADTRAN-DTCC program package. In western Germany, Austria, Luxembourg, Italy, the Scandinavian countries and the former Warsaw Pact, the co-ordinates may be given as x and y values (sometimes designated R and H for the German 'rechts' and 'hoch') within a given 1:5000 or 1:25000 map sheet. For historical reasons, cartographers outside France use X for north-south and Y for east-west co-ordinates. In France, the kilometre digits of the Lambert co-ordinate system appear along the border of the map, but either the name or number or conic segment of the map must also be supplied to prevent ambiguity. The borders of the French 1:25000 maps give geographic co-ordinates in "gon", 1/400th of a circle, but the conversion by multiplying by 0.9 to get degrees is trivial. The map border markings are based on the meridian of Paris rather than Greenwich on the older maps, so that a constant has to be subtracted as well. Newer maps have both Greenwich and Paris meridian markings.

### Map naming conventions:

Maps at scales of 1:100000 or larger have widely different nomenclatures, depending on the country of origin.

A few examples:

Austria bases the zero meridian of its GK mapping system according to the convention of 1634 in the Habsburg monarchy in the westernmost Canary island Ferro and numbers the 1:25000 maps in 1:50000 squares from 1 to 4 on the military grid. Since 1976, the 1:25000 maps are merely enlargements of the 1:50000 and have a sequential numbering system starting at the northernmost tip of Austria and proceeding to the south from west to east.

In Britain, the maps are given the two letter combination from the 100 Km national grid, plus the numbers of the lower left co-ordinates on that grid. Ireland uses a similar scheme but with different letters and with some squares not implemented.

In Belgium the newer 1:50000 maps are numbered sequentially from north to south on with 1:25000 subdivisions with sub numbering. The Lambert projection is a national one and is not compatible with the French projection. The 1924 international ellipsoid is used instead of the Clarke 1880 ellipsoid used in France.

Denmark uses Arabic numbers for the 1:100000, Roman numerals for the numbering of the 1:50000, and for the 1:25000 sub numbers these as Northwest, Northeast, Southwest and Southeast, using a UTM model although the standardised UTM notation is not followed for the map names.

In France, the 1:50000 maps are numbered with Arabic numerals for the rows and Roman numerals for the columns, and these are further divided into four sheets of two half-sheets each at 1:25000, numbered from 1 to 8, each bearing the name of the principal town in the 1:50000 sheet.

The 1:25000 (TK) maps in Germany are bounded by 6' and 10' latitude and longitude, but the GK grid is shown. The 1:5000 maps use the GK grid as a border and are numbered using the first four digits of the x and y co-ordinates in that system. The German 1:25000 maps for historical reasons are numbered separately with four digits starting in the north and terminating in the south of the country. The author's AirPhoto program offers conversions between TK numbers and the GK grid.

Luxembourg numbers its maps from 1 to 30 and uses a GK system passing through the meridian of the city of Luxembourg.

In Switzerland, maps at 1:100000 have two, 1:50000 three, and 1:25000 four digit numbers. The mapping transformation is a unique variant of GK (see below).

### Meridian and latitude strips:

Given this map maker's Tower of Babel, it is not surprising that archaeological databases on or across international boundaries are not easily implemented. Even within a given country, searches for sites on adjacent but non-numerically named maps is a major chore. The situation is made even more complicated by the fact that a given transformation may introduce too much distortion in a large country. Therefore both the transverse Mercator and the Lambert Conformal methods break the ellipsoidal surface up into a series of meridian or latitude strips. This is shown for the GK case in:

for the whole globe. Seen in detail for a given country like Germany, the strip boundaries at 3° intervals are shown in:

The strips are numbered beginning at the Greenwich meridian according to the central meridian of the strip, so that West Germany has three strips numbered 2 to 4 corresponding to the 6°, 9° and 12° strips. At the strip boundaries, the map co-ordinate numbers jump abruptly and no database query system can find adjacent map co-ordinates by simple arithmetic. In fact, none of the requirements stated above for distance and co-ordinate measurements can be met near such strip boundaries. For the Gauss-Krüger system used in eastern Europe and Russia, the strips are at 6° intervals. To add to confusion, the GK maps used in the former German Democratic Republic have 3° interval strips but use the former Warsaw Pact ellipsoid and datum so that co-ordinates are incompatible with those across the former Iron Curtain to the west.

Similarly, the UTM map has an overlapping wedge shaped boundary zone every 6° as shown in:

Fortunately for the Danes, their country is contained almost entirely within one UTM strip. Britain is also fortunate in being elongated from north to south, so that only one strip is used tangentially to 2° west of Greenwich. Although computed using the UTM method, the map lettering is incompatible with standard UTM nomenclature. Austria has three strips, with the zero meridian at Ferro as mentioned above and is compatible only with older maps of Spain (the Habsburg connection!). The Swiss system with a inclined transverse cylinder centred on the old Observatory in Bern is independent of all its European neighbours. Like that of the Netherlands, it is unique in Europe.

The author's AirPhoto program offers support for the national grids of:

Australia:

The Australian grid (ANG84) is standard UTM using the WGS84 ellipsoid and a 10,000,000 offset added to the 'northing', to prevent negative numbers, since all of Australia lies south of the equator. The older AGD66 grid which is used on the maps of Tasmania is not supported, since it cannot be readily converted to WGS84.

Austria:

The Austrian calculations use the Bessel ellipsoid of 1841, but the geographic co-ordinates are shifted to 17° 40' west (Ferro island in the Canaries). A modified Gauss-Krüger calculation is used for the three meridian strips covering Austria at 28°, 31° and 34° east of Ferro. The Austrian GK has no added constants, so the R values may be negative and are duplicated in each strip. Therefore either the strip or 1:50,000 map number must be specified to compute lat/lon from the Austrian co-ordinates. Austrian co-ordinates should only be used within Austria's national boundaries.

Lloyd A. Brown, "The Story of Maps," p. 283 writes: "In 1634 Louis XIII decreed the island of Ferro (Hierro) in the Canaries as the prime meridian to be used on all French maps, and so it remained until about 1800." Also, in the Encyclopaedia Britannica, 11th-13th ed., p. 985, under "Time, Measurement of": "... a scientific congress, assembled by Richelieu at Paris in 1630, selected the island of Ferro for this purpose [prime meridian] .... Ferro, which received the authorisation of Louis XIII. on the 25th of April 1634, gradually superseded the others." (personal communication from the late J. P. Snyder to the author).

Belgium:

The Belgian grid uses a modified single zone Lambert Conformal mapping with the International Ellipsoid of 1924. Offsets are added as described above.

Finland:

The Finnish grid uses a 3° Gauss-Krüger system with the International Ellipsoid of 1924, but with a special Finnish Datum, abbreviated KKJ. The KKJ Datum is a slightly rotated, scaled and translated version of the European Datum of 1950 datum. For compatibility with neighbouring countries grid systems, the KKJ datum is first transformed to ED50 before carrying out the Lat/Lon to ED50 calculations, and the inverse transformation is used to convert Finnish co-ordinates to equivalent ED50 co-ordinates before returning Lat/Lon in the WGS84 datum. There are three strips, and zone numbers 1-4 are inserted in front of the 500000 central meridian for each, such that the strip meridians are at 15000000, 25000000 etc. Details are given in the Ollikainen on-line reference below.

France and Corsica:

The French calculations use the Clarke Ellipsoid of 1880. The Lambert zone number (1-4) must be specified when computing lat/lon from Lambert co-ordinates. French co-ordinates are sometimes also used in Luxembourg and Belgium, but in the latter country, they will not be identical with the Belgian Lambert co-ordinates which appear on the 1:25000 maps, and they are totally incompatible with the national grid system of Luxembourg.

Gauss-Krüger 3°:

Gauss-Krüger conversion is based on the original equations due to Krüger (1912) as implemented by Reit (personal communication) and modified for fast Clenshaw summation of series expansion following a little known method worked out at the Danish Cadastre in the author's program. Since 8th order coefficients are used, accuracy is in the sub-millimetre range. The GK calculations use the Bessel Ellipsoid of 1841. It should always be used in the former west Germany and may be used anywhere else in the world. In Colombia, South America, offsets based on a datum at the Observatory in Bogota are added to the north and east values.

Gauss-Krüger 6°:

The GK6 calculations use the Krassovsky Ellipsoid of 1942. The method is the same as that used for the GK3 calculations modified for a 6° meridian strip. It should be used in all former of the Warsaw Pact countries and may be used elsewhere if desired, although the errors near the strip boundaries are slightly higher than in the UTM system. Please note that this grid was not used in the former German Democratic Republic, where the GK 3° grid appears on the 1:10000 maps. The datum there is, however, a Krassovsky ellipsoid with origin at-Pulkovo near St. Petersburg and not the Bessel ellipsoid as used with GK3 in the rest of western Germany. The result is that there may be differences of hundreds of metres or more in the area along the old Iron Curtain where the two systems are adjacent. Thus, within the reunited Germany, there is a serious discordance in the local maps.

Ireland:

The Irish national grid calculations are also a variant of UTM with the origin shifted to 8° W longitude, 53.5° N latitude and a modified Airy Ellipsoid, with false easting of 200,000 and false northing of 250,000 and modified central scale factor of 1.000035. Irish co-ordinates should only be used within the Republic of Ireland. Although both the UK and the Republic use an Airy Ellipsoid, the two are not quite the same, nor is the datum point, so that there is a discordance along the border with Ulster of considerable importance.

Italy (Gauss-Boaga):

Italy uses a standard UTM mapping, but with a datum point located at Mount Mario near Rome based on the Hayford Ellipsoid and offsets of 1,500,000 and 2,500,000 added to the 'eastings' for the two strips which cover the whole country and its adjacent islands. The strips are given names, called Fuso Ouest and Fuso Est, not numbers. An older system used in Sardinia is not supported.

Luxembourg:

The Luxembourg single strip Gauss-Krüger system is centred on 49° 50' north latitude, 6° 10' east longitude using the International ellipsoid of 1924. Numerical offsets of 80000 and 100000 meters are added to the east-west and north-directions respectively.

Netherlands:

The Dutch grid system is based on a Schreiber - Gauss conformal mapping of the Bessel 1841 Ellipsoid to a sphere, followed by a sterographic projection. The zero point is based on Amersfoort, with offsets of 155000.0 and 463000.0. A bivariate quintic polynomial is used for the conversions with constants as given in the on-line reference due to Schut below.

Norway:

Recent Norwegian maps use the WGS84 GPS datum and a UTM Grid. Older Norwegian maps prior to 1993 use UTM with the ED50 Hayford Ellipse and Potstdam datum. If required, this can be chosen by selecting UTM International instead of UTM WGS84.

Switzerland:

The Swiss calculations use the Swiss Bessel Ellipsoid of 1903. The co-ordinate system is based on a conformal mapping of the ellipsoid to a sphere, followed by an oblique transverse Mercator mapping centred on the Old Observatory at Bern with zero meridian passing through Bern, offset by 200000 and 600000 in the longest north-south and east-west directions in Switzerland. They are based on Odermatt's modification of the original Rosenmund method. The author's program uses the official equations provided by the Swiss Federal land survey department.

Sweden:

Sweden uses a Bessel Ellipsoid with a single strip having a central meridian at 15° 48' 29.8" and a false Easting of 1500000. This is the RT90 grid. Routines are offered for conversion to latitude and longitude with WGS84 compatibility for use with GPS co-ordinates. In this case, a special unpublished set of offsets and scaling supplied by Bo Gunnar Reit of the National Land Survey of Sweden is used avoiding the need for a Helmert transform.

United Kingdom:

The UK national grid calculations are a variant of the UTM calculation with the origin shifted to 49° N Latitude, 2° W longitude and false easting of 500,000 and northing of - 100,000 added to the constant computed to obtain the latitude shift. The Airy Ellipsoid is used with a Central scale factor = 0.9996012717. The UK 100 km map sheet names maybe specified and the 100 km digit dropped when computing lat/lon from the UK grid co-ordinates, or the full grid co-ordinates may be given instead. UK co-ordinates should only be used within the United Kingdom. Map names are quickly found via binary search on a sorted array. Although the official calculations used in the UK are based on equations due to Hotine (Snyder 1982), the author's program uses the more accurate Swedish-Danish technique based on 8th order GK method.

UTM, WGS84, International and Clarke 1866 Ellipsoids:

The UTM conversions are based on the original Krüger (1912) method, using the International Ellipsoid of 1924 or the Clarke ellipsoid of 1866 or the WGS ellipsoid of 1984. In the USA, when a transverse Mercator projection is used, it is used with the Clarke ellipsoid of 1866. The datum used for UTM International calculations is the European Datum of 1950. With the Clarke calculations, the datum used is the North American of 1927 (NAD27). For WGS84, only the ellipsoid parameters are used. GPS data may be entered directly however and mapped in UTM co-ordinates.

The zone number must be specified when computing lat/lon from UTM. Since 10,000,000 is added to a UTM northing if a point is south of the equator, a flag must be set true when this is the case. It should be used in Scandinavia, the USA and almost any place else in the world. Again, be warned that UTM here is based on the International Ellipsoid, but that some USA state maps use the Clarke Ellipsoid of 1866 with different offsets.

Other grid systems and datum transformations are added to AirPhoto and the underlying Maplib32.dll as need arises.

Translating from one national or international grid system with different datums to another:

There are no direct methods available for translating from one national grid system to another. However, one can be implemented via a series of intermediate steps when appropriate constants are available.

1. Using the appropriate inverse transformation equations for the first grid system, the latitude and longitude of a point is obtained in the local datum.
2. This latitude and longitude is transformed into the latitude and longitude of the GPS WGS84 system using either a Helmert, Molodensky or Bursa-Wolff transformation, depending on available constants.
3. This GPS latitude and longitude is transformed into the latitude and longitude of the second national grid system again using one of the three transformations with appropriate new constants.
4. Finally, the new latitude and longitude is transformed back into the second grid system using the formulae for the second grid.

The accuracy achievable depends on the precision of the constants available. The error introduced by all these intermediate steps is usually a bit more than the uncertainty in the available constants, but it is far better than simply using the raw latitudes and longitudes from one system for the other.

In AirPhoto, all the latitudes and longitudes of the calibration points for a map are transformed in a single operation to those of the GPS system. A new grid system is then chosen, and the transformation from GPS to that system is made, from which point on, the program then runs as if it were using the second system. The time required on a modern machine is so small as to appear instantaneous.

### Printed References:

• DMA (Defence Mapping Agency) Technical Manual 8358.1, 1990, Datums, Ellipsoids, Grids, and Grid Reference Systems. USA Department of Defence, Washington, DC.
• DMA Technical Report TR 8350.2, DoD World Geodetic System September, 1991, USA Department of Defence, Washington, D.C.
• Jordan, W., Eggert, O., Knießel, M. 1958, Handbuch der Vermessungskunde, 10te. Aufl., 4 1958 478-489 J.B. Metzler Verlag, Stuttgart
• Krüger, L. 1912, Konforme Abbildung des Erdellipsoids in der Ebene., Veröffentlichung des Kgl. Preuß. Geodätischen Instituts, N.F, 52 1912 Teubner Verlag, Leipzig
• Reit, B.G. 1997, A simple way of introducing a global reference frame in surveying and mapping. Survey Review, 34(264) 1997 87-90.
• Schödlbauer, A. Rechenformeln und Rechenbeispiele zur Landesvermessung, Herbert Wichmann Verlag, Karlsruhe, Teil 1 ISBN 3-87907-120-9 1981 and Teil 2 ISBN 3-87907-121-7 1982
• Scollar, I. 1989, Geodetic and cartographic problems in archaeological data bases at and within the boundaries of some countries, Computer Applications in Archaeology, 548 1989 251-273 BAR, Oxford
• Scollar, I., Tabbagh, A., Hesse, A., Herzog, I. 1990, Archaeological Prospecting and Remote Sensing in Archaeology, Cambridge University Press, Cambridge ISBN 0 521 32050 X, 673 p.
• Snyder, J.P. 1982, Map Projections Used by the US Geological Survey, 2nd ed., Geological Survey Bulletin 1532, U.S. Government Printing Office I 19.3:1532, Washington, D.C.
• Snyder, J.P. 1985, Computer - Assisted Map Projection Research, Geological Survey Bulletin 1629, U.S. Government Printing Office I 19.3:1629, Washington, D.C.
• Snyder, J. P., Map Projections - A Working Manual, U.S. Geological Survey Professional Paper 1395, 1987, corrected reprint 1994
On-Line sources:

For experienced programmers who wish to use the AirPhoto Maplib32.dll, the computational component of AirPhoto for co-ordinate and datum transformation, the documentation for the interface can be examined and printed from the help file if desired. The dynamic link library may be used freely without AirPhoto from any compatible programming language or database which can use standard Windows dynamic link libraries for other purposes like making co-ordinate databases for sites spread across national boundaries. In AirPhoto, when a scanned map has been calibrated, coordinates are shown in the calibrated system or in any transformed system instantaneously with each mouse movement. When the scanned archaeological airphoto has been transformed by the program to the map, the coordinates of any features in the combined image may this be read directly with accuracy limited only by the precision of the scanner used.

AirPhoto can be obtained by downloading from the author's Web site or from mirror sites in Edinburgh, Scotland and Storrs and Buffalo in the USA

or via anonymous ftp:
• ftp.uni-koeln.de /pc/basp
• mirrored at the University of Edinburgh ftp site: super3.arcl.ed.ac.uk, /ftp/pub/baspmirror

Irwin Scollar
AL001@MAIL1.RRZ.UNI-KOELN.DE
In der Au 9
D 53424 Remagen
Germany