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Chapter 6.
Vertical Photographs

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6-1. Geometry of Vertical Photographs

 Photographs taken from an aircraft with the optical axis of the camera vertical or as nearly
vertical as possible are called vertical photographs

 If the optical axis is exactly vertical, the resulting photograph is termed truly vertical

 In this chapter, equations are developed assuming truly vertical photographs

 In spite of precautions taken to keep the camera axis vertical, small tilts are invariably present

 For photos intended to be vertical, however, tilts are usually less than 1° and rarely exceed 3°

 Photographs containing these small unintentional tilts are called near-vertical or tilted
photographs, and for many practical purposes these photos may be analyzed using the
relatively simple "truly vertical" equations of this chapter without serious error

In this chapter, besides assuming truly vertical photographs, other assumptions are that
the photo coordinate axis system has its origin at the photographic principal point and
that all photo coordinates have been corrected for shrinkage, lens distortion,
and atmospheric refraction distortion

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6-1. Geometry of Vertical Photographs

 Figure 6-1 illustrates the geometry of a vertical

photograph taken from an exposure station 𝐿

 The negative, which is a reversal in both tone and

geometry of the object space, is situated a distance
equal to the focal length (distance 𝑜′𝐿 in Fig. 6-1) above
the rear nodal point of the camera lens

 The positive may be obtained by direct emulsion-to-

emulsion "contact printing" with the negative

Figure 6-1. The geometry of a vertical

photograph.

 This process produces a reversal of tone and geometry from the negative, and therefore the
tone and geometry of the positive are exactly the same as those of the object space

 Geometrically the plane of a contact-printed positive is situated a distance equal to the focal
length (distance 𝑜𝐿 in Fig. 6-1) below the front nodal point of the camera lens

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6-1. Geometry of Vertical Photographs

 The same is true for an image obtained with a frame-

type digital camera

 The reversal in geometry from object space to negative

is readily seen in Fig. 6-1 by comparing the positions of
object points 𝐴, 𝐵, 𝐶, and 𝐷 with their corresponding
negative positions 𝑎′, 𝑏′, 𝑐′, and 𝑑′

 The correspondence of the geometry of the object

space and the positive is also readily apparent

 The photographic coordinate axes 𝑥 and 𝑦 are shown

Figure 6-1. The geometry of a vertical on the positive of Fig. 6-1
photograph.

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6-2. Scale
 Map scale is ordinarily interpreted as the ratio of a map distance to the corresponding distance
on the ground
 In a similar manner, the scale of a photograph is the ratio of a distance on the photo to the
corresponding distance on the ground
 Due to the nature of map projections, map scale is not influenced by terrain variations
 A vertical aerial photograph, however, is a perspective projection, and its scale varies with
variations in terrain elevation
 Scales may be represented as unit equivalents, unit fractions, imensionless representative
fractions, or dimensionless ratios
 For example, if 1 inch (in) on a map or photo represents 1000 ft (12,000 in) on the ground, the
scale expressed in the aforementioned four ways is
* By convention, the first term in a scale expression is always chosen as 1
1. Unit equivalents: 1 in = 1000 ft
2. Unit fraction: 1 in/1000 ft
3. Dimensionless representative fraction: 1/12,000
4. Dimensionless ratio: 1:12,000
 A large number in a scale expression denotes a small scale, and vice versa;
⇨ for example, 1:1000 is a larger scale than 1:5000.

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6-3. Scale of a Vertical Photograph Over Flat Terrain

 Figure 6-2 shows the side view of a vertical photograph

taken over flat terrain
 The scale of a vertical photograph over flat terrain is
simply the ratio of photo distance 𝑎𝑏 to corresponding
ground distance 𝐴𝐵

 That scale may be expressed in terms of camera focal

length ƒ and flying height above ground 𝐻′ by equating
Figure 6-2. Two-dimensional view of a vertical
photograph taken over flat terrain. similar triangles 𝐿𝑎𝑏 and 𝐿𝐴𝐵 as follows:
𝑎𝑏 𝑓
𝑆= = ′ (6-1)
𝐴𝐵 𝐻

 From Eq. (6-1) it is seen that the scale of a vertical photo is directly proportional to camera
focal length (image distance) and inversely proportional to flying height above ground (object
distance)

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6-4. Scale of a Vertical Photograph Over Variable Terrain

 For any given vertical photo scale increases with increasing terrain elevation and decreases
with decreasing terrain elevation
 Suppose a vertical aerial photograph is taken over variable
terrain from exposure station 𝐿 of Fig. 6-3

 Ground points 𝐴 and 𝐵 are imaged on the positive at 𝑎 and 𝑏,

respectively

 Photographic scale at ℎ, the elevation of points 𝐴 and 𝐵, is

equal to the ratio of photo distance 𝑎𝑏 to ground distance 𝐴𝐵

 By similar triangles 𝐿𝑎𝑏 and 𝐿𝐴𝐵, an expression for photo

Figure 6-3. Scale of a vertical photograph
scale 𝑆𝐴𝐵 is 𝑎𝑏 𝐿𝑎
over variable terrain. 𝑆𝐴𝐵 = = (a)
𝐴𝐵 𝐿𝐴

 Also, by similar triangles 𝐿𝑂𝐴 𝐴 and 𝐿𝑜𝑎, 𝐿𝑎 𝑓

= (b)
𝐿𝐴 𝐻 − ℎ

Substituting Eq. (b) into Eq. (a) gives

𝑎𝑏 𝐿𝑎 𝑓
𝑆𝐴𝐵 = = = (c)
𝐴𝐵 𝐿𝐴 𝐻 − ℎ

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6-4. Scale of a Vertical Photograph Over Variable Terrain

 Considering line 𝐴𝐵 to be infinitesimal, we see that Eq. (c) reduces to an expression of photo
scale at a point

 In general, by dropping subscripts, the scale at any point whose elevation above datum is ℎ
may be expressed as
𝑓
𝑆= (6-2)
𝐻−ℎ

 In Eq. (6-2), the denominator 𝐻– ℎ is the object distance

 In this equation, as in Eq. (6-1), scale of a vertical photograph is seen to be simply the ratio of
image distance to object distance
𝑎𝑏 𝑓
𝑆= = ′ (6-1)
𝐴𝐵 𝐻

 The shorter the object distance (the closer the terrain to the camera), the greater the photo
scale, and vice versa

 For vertical photographs taken over variable terrain, there are an infinite number of different
scales
⇨ The principal differences between a photograph and a map

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6-5. Average Photo Scale

 It is often convenient and desirable to use an average scale to define the overall mean scale of
a vertical photograph taken over variable terrain

 Average scale is the scale at the average elevation of the terrain covered by a particular
photograph and is expressed as
𝑓
𝑆𝑎𝑣𝑔 = (6-3)
𝐻 − ℎ𝑎𝑣𝑔

 When an average scale is used, it must be understood that it is exact only at those points that
lie at average elevation, and it is an approximate scale for all other areas of the photograph

 In each of Eqs. (6-1), (6-2), and (6-3), it is noted that Flying height appears in the denominator ⇨
Thus, for a camera of a given focal length, if flying height increases, object distance
𝐻– ℎ increases and scale decreases

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6-5. Average Photo Scale

 Figures 6-4a through 𝑑 illustrate this principle
vividly

 Each of these vertical photos was exposed using

the very same 23-cm format and 152-mm-focal-
length camera

 The photo of Fig.6-4a had a flying height of 460m

above ground, resulting in an average photo scale
of 1:3000

 The photos of Fig. 6-4b, c, and d had flying

heights above average ground of 910 m, 1830 m,
and 3660 m, respectively, producing average
photo scales of 1:6000, 1:12,000, and 1:24,000,
Figure 6-4. Four vertical photos taken over Tampa, respectively.
Florida, illustrating scale variations due to changing
flying heights. (Courtesy Aerial Cartographics of
America, Inc.)

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6-6. Other Methods of Determining Scale of Vertical Photographs

 A ground distance may be measured in the field between two points whose images appear
on the photograph
 After the corresponding photo distance is measured, the scale relationship is simply the ratio
of the photo distance to the ground distance
 The resulting scale is exact only at the elevation of the ground line, and if the line is along
sloping ground, the resulting scale applies at approximately the average elevation of the two
endpoints of the line
 The scale of a vertical aerial photograph may also be determined if a map covering the same
area as the photo is available
 In this method it is necessary to measure, on the photograph and on the map, the distances
between two welldefined points that can be identified on both photo and map

 Photographic scale can then be calculated from the following equation:

𝑝ℎ𝑜𝑡𝑜 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑆= × 𝑚𝑎𝑝 𝑠𝑐𝑎𝑙𝑒 (6-4)
𝑚𝑎𝑝 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒

 The scale of a vertical aerial photograph can also be determined without the aid of a
measured ground distance or a map if lines whose lengths are known by common knowledge
appear on the photo

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6-7. Ground Coordinates from a Vertical Photograph

 The ground coordinates of points whose images appear in a vertical photograph can be
determined with respect to an arbitrary ground coordinate system.

 The arbitrary 𝑋 and 𝑌 ground axes are in the same vertical

planes as the photographic 𝑥 and 𝑦 axes, and the origin of
the system is at the datum principal point (point in the
datum plane vertically beneath the exposure station)
 Figure 6-5 shows a vertical photograph taken at a flying
height 𝐻 above datum

 Images 𝑎 and 𝑏 of the ground points 𝐴 and 𝐵 appear

on the photograph, and their measured photographic
Figure 6-5. Ground coordinates from a vertical coordinates are 𝑥𝑎 , 𝑦𝑎 , 𝑥𝑏 , 𝑦𝑏
photograph.

 The arbitrary ground coordinate axis system is 𝑋 and 𝑌, and the coordinates of points 𝐴 and 𝐵
in that system are 𝑋𝐴 , 𝑌𝐴 , 𝑋𝐵 , 𝑌𝐵

 From similar triangles 𝐿𝑎′𝑜 and 𝐿𝐴′𝐴𝑜 , the following equation may be written:
𝑜𝑎′′ 𝑓 𝑦𝑎 𝐻 − ℎ𝐴
= = from which 𝑋𝐴 = 𝑥𝑎 (6-5)
𝐴𝑜 𝐴′′ 𝐻 − ℎ𝐴 𝑋𝐴 𝑓

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6-7. Ground Coordinates from a Vertical Photograph

 Also, from similar triangles 𝐿𝑎′′𝑜 and 𝐿𝐴′′𝐴𝑜 ,
𝑜𝑎′′ 𝑓 𝑦𝑎 𝐻 − ℎ𝐴
= = from which 𝑌𝐴 = 𝑦𝑎
𝑓
(6-6)
𝐴𝑜 𝐴′′ 𝐻 − ℎ𝐴 𝑌𝐴

 Similarly, the ground coordinates of point 𝐵 are

𝐻 − ℎ𝐵
𝑋𝐵 = 𝑥𝑏 (6-7)
𝑓

𝐻 − ℎ𝐵
𝑌𝐵 = 𝑦𝑏 (6-8)
𝑓

 Upon examination of Eqs. (6-5) through (6-8), it is seen that 𝑋 and 𝑌 ground coordinates of any
point are obtained by simply multiplying 𝑥 and 𝑦 photo coordinates by the inverse of photo
scale at that point

 From the ground coordinates of the two points 𝐴 and 𝐵, the horizontal length of line 𝐴𝐵 can be
calculated, using the pythagorean theorem, as
𝐴𝐵 = (𝑋𝐵 − 𝑋𝐴 )2 +(𝑌𝐵 − 𝑌𝐴 )2 (6-9)

 Also, horizontal angle 𝐴𝑃𝐵 may be calculated as

𝑋𝐵 𝑌𝐴
𝐴𝑃𝐵 = 90° + tan−1 + tan−1 (6-10)
𝑌𝐵 𝑋𝐴

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6-7. Ground Coordinates from a Vertical Photograph

𝐻 − ℎ𝐴 𝐻 − ℎ𝐴 𝐻 − ℎ𝐵 𝐻 − ℎ𝐵
𝑋𝐴 = 𝑥𝑎 (6-5) 𝑌𝐴 = 𝑦𝑎 (6-6) 𝑋𝐵 = 𝑥𝑏 (6-7) 𝑌𝐵 = 𝑦𝑏 (6-8)
𝑓 𝑓 𝑓 𝑓

 To solve Eqs. (6-5) through (6-8) it is necessary to know the camera focal length, flying height
above datum, elevations of the points above datum, and photo coordinates of the points
 The photo coordinates are readily measured, camera focal length is commonly known from
camera calibration, and flying height above datum is calculated by methods
 Elevations of points may be obtained directly by field measurements, or they may be taken from
available topographic maps

 Ground coordinates calculated by Eqs. (6-5) through (6-8) are in an abritrary rectangular
coordinate system, as previously described

 If arbitrary coordinates are calculated for two or more "control" points (points whose
coordinates are also known in an absolute ground coordinate system such as the state plane
coordinate system), then the arbitrary coordinates of all other points for that photograph can
be transformed to the ground system

 Using Eqs. (6-5) through (6-8), an entire planimetric survey of the area covered by a vertical
photograph can be made

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6-8. Relief Displacement on a Vertical Photograph

 Relief displacement is the shift or displacement in the photographic position of an image caused
by the relief of the object, i.e., its elevation above or below a selected datum

 With respect to a datum, relief displacement is outward for points whose elevations are above
datum and inward for points whose elevations are below datum
 The concept of relief displacement is illustrated in
Fig. 6-6, which represents a vertical photograph
taken from flying height 𝐻 above datum
 Camera focal length is 𝑓 , and 𝑜 is the principal point
 The image of terrain point 𝐴, which has an elevation
ℎ𝐴 above datum, is located at a on the photograph
 An imaginary point 𝐴′ is located vertically beneath 𝐴
in the datum plane, and its corresponding imaginary
Figure 6-6. Relief displacement on a vertical
image position is at 𝑎′
photograph.
 On the figure, both 𝐴′𝐴 and 𝑃𝐿 are vertical lines, and therefore 𝐴′𝐴𝑎𝐿𝑜𝑃 is a vertical plane, and
Plane 𝐴′𝑎′𝐿𝑜𝑃 is also a vertical plane which is coincident with 𝐴′𝐴𝑎𝐿𝑜𝑃
 Since these planes intersect the photo plane along lines 𝑜𝑎 and 𝑜𝑎′, respectively, line 𝑎𝑎′ (relief
displacement of point 𝐴 due to its elevation ℎ𝐴 ) is radial from the principal point
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6-8. Relief Displacement on a Vertical Photograph

 An equation for evaluating relief displacement may be
obtained by relating similar triangles
 First consider planes 𝐿𝑎𝑜 and 𝐿𝐴𝐴𝑜 in Fig. 6-6:
𝑟 𝑓
=
𝑅 𝐻 − ℎ𝐴
(d) Rearranging gives 𝑟 𝐻 − ℎ𝐴 = 𝑓𝑅

 Also, from similar triangles 𝐿𝑎′𝑜 and 𝐿𝐴′𝑃 ,

𝑟′ 𝑓
= (e) or 𝑟 ′ 𝐻 = 𝑓𝑅
𝑅 𝐻
 Equating expressions (d) and (e) yields
Figure 6-6. Relief displacement on a vertical
𝑟 𝐻 − ℎ𝐴 = 𝑟 ′ 𝐻 photograph.

 Rearranging the above equation, dropping subscripts, and substituting the symbol 𝑑
for 𝑟– 𝑟 ′ gives
𝑟ℎ
𝑑= (6-11)
𝐻
where 𝑑=relief displacement
ℎ = height above datum of object point whose image is displaced
𝑟 = radial distance on photograph from principal point to displaced image
(The units of 𝑑 and 𝑟 must be the same.)
𝐻 = flying height above same datum selected for measurement of ℎ

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6-8. Relief Displacement on a Vertical Photograph

𝑟ℎ
𝑑= (6-11)
𝐻
where 𝑑=relief displacement
ℎ = height above datum of object point whose image is displaced
𝑟 = radial distance on photograph from principal point to displaced image
(The units of 𝑑 and 𝑟 must be the same.)
𝐻 = flying height above same datum selected for measurement of ℎ

 Equation (6-11) is the basic relief displacement equation for vertical photos

 Examination of this equation shows that relief displacement increases with increasing radial
distance to the image, and it also increases with increased elevation of the object point above
datum

 On the other hand, relief displacement decreases with increased flying height above datum

 It has also been shown that relief displacement occurs radially from the principal point

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6-8. Relief Displacement on a Vertical Photograph

 Figure 6-7 is a vertical aerial photograph which vividly illustrates
relief displacement

 Note in particular the striking effect of relief displacement on the

tall buildings in the upper portion of the photo

 Notice also that the relief displacement occurs radially from the
center of the photograph (principal point)

 This radial pattern is also readily apparent for the relief

Figure 6-7. Vertical photograph of Tampa,
Florida, illustrating relief displacements.
displacement of all the other vertical buildings in the photo
(Courtesy US Imaging, Inc.)

 The building in the center is one of the tallest imaged on the photo (as evidenced by the length
of its shadow); however, its relief displacement is essentially zero due to its proximity to the
principal point.

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6-8. Relief Displacement on a Vertical Photograph

 Figure 6-7 is a vertical aerial photograph which vividly illustrates
relief displacement

 Note in particular the striking effect of relief displacement on the

tall buildings in the upper portion of the photo

 Notice also that the relief displacement occurs radially from the
center of the photograph (principal point)

 This radial pattern is also readily apparent for the relief

Figure 6-7. Vertical photograph of Tampa,
Florida, illustrating relief displacements.
displacement of all the other vertical buildings in the photo
(Courtesy US Imaging, Inc.)

 Relief displacement often causes straight roads, fence lines, etc., on rolling ground to appear
crooked on a vertical photograph

 The severity of the crookedness will depend on the amount of terrain variation

 Relief displacement causes some imagery to be obscured from view

 Several examples of this are seen in Fig. 6-7; e.g., the street in the upper portion of the photo is
obscured by relief displacement of several tall buildings adjacent to it

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6-8. Relief Displacement on a Vertical Photograph

 Vertical heights of objects such as buildings, poles, etc., appearing on aerial photographs can
be calculated from relief displacements

 For this purpose, Eq. (6-11) is rearranged as follows:

𝑟ℎ 𝑑𝐻
𝑑=
𝐻
(6-11) ⇨ ℎ=
𝑟
(6-12)

 To use Eq. (6-12) for height determination, it is necessary that the images of both the top
and bottom of the vertical object be visible on the photograph, so that 𝑑 can be measured

 Datum is arbitrarily selected at the base of the vertical object

 Equation (6-12) is of particular import to the photo interpreter, who is often interested in
relative heights of objects rather than absolute elevations

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6-9. Flying Height of a Vertical Photograph

 It is apparent that flying height above datum is an important quantity which is often needed for
solving basic photogrammetric equations

 For rough computations, flying height may be taken from altimeter readings

 Flying heights also may be obtained by using either Eq. (6-1) or Eq. (6-2) if a ground line of
known length appears on the photograph

 In general, the greater the difference in elevation of the endpoints, the greater the error in the
computed flying height
⇨ Therefore the ground line should lie on fairly level terrain

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6-9. Flying Height of a Vertical Photograph

 Accurate flying heights can be determined even though the endpoints of the ground line lie at
different elevations, regardless of the locations of the endpoints in the photo
 This procedure requires knowledge of the elevations of the endpoints of the line as well as of
the length of the line
 Suppose ground line 𝐴𝐵 has its endpoints imaged at 𝑎 and 𝑏 on a vertical photograph

 Length 𝐴𝐵 of the ground line may be expressed in terms of ground coordinates, by the
pythagorean theorem, as follows:
(𝐴𝐵)2 = (𝑋𝐵 − 𝑋𝐴 )2 +(𝑌𝐵 − 𝑌𝐴 )2

Substituting Eqs. (6-5) through (6-8) into the previous equation gives
𝑥𝑏 𝑥𝑎 𝑦𝑏 𝑦𝑎
(𝐴𝐵)2 = [ 𝐻 − ℎ𝐵 − 𝐻 − ℎ𝐴 ]2 +[ 𝐻 − ℎ𝐵 − 𝐻 − ℎ𝐴 ]2 (6-13)
𝑓 𝑓 𝑓 𝑓
 The only unknown in Eq. (6-13) is the flying height 𝐻

 When all known values are inserted into the equation, it reduces to the quadratic form of
𝑎𝐻 2 + 𝑏𝐻 + 𝑐 = 0

 The direct solution for 𝐻 in the quadratic is

−𝑏 ± 𝑏 2 − 4𝑎𝑐
𝐻= (6-14)
2𝑎

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6-10. Error Evaluation

 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

 Some of the more significant sources of errors in calculated values 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

 Sources1 and 2 can be minimized if precise, properly calibrated equipment and suitable
caution are used in making the measurements

 Source3 can be practically eliminated by making corrections

 Magnitudes of error introduced by Source4 depend upon the severity of the tilt

 For the methods described in this chapter, errors caused by lens distortions and atmospheric
refraction are relatively small and can generally be ignored

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6-10. Error Evaluation

 A simple and straightforward approach to calculating the combined effect of several random
errors is to use statistical error propagation

 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:
𝐴𝐵 1524
𝐻′ = 𝑓 = 152.4 = 1829𝑚
𝑎𝑏 127.0

 It is required to calculate the expected error 𝑑𝐻′ caused by errors in measured quantities 𝐴𝐵
and 𝑎𝑏
 Suppose that the error 𝜎𝐴𝐵 in the ground distance is ±0.50 m and that the error𝜎𝑎𝑏 in the
measured photo distance is ±0.20 mm

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6-10. Error Evaluation

 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
𝜕𝐻′ 𝑓 152.4𝑚𝑚
= = = 1.200
𝜕𝐴𝐵 𝑎𝑏 127.0𝑚𝑚
 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
𝜕𝐻′ −𝑓(𝐴𝐵) −152.4𝑚𝑚(1524𝑚)
= = = −14.40𝑚/𝑚𝑚
𝜕𝑎𝑏 (𝑎𝑏)2 (127.0𝑚𝑚)2
 A useful interpretation of these derivative terms is that an error of 1m in ground distance 𝐴𝐵 will
cause an error of approximately 1.2m in the flying height, whereas an error of 1mm in image
distance 𝑎𝑏 will cause an error of approximately 14m in the flying height
 Substitution of these derivative terms into the error propagation along with the error terms 𝜎𝐴𝐵
and 𝜎𝑎𝑏 gives
𝜎𝐻 ′ = (1.200)2 (0.50𝑚)2 + −14.40𝑚/𝑚𝑚 2 (0.20𝑚𝑚)2 = 0.36𝑚2 + 8.29𝑚2 = ±2.9𝑚

 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
Vertical Photographs