Answers obtained in solving the various equations presented in this chapter will inevitably contain errors. It is important to have an awareness of the presence of these errors and to be able to assess their approximate magnitudes. Errors in computed answers are caused partly by random errors in measured quantities that are used in computation, and partly by the failure of certain assumptions to be met. Some of the more significant sources of errors in calculated values using the equations of this chapter are
1. Errors in photographic measurements, e.g., line lengths or photo coordinates 2. Errors in ground control
3. Shrinkage and expansion of film and paper
4. Tilted photographs where vertical photographs were assumed
Sources 1 and 2 can be minimized if precise, properly calibrated equipment and suitable caution are used in making the measurements. Source 3 can be practically eliminated by making corrections as described in Sec. 4-9. Magnitudes of error introduced by source 4 depend upon the severity of the tilt.
Generally if the photos were intended to be vertical and if paper prints are being used, these errors are compatible with the other sources. If the photo is severely tilted, or if the highest accuracy is desired, analytical methods of Chap. 11 should be used. For the methods described in this chapter, errors caused by lens distortions and atmospheric refraction are relatively small and can generally be ignored.
A simple and straightforward approach to calculating the combined effect of several random errors is to use statistical error propagation, as discussed in Sec. A-4. This approach involves calculating rates of change with respect to each variable containing error and requires the use of differential calculus. As an example of this approach, assume that a vertical photograph was taken with a camera having a focal length of 152.4 mm. Assume also that a ground distance AB on flat terrain has a length of 1524 m and that its corresponding photo distance ab measures 127.0 mm.
Flying height above ground may be calculated, using Eq. (6-1), as follows:
Now it is required to calculate the expected error dH′ caused by errors in measured quantities AB and ab. This is done by taking partial derivatives with respect to each of these quantities containing error. Suppose that the error σAB in the ground distance is ±0.50 m and that the error σab in the measured photo distance is ±0.20 mm. The rate of change of error in H′ caused by the error in the ground length can be evaluated by taking the partial derivative ∂H′/∂AB as
In a similar manner, the rate of change of error in H′ caused by the error in the measured image length can be evaluated by taking the partial derivative ∂H′/∂ab as
A useful interpretation of these derivative terms is that an error of 1 m in ground distance AB will cause an error of approximately 1.2 m in the flying height, whereas an error of 1 mm in image distance ab will cause an error of approximately 14 m in the flying height. Substitution of these derivative terms into the error propagation Eq. (A-2) along with the error terms σAB and σab gives
Note that the error in H′ caused by the error in the measurement of photo distance ab is the more severe of the two contributing sources. Therefore, to increase the accuracy of the computed value of H′, it would be more beneficial to refine the measured photo distance to a more accurate value. Errors in computed answers using any of the equations presented in this chapter can be analyzed in the manner described above, and the method is valid as long as the contributing errors are small.
References
American Society of Photogrammetry: Manual of Photogrammetry, 3d ed., Bethesda, MD, 1966, chap.
2.
———: Manual of Photogrammetry, 4th ed., Bethesda, MD, 1980, chap. 2.
Ghilani C.: Adjustment Computations: Spatial Data Analysis, Wiley and Sons, Hoboken, NJ, 2010.
Problems
Express answers for scale as dimensionless ratios and answers for distance in meters, unless otherwise specified.
6-1. The photo distance between two image points a and b on a vertical photograph is ab, and the corresponding ground distance is AB. What is the photographic scale at the elevation of the ground
(c) ab = 195.7 mm; AB = 1424 m (d) ab = 130.2 mm; AB = 1.00 mi
6-3. On a vertical photograph a section line measures 80.3 mm. What is the photographic scale at the elevation of the section line?
6-4. On a vertical photograph a college football field measures 39.0 mm from goal line to goal line (100.0 yd). What is the scale of the photograph at the elevation of the football field?
6-5. A semitractor and trailer combination which is known to be 18 m long measures 11.3 mm long on a vertical aerial photo. What is the scale of the photo at the elevation of the semi?
6-6. Repeat Prob. 6-5, except that a railroad boxcar of 25.0-m known length measures 10.4 mm on the photo.
6-7. An interstate highway pavement of known 24.0-ft width measures 3.57 mm wide on a vertical aerial photo. What is the flying height above the pavement for this photo if the camera focal length was 152.4 mm?
6-8. In the photo of Prob. 6-7, a rectangular building near the highway has photo dimensions of 6.1 mm and 9.2 mm. What is the actual size of the structure?
6-9. Repeat Prob. 6-8, except that in the photo of Prob. 6-7 a bridge appears. If its photo length is 28.9 mm, what is the actual length of the bridge?
6-10. A vertical photograph was taken, with a camera having a 152.9-mm focal length, from a flying height 2160 m above sea level. What is the scale of the photo at an elevation of 385 m above sea level? What is the datum scale?
6-11. Aerial photographs are to be taken for highway planning and design. If a 152-mm-focal-length camera is to be used and if an average scale of 1:6000 is required, what should be the flying height above average terrain?
6-14. Vertical photography for military reconnaissance is required. If the lowest safe flying altitude over enemy defenses is 5000 m, what camera focal length is necessary to achieve a photo scale of 1:24,000?
6-15. A distance ab on a vertical photograph is 65.0 mm, and the corresponding ground distance AB is 1153 m. If the camera focal length is 153.19 mm, what is the flying height above the terrain upon which line AB is located?
6-16. Vertical photography at an average scale of is to be acquired for the purpose of constructing a mosaic. What is the required flying height above average terrain if the camera focal length is 152.9 mm?
6-17. The distance on a map between two road intersections in flat terrain measures 43.6 mm. The distance between the same two points is 83.0 mm on a vertical photograph. If the scale of the map is 1:50,000, what is the scale of the photograph?
6-18. For Prob. 6-17, the intersections occur at an average elevation of 278 m above sea level. If the camera had a focal length of 209.6 mm, what is the flying height above sea level for this photo?
6-19. A section line scales 88.6 mm on a vertical aerial photograph. What is the scale of the photograph?
6-20. For Prob. 6-19, the average elevation of the section line is at 395 m above sea level, and the camera focal length is 152.4 mm. What would be the actual length of a ground line that lies at elevation 303 m above sea level and measures 53.1 mm on this photo?
6-21. A vertical aerial photo is exposed at 3910 ft above mean sea level using a camera having an 88.9-mm focal length. A triangular parcel of land that lies at elevation 850 ft above sea level appears on the photo, and its sides measure 39.5 mm, 28.9 mm, and 27.7 mm, respectively. What is the approximate area of this parcel in acres?
6-22. On a vertical aerial photograph, a line which was measured on the ground to be 536 m long scales 34.6 mm. What is the scale of the photo at the average elevation of this line?
6-23. Points A and B are at elevations 323 m and 422 m above datum, respectively. The photographic coordinates of their images on a vertical photograph are xa = 68.27 mm, ya = –32.37 mm, xb = –87.44 mm, and yb = 26.81 mm. What is the horizontal length of line AB if the photo was taken from 1535 m above datum with a 152.35-mm-focal-length camera?
6-24. Images a, b, and c of ground points A, B, and C appear on a vertical photograph taken from a flying height of 2625 m above datum. A 153.16-mm-focal-length camera was used. Points A, B, and C have elevations of 407 m, 311 m, and 379 m above datum, respectively. Measured photo coordinates of the images are xa = –60.2 mm, ya = 47.3 mm, xb = 52.4 mm, yb = 80.8 mm, xc = 94.1 mm, and yc = – 79.7 mm. Calculate the horizontal lengths of lines AB, BC, and AC and the area within triangle ABC in hectares.
6-25. The image of a point whose elevation is 832 ft above datum appears 58.51 mm from the principal point of a vertical aerial photograph taken from a flying height of ft above datum. What would this distance from the principal point be if the point were at datum?
6-26. The images of the top and bottom of a utility pole are 113.6 mm and 108.7 mm, respectively, from the principal point of a vertical photograph. What is the height of the pole if the flying height above the base of the pole is 834 m?
6-27. An area has an average terrain elevation of 335 m above datum. The highest points in the area are 412 m above datum. If the camera focal plane opening is 23 cm square, what flying height above
datum is required to limit relief displacement with respect to average terrain elevation to 5.0 mm?
(Hint: Assume the image of a point at highest elevation occurs in the corner of the camera format.) If the camera focal length is 152.7 mm, what is the resulting average scale of the photography?
6-28. The datum scale of a vertical photograph taken from 915 m above datum is . The diameter of a cylindrical oil storage tank measures 6.69 mm at the base and 6.83 mm at the top. What is the height of the tank if its base lies at 213 m above datum?
6-29. Assume that the smallest discernible and measurable relief displacement that is possible on a vertical photo is 0.5 mm. Would it be possible to determine the height of a telephone utility box imaged in the corner of a 23-cm-square photo taken from 870 m above ground? (Note: Telephone utility boxes actually stand 1.2 m above the ground.)
6-30. If your answer to Prob. 6-29 is yes, what is the maximum flying height at which it would be possible to discern the relief displacement of the utility box? If your answer is no, at what flying height would the relief displacement of the box be discernible?
6-31. On a vertical photograph, images a and b of ground points A and B have photographic coordinates xa = –12.68 mm, ya = 70.24 mm, xb = 89.07 mm, and yb = –92.41 mm. The horizontal distance between A and B is 1317 m, and the elevations of A and B are 382 m and 431 m above datum, respectively. Calculate the flying height above datum if the camera had a 152.5-mm focal length.
6-32. Repeat Prob. 6-31, except that the horizontal distance AB is 1834 m and the camera focal length is 88.92 mm.
6-33. In Prob. 6-13, assume that the values given for focal length, photo distance, and flying height contain random errors of ±0.10 mm, ±0.05 mm, and ±0.30 km, respectively. What is the expected error in the computed diameter of the crater?
6-34. In Prob. 6-15, assume that the values given for focal length, photo distance, and ground length contain random errors of ±0.005 mm, ±0.50 mm, and ±0.30 m, respectively. What is the expected error in the computed flying height?
6-35. In Prob. 6-26, assume that the random error in each measured photo distance is ±0.10 mm and that the error in the flying height is ±2.0 m. What is the expected error in the computed height of the utility pole?