Thursday, October 25, 2007

Molodensky: A Short History of Man's Contribution to Geodesy

As internet mapping becomes more and more prevelant and GPS is being used more we keep on hearing about the Molodensky Transformation, but who was Molodensky?
A Short History
Mikhail Sergeevich Molodenksy (1909 - 1991) was a promiennt geodesist and geophysicists who many consider a reformer in the theory of the figure of the Earth and the study of the Earth's rotation and oscillations.
He was born on June 15, 1909, in Epiphan, a small town in the Russian province of Tula. He did his initial studies at the Astronomic Department of the Mechanics and Mathematics Faculty of Moscow State University.
He was later invited by F. N. Krasovsky (there is an ellipsoid that bears this geodesist's name - more on a later blog about Russian mapping) to join the staff at the Central Research Institute of Geodesy, Aerophotogrammetry and Cartography (TsNIIGAik). It was here that he worked for over 25 years.
Molodensky's early steps in geodetic research and geodetic surveys date back to 1929 when he did some inital work for the Institute of General Geodetic Surveys (IOGR) and prior to this survey, he was given a proposal from the director of Astronomy - Geodetic Research Institute (AGNII) at State University in Moscow to do geodetic research.
There came a famous decree called the "Soviet of Labor and Defense" (May 6, 1927) which set about the establishment of a general gravimetric survey over the whole country. Lenin was laying out his vision for the country and defining a "Soviet". With this decree, there became an increase in the development of gravimetric surveying throughout the whole country.
Molodensky, who was avidly interested in gravimetry, participated in these surveys with his work at TsNIIGAik. While here and conducting these surveys, a young Molodensky, in 1933, headed up an expedition to the Crimea to perform gravimetric surveys (again under the above decree).
A Rigourous Solution
A year later, in 1934, Molodensky was beginning to make a name for himself by presenting a report at the 7th Baltic Geodetic Commission Conference in Moscow. His topic, which geodesists considered urgent at the time, is the co-swinging influence on double pendulums. Before his presentation, all solutions were seen as impossible because the accuracy of the solutions could not be determined. Molodensky provided a rigourous solution. In turn, this presentation was heard by scientists from Denmark, Finland, Germany, Poland, Sweden, the USSR and the members of the International Association of Geodesy. He essentially turned the world of geodesy upside down concerning astronomic-gravimetric leveling. His final report was published in Helsinki at the 1937 meeting of the Baltic Geodetic Commission.
Molodensky made significant developments to Soviet geodesy and gravimetry, in theory and in practice, especially when developing and applying survey methods combined with the design of gravimetric instruments. In the 1930's the only gravity measuring devices that could be found in the former Soviet Union were those of foreign manufacture. Therefore, the state saw immediately that there was an immediate task facing geodesy in the Soviet Union - the manufacture of gravity measuring instruments. A small batch were initially made, based on the German "Bamberg" instruments and several designs of original instruments had failed. It was not until 1938 that the Soviet Union had developed their own gravity measuring devices.
April, 1943, lead to the appointment of Molodensky as chief of the gravimetric laboratory at TsNIIGAik. He held this post until July 1956 when, against his will, he was appointed director of the Geophysical Institute of the Academy of Sciences (GEOFIAN). At this point he became responsible for the realization of the technical policies related to geodesy in the Soviet Union.
During this time he continued to make significant contributions to geodesy and the state wanted to recognize him for his efforts. In 1946, he was awarded the USSR State Prize, then he recieved the high degree of a Doctor of Technical Sciences and was then elected a member of the USSR Academy of Sciences.
The Geoid and Plumb-Lines
As you may recall in my previous blog, about the geoid, I discussed the "deflection of the vertical", well Molodensky's work played into this realm as well. Molodensky put forward the possibility of using gravimetric survey data for interpolation of plumb-line deflection between astronomic points of astronomic-geodetic networks. The result of this work permitted the integration of isolated sections of astronomic-geodetic networks into the main systems of co-ordinates. This made it therefore possible to map vast areas which had never been surveyed before.
Nowadays a similar process is in progress in Africa and South America and Canada with respects to gravity models which will help in determining Orthometric heights. Through Molodensky, lead to a determination of geoid heights.
In 1957, TsNIIGAik began to change direction in what it saw as important, and focused on solving more complex problems; such as the figure of the Earth, space exploration, defence problems, and the development of triangulation methods for large territories.
Famous Equations
Molodensky, though is more famous for a set of equations, that relate to datum transformation.
As we all know spatial data can have co-ordinates with different underlying ellipsoids or the underlying ellipsoids have different datums. The latter means that, apart from different ellipsoids, the centres or the rotation axes of the ellipsoids do not coincide. To relate these data one may need a so-called datum transformation.
In the early days of satellite surveying, when relationships between datums were not well defined and the data itself was not very precise, it was usual to apply a three parameter dX, dY, dZ shift to the X,Y,Z coordinate set in one datum to derive those in the second datum.
This assumed, generally erroneously, that the axial directions of the two ellipsoids involved were parallel. For localized work in a particular country or territory, the consequent errors introduced by this assumption were small and generally less than the observation accuracy of the data. As we collected more and more information about the shape and form of the Earth, and based on what Molodensky presented, among others, our knowledge and the amount of data that has been built up and as our surveying methods became more and more accurate, it became evident that a three parameter transformation is neither appropriate for world wide use, nor for widespread national use if one is seeking the maximum possible accuracy from the satellite surveying and a single set of transformation parameters.

The simplest transformation to implement involves applying shifts to the three geocentric coordinates. Molodensky developed a transformation which applies the geocentric shifts directly to geographical coordinates. This method assumes that the axes of the source and target systems are parallel to each other.
From a mathematical point of view a datum transformation is possible via 3 dimensional geocentric co-ordinates, thus implying a 3D similarity transformation defined by 7 parameters: 3 shifts, 3 rotations and a scale difference. This transformation is combined with transformations between the geocentric co-ordinates and ellipsoidal latitude and longitude co-ordinates in both datum systems.

The transformation from the latitude and longitude co-ordinates into the geocentric co-ordinates is rather straightforward and turns ellipsoidal latitude , longitude and height into X,Y and Z, using 3 direct equations that contain the ellipsoidal parameters a and e.

The inverse equations are more complicated and require either an iterative calculation of the latitude and ellipsoidal height.
A very good approximation of this datum transformations makes use of the Molodensky and the regression equations, relating directly the ellipsoidal latitude and longitude, and in case of Molodensky also the height, of both datum systems.
Various software uses the formulation put forward by Molodensky, whether used in cs2cs and ESRI software, or even in Oracle, the foundations where laid out by this man who has made significant contribution to our understanding of the shape and form of the Earth.