COMPUTER EVALUATION OF DOUBLE-THEODOLITE DATA by W. Gale Biggs Technical Report No. 4 ORA Project 03632 NATIONAL SCIENCE FOUNDATION GRANT G-11404 College of Engineering Department of Engineering Mechanics Meteorological Laboratories and Institute of Science and Technology Great Lakes Research Division Special Report No. 13 THE UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN April 1962

KAU 96V63t

ABSTRACT Conventional methods of evaluating double-theodolite data involve numerical and/or graphical techniques that are laborious and time consuming. The following presentation offers a method whereby the electronic digital computer first computes the position of the balloon in space and then from this determines the wind vector. The program also makes provision for the problem of missing data. This computing technique is highly valuable when a large number of pilot balloon ascents must be processed.

While conducting field operations at the Enrico Fermi Nuclear Reactor Site during the summer of 1961, it was necessary to take double-theodolite balloon ascents as a means of evaluating the wind speed in the lower few thousand feet of the atmosphere. The basic approaches to evaluating double-theodolite data have been: 1) numerically to calculate the position of the balloon in space and time, and then to plot it on a winds-aloft plotting board to determine wind speed and direction; or 2) to use graphical methods throughout to determine wind speed and direction. Either of these methods requires the use of winds-aloft plotting boards and both methods are laborious and time consuming. Weedfall and Jagodzinski [1961] describe a graphical method in which they can evaluate a 30-interval run in about 25 min. The United States Weather Bureau Circular "0" describes a method which takes 70 min for a 30-interval run. When a large number of observations are to be evaluated, the high speed electronic digital computer offers a much more convenient and rapid technique of analysis. The following method of analysis requires no plotting of data. The only work is in the conversion of the data from

tabular form to punched cards or any other convenient form of computer input. The data deck consists of one card containing the length of the baseline and the number of observations, followed by four sets of cards. The first of these contains the azimuth angles from station A (the balloon release point), the second contains the elevation angles from station A, the third contains the azimuth angles from station B (the satellite station), and the fourth contains the elevation angles from station B. These angles are read into a three dimensional matrix A.., where in this case i is the number of layers, j is the number of rows, and k is the number of columns. The output of the computer is in the desired form of the horizontal wind vector and the altitude of the balloon for each sounding. Also printed out are the horizontal radial distance of the balloon from the launch site, the u and v components of the wind vector, and the x and y coordinates of the balloon. The mathematical basis for the method presented is simple and is easily handled by the digital computer. The use of two theodolites necessitates a baseline which is carefully surveyed. The baseline used was 2000 ft long and lay in an east-west direction. Since the terrain

is relatively flat, the two stations were at equal elevations, except for the two kilometer inland site which had seven feet difference in elevation. The other sites had less than a foot difference in elevation. Communications were maintained through the use of two transistorized walkie-talkies. Five men were used, three at the balloon release station and two at the satellite station (figure 1). At the balloon release station one man tracked the balloon and read the two angles, one man recorded, and one man kept time. At the satellite station the radio was turned up so both could hear. One man tracked the balloon and read the angles while the second man recorded the data. This arrangement allowed for reading the theodolites at ten second intervals. Three sites were surveyed with 2000 ft baselines, one at the plant site, one at two kilometers inland, and one at four kilometers inland. Provisions were also made to survey an inland site at six kilometers which will be done during the spring of 1962. The two theodolites were adjusted so that both were reading north at 3600. As long as the balloon is not too near the vertical plane containing the baseline, the horizontal triangle ABC (figure 2) is solved first, to obtain the projection of the balloon and its horizontal distance

0 a') 0 0'3 (0 00 %~~~-~~ 0~Q Hd (:4n 4~~~~~~~~~~~~ca

Figure 2. Perspective of two theodolite triangulation. from the two stations. AB is the baseline of length b; P is the position of the balloon, and C is its projection on the horizontal plane through A. The angles measured from A are a and e, those measured from B, a' and E' ~ After the projection of the balloon (C) has been obtained, its height above the level of A is calculated by the relation Z = AC tan e, where e is the elevation angle measured at A. The elevation above B may be calculated as a check; it is given by Z' = BC tan'. It is seen from figure 2 angle ACB = a - a'

and by the sine law AC b sin a' sin (a - a') therefore b sin a' AC = sin (a - a') Similarly b sin a BC = sin (a - a') also b sin a' tan e sin (a - a') and z b sin a tan E' sin (a - a') Since in all but one case, A and B are at the same elevation, Z and Z' should be equal. At the two kilometer site where A and B are not in the same horizontal plane, all the above formulae hold except for Z' where it will differ by h, where h is the difference in elevation of A and B ~ The analysis so far has described the position of the balloon in space in a cylindrical coordinate system (r, e, z), where r = AC, e = a, and z = z. For the

following approach this needs to be converted to a Cartesian coordinate system (x, y, z). This is done as follows: x = r cos e y = r sin e z = z The Cartesian coordinate system is set up by requiring that the x-axis coincide with the baseline used for the two theodolites and that the origin be at the theodolite where the balloon is released. The baseline was located on an east-west line thus making the y-axis on a north-south line (figure 3). y (XiI, Yi+l) N (xi,yi) W S B A OS i<n Figure 3. Diagram showing coordinate system used in equations.

The position of the balloon in the x, y plane is designated as xo, yo for the release position; xl th Yi as the first position, etc., to x, y as the n n n position; then in order to find the distance the balloon traveled in going from xi., Yi to xi+. yi the following formulae are used r =i+- Xi) 2 + (y i+ yi)2 - = tan' [i+l i Xi+l - Xi O < i < n Thus, knowing r, the distance traveled, and t, the time between successive readings (for this program it was every 10 sec), the average wind speed U is simply r/t. The height of the balloon was previously calculated by the computer in order to describe its position in space. The U. S. Weather Bureau in computing its pibal soundings uses every other point in obtaining a wind speed and direction; this is very simply done by using xi+2, Yi+ instead of Xi+l' Yi+l

When the computer comes to a missing point, an average is taken by using the next point. This is also true if two consecutive points are missing. However, if three or more consecutive points are missing, then an average is not taken and the computer continues to test for missing data until points are again found and the averaging process starts again. Thus a segment of the run is considered missing only if three or more consecutive readings are missing. A problem also arises when the balloon crosses the baseline or travels along it. A small error in a or a' will cause a large error in the computed position of the balloon. In this case it is better to solve the vertical triangles ABP and ACP (figure 4). It follows that A BC Figure 4. Solution of vertical triangles.

sin E cos 6 AC = b sin (e +') cos' sin E BC = b sin (e ~ 6E) sinE' Sin' Z = b sin (E + e') The positive sign will be used when C lies in AB, the negative sign when it lies in AB produced. Once the position of the balloon in space is computed, the coordinates are converted to Cartesian and the wind speed and direction are computed as before. Since the angles as recorded by the theodolites are used only in sine and cosine functions, the fact that they go through 360~ makes no differenceo The sine and cosine functions are symmetric and periodic, thus the sine 362~0 sine 2~, and the same is true for the cosine. Since the arc tangent is multivalued, there is a question of which value to choose. This problem can be easily y solved by looking at the combination of signs on the x and y values used to compute the arc tangent. If both are positive ive then the angle lies in the first quadrant; if y is positive and x is negative then the angle lies in the second quadrant; etc. This was even less of a problem for this program since 10

the computer has a calling subroutine that will compute the arc tangent for values between 0 - 2 v. Just before the results are printed out by the computer, the complement of each angle is taken and then rotated 2700. The complement is taken since the program computes e in a counterclockwise direction, whereas it is standard practice to use e in a clockwise direction to express wind direction. The angle is then rotated 2700 because in the program 360~ is east, and the angle e defines the direction in which the balloon is traveling. Thus e is rotated 90~ to correspond to the concept of north being 3600 and then 180~ more to correspond to the concept of wind direction being given as the direction from whence the wind is blowing. Table 1 shows a comparison between the computer evaluation and the hand calculated values. As a means of showing the computer evaluation with missing data, two values were read in as zeros and the results compared. The first pair of columns presents the radial horizontal distance from the release point to the balloon (in feet). The second pair gives the height of the balloon above the theodolite (in feet) and the third gives the local wind speed (in feet per second). The fourth pair of columns shows the local wind direction (in 11

TABLE 1 Comparison of values obtained by the computer with those obtained by conventional methods. R Z U e u v (ft) (ft) (ft/sec) (deg) (ft/sec) Comp HC Comp HC Comp HC: Comp HC Comp Comp 0 0 0 0 7 7 69 69 6.6 2.5 70 70 42.3 42.3 5 5 94 92 4.8 -0.3 116 116 64.3 64.3 6 6 75 75 6.0 1.6 178 178 96.3 96.3 0 - 0 - 18 18 75 76 17.5 4.5 0 - 0 - 722 722 329.0 329.0 15 14 73 67 14.7 4.4 875 875 382.3 382.3 11 10 64 62 10.1 4.9 1118 1118 422.3 422.3 13 14 68 67 12.3 4.9 1306 1306 453.8 453.8 19 20 67 67 17.4 7.5 1438 1438 509.3 509.3 13 12 65 65 12.0 5.6 1588 1588 537.6 537.6 R - Radial horizontal distance of balloon from launch site (ft) Z - Height of balloon (ft) U - Wind speed (ft/sec) e - Azimuth of wind (deg) u - x-comp of wind (as obtained by computer) (ft/sec) v - y-comp of wind (as obtained by computer) (ft/sec) Comp - Computer values HC - Hand calculated values 12

degrees), and the last two columns give the local u and v components of the wind speed as evaluated by the computer. The method of evaluation used to obtain the hand calculated figures was as follows: first the position of the balloon in space was numerically calculated, and then this was plotted on a winds-aloft plotting board and a graphical analysis used to determine the wind speed and direction. The program, since it is a computer analysis, suffers from none of the human errors involved in a graphical analysis and is therefore exact. Thus the accuracy of the final result is determined only by the accuracy with which the original angles were recorded. The program has (as an external function) a system whereby a smoothing subroutine is incorporated in it. Any method of smoothing may be used in the subroutine, with the present smoothing function given by A. = / B. + B. + 1/ B where 0 < i < n The smoothing subroutine also takes care of missing data. In the present program only the wind speed is smoothed. 13

The merit of the program is the great speed with which a large number of data can be analyzed. One program consisted of 10 pibal soundings, each sounding 10 min long with readings taken every 10 sec for a total of 600 points. The computer time involved was 1.5 min for the IBM 709; it required about 45 min of one person's time for punching the data on cards. This is to be compared with time for the method presented by Weedfall and Jagodzinski [1961] which required about 25 min to complete one run of only 30 points and that for the Circular "0" method which requires 70 min for a 30-point run. The program was written in the MAD* language and is reproduced here in that form in the Appendix. The program can be broken into four main groups —the first group consists of reading in the data and converting to radians. The second group computes the location of the balloon in space. The third group tests for missing data, computes wind speed and direction, and smoothes the wind speed. The last group consists of input and output statements necessary for the MAD translator. Michigan Algorithm Decoder 14

Since the MAD language may not be familiar to everyone, each MAD statement is given with its SAP* translation. Thus the program may be used in most of the standard computers available today. Share Assembly Program 15

REFERENCES 1. Middleton, W. E. K., and A. F. Spilhaus, 1953: Meteorological Instruments (3rd ed.). Toronto, University of Toronto Press, 186-187. 2. Weedfall, R. O., and W. M. Jagodzinski, 1961: Comments on Double-Theodolite Evaluations. Bull. Amer. Meteor. Soc., 42, 322-324. 3. Hansen, F. V, and N. H. Taft, 1959: Another Method of Evaluatinq Double-Theodolite Runs. Bull. Amer. Meteor. Soc., 40, 221-224. 4. U. S. Weather Bureau Circular "O." 16

APPENDIX

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