Geodetic Models

Geodetic Data

Geodetic or geospatial data are composed of point, lines and area relations. These data are rich with attributes. As in many other sciences, geodetic problems are located in a real world.

The concept – the problem need to be modeled.

The Solution – have to be put in a geometrical and physical formulation by mathematical means

The purpose is either description or explanation and allow to

i) estimate non-observable state and

ii) predict future state

Typical types of models in geodesy are:-

i) Measurement model (observation model)

ii) Geodetic model (theoretical model)

Both models are connected via the geodetic observable.

The task of measurement – to correct and evaluate the raw observation (true observation) to adapt them to the geodetic model

Introduction to Least Squares

Least Squares – Original and Advancement

In history....original method

1795 : Karl Friederich Gauss & Adrien Marie Legendre (refer : Gauss, K.F. (1963). Theory of Motion of the Heavenly Bodies. Dover, New York

Advancements made to the original method
1905 : Kruger (refer : Kruger, L. (1905). Uber die Ausgleichung von bedingten Beobachtungen in zwei Gruppen, Postdam.

1930 : Tobey (refer : Tobey, W.M. (1930). The Differential Adjustment of Normally Distributed Observations. Geodetic Survey of Canada Pub. 27. Ottawa

1950 : Tienstra (refer : Tienstra, J.M. (1956). Theory of the Adjustment of Normally Distributed Observations. Argus, Amsterdam

- Tienstra explains that a least squares problem can be rigorously solved in “phases” – by treating the already estimated parameter and corrected observations of a previous phase as quasi-observation in the subsequent phase.

Advancement – Kalman Filtering
1960 : Kalman (refer : Kalman, R.E. (1960). A New Approach to Linear Filtering and Prediction. Journal of Basic Engineering, ASME, 82D

- extended to i) adding a second model – vector parameters to vary with time, ii) inserting covariance matrix in second model and iii) solving the problem in part

Advancement – Bayesian Procedure

1972 : Morrison ( refer : Morrison, N. (1972). Introduction to Sequential Smoothing and Prediction. McGraw-Hill Book Company, New York

Advancement – Sequential Procedure

1965 : Schmid & Schmid (refer : Schmid, H.H. & Schmid, E. (1965). A Generalized Least Squares Solution for a Hybrid Measuring System. The Canadian Surveyor, XIX, No. 1, Ottawas

- formulated with weight ranging from zero to infinity and rigorously solved “sequentially”. Sequential procedure achieved by updating the original estimate by a corrective term.

Advancement – Collocation Procedure

1969 & 1970 : Krarup ( refer : Krarup, L. (1969). A Contribution to the Mathematical Foundation of Physical Geodesy. Publication No. 44 of the Danish Geodetic Institute, Copenhagen & Krarup, L. (1970). The Method of Least Square Collocation. Studi Geophysica, No. 2, Roc. 14.

1972 : Moritz (refer : Moritz, H. (1972). Advanced Least Squares Method. Report of the Department id Geodetic Science, No. 75, The Ohio State University, Columbus.

Advancement – Stepwise & Sequential Procedure

1973 : Moritz (refer : Moritz, H. (1973). Stepwise and Sequential Collocation. Report of the Department of Geodetic Science. No. 203, The Ohio State University, Columbus

History....Documentation

1. The best-known writting on surveying was by Frontinus.

The engineering ability of the Romans was demonstrated by their construction work throughout the empire.

The famous instruments is groma. This instrument used for sighting (Wolf & Ghilani, 2002).

note: others information about groma.

a) . http://www.legiiavg.org.uk/military/groma.html

b) . Source from Wikipedia - The Groma or gruma (altered from Greek gnmon γνόμων "indicator", possibly through Etruscan) was the principal Roman surveying instrument. It was composed of a vertical staff with horizontal cross pieces mounted on a bracket. Each cross piece had a plumb line and plumb bob hanging vertically. Its main use was to survey straight lines, squares, and rectangles.

c) . Britannica - use in ancient Egypt
There is some evidence that, in addition to a marked cord, wooden rods were used by the Egyptians for distance measurement.

2. Oldest Latin manuscript - Codex Acerianus, contaion an account of surveying as practised by the Romans and includes several pages from Frontinus work.

This manuscript found in the tenth century by Gerbert.

3. Middles Ages - Arabs in Islam world was making more progress in art of surveying - "practical geometry".

4. In 13th Century - Von Piso wrote Practica Geometria which contained instruction of surveying.

He also authored Liber Quadratorum - dealing with the quadran, a square brass frame having 90degree angle and other graduated scales

5. Instruments - Astrolabe.- a metal circle with apointer hinged at its center and held by a ring at the top, and the cross staff, a wooden rod about 4-ft long with an adjustable cross-arm ar right angles to it (Wolf & Ghilani, 2002). This instruments widely used in Islamic empire.

source of image: http://www.agmgifts.co.uk/resources/astrolabe_5.jpg

References

1. Wolf, P.R. & Ghilani, C.D. (2002) Elementary Surveying An Introduction to Geomatic. Prentice Hall, Upper Saddle River, New Jersey

2. Wikipedia : An astrolabe is a historical astronomical instruments used by classical astronomer, navigator and astrologers. Its many uses include locating and predicting the positions of the Sun, Moon, planets and stars determining local time given local latitude and vice-versa; surveying; and triangulation.
In the medieval Islamic world, they were used primarily for astronomical studies, as well as in other areas as diverse as astrology, navigationm, surveying, timekeeping, Salah prayer and Qibla. Astrologers of the European nations used astrolabes to construct horoscopes.
There is often confusion between the astrolabe and the mariner's astrolabe. While the astrolabe could be useful for determining latitude on land, it was an awkward instrument for use on the heaving deck of a ship or in wind. The mariner's astrolabe was developed to address these issues.

History...The Dioptra

The oldest historical record - begin in Egypt, Hetorodotus recorded

i. Sesostris (about 1400 B.C) - Devided the land of Egypt into plots for the purpose of taxation

ii. Early surveyor were called ropestretcher - measurement were made with ropes

iii. Greek thinker - Heron, the author The Dioptra, which related to method of surveying a field, drawing a plan and making related calculations.


(note : wikipedia source - A dioptre, or diopter, is a unit of measurement of the optical power of a lens or curved mirror, which is equal to the reciprocal of the focal length measured in metres (that is, 1/metres). For example, a 3 dioptre lens brings parallel rays of light to focus at 1/3 metre. )


Taken from T. P. Tassios , Ancient Greek Technology - Measuring Instruments, FIG Working Week in Athens, May 22-27, 2004 (http://www.fig.net/news/news_2004/athens/ancient_exhibit.htm)



The "dioptra" devised by Heron is a portable surveying instrument for making accurate geodetic measurements. It measures azimuths, heights, lengths and angular distances.
The apparatus, which was mounted on a tripod about 50 cm tall, pivoted on a circular plate and was moved around horizontally by a worm screw , driven by a small handle. The top plate was mounted on a toothed semicircular plate and was adjusted in the vertical plane by another worm drive. Angles could be read off the top disc, which was scored with two lines crossing at right angles. The sighting and reading elements were made of bronze.

Definition...

Just for introduction...

Surveying has been important since the begining of civilization. In early applications were in measuring and marking boundaries of property ownerships.

Surveying, recently also been interchangeably called geomatics, has traditionally been defined as the science, art and technology of determining the relative positions of point above, on, or beneath the earth's surface or of establishing such points (Wolf & Ghilani, 2002).

Surveyor traditionally and historically made their measurement using gorund-based method such as transit and tape. This is a surveyor primary instruments. The product such as computation sheet, analyses, and the report and maps they delivered to their client were prepared through manual process.

Todays, modern surveyors start to measuring and collecting using electronic instrumenatation, satellite surveying system and modern aerial imaging. The data process with high speed computer system.