11.1 Aerial Photogrammetry, Photographic Scale, Relief Displacement, Flight Planning, and Stereoscopy

11.1 Aerial Photogrammetry, Photographic Scale, Relief Displacement, Flight Planning, and Stereoscopy

Aerial photogrammetry is the science and technology of obtaining reliable spatial measurements and physical terrain data from aerial photographs. For Geodetic Engineers in the Philippines, mastering aerial photogrammetry is essential for cadastral mapping, topographic surveys, coastal resource mapping, and infrastructure development. The discipline spans geometric optic principles, flight mission design, stereo vision mechanics, and rigid spatial transformations.

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1. Geometry of Aerial Photography

Aerial photographs are perspective projections taken from airborne platforms using precision photogrammetric cameras. Unlike map projections (which are orthogonal), aerial photos inherit perspective distortion where rays of light pass through a single optical center (the lens projection center) onto a focal plane.

Classification by Optical Axis Tilt

1. Vertical Aerial Photography: The camera optical axis is aligned intentionally perpendicular to the Earth's surface. True vertical photographs are rare due to aircraft dynamics; in practice, photos with a tilt angle $\theta \le 3^\circ$ are classified as near-vertical photogrammetric photographs.
2. Oblique Aerial Photography: The optical axis is intentionally tilted away from the vertical plumb line:
   • Low Oblique Photography: Tilted optical axis where the horizon is not visible in the photograph.
   • High Oblique Photography: Tilted optical axis where the Earth's apparent horizon is visible in the photograph.

Critical Geometric Points on a Tilted Aerial Photograph

• Principal Point ($o$): The point where the camera's optical axis intersects the image focal plane. It is located at the intersection of lines joining opposing fiducial marks.
• Nadir Point ($n$): The point on the photograph where a true vertical plumb line passing through the camera lens center intersects the image plane. It represents the point directly beneath the camera.
• Isocenter ($i$): The point on the photo along the principal line halfway between the principal point and the nadir point ($oi = ni = f \cdot \tan(\theta / 2)$). Tilt displacement radiates from the isocenter.

Crab and Drift

Flight execution frequently encounters atmospheric wind currents, producing two distinct navigational aberrations:

• Drift: The lateral displacement of the aircraft off its intended flight path due to crosswinds. Drift results in improper side lap and potential coverage gaps between flight lines.
• Crab: The angular rotation of the aircraft fuselage relative to the flight direction to correct for crosswinds. If uncorrected in camera mounting, the sides of the photographs are not parallel to the flight line, reducing the effective width of stereoscopic coverage.

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2. Photographic Scale Equations and Terrain Effects

Photographic scale ($S$) expresses the ratio of a distance measured on the aerial photograph ($d$) to the corresponding horizontal distance measured on the ground ($D$). Because aerial photographs are perspective projections, scale varies inversely with the flying height above ground level ($H' = H - h$).

$$S = \frac{d}{D} = \frac{f}{H - h}$$

Where:

• $f = \text{calibrated focal length of the camera lens (m or mm)}$
• $H = \text{flying height of the aircraft above mean sea level (MSL) or reference datum (m)}$
• $h = \text{elevation of the terrain or ground point above reference datum (m)}$
• $H - h = \text{flying height above terrain (m)}$

Scale Expressions

1. Point Scale ($S_h$): The exact local photographic scale at a specific point on the ground having elevation $h$: $$S_h = \frac{f}{H - h}$$
2. Average Scale ($S_{avg}$): The nominal scale over an entire photograph or flight block based on average terrain elevation ($h_{avg}$): $$S_{avg} = \frac{f}{H - h_{avg}}$$
3. Representative Fraction (RF): Scale expressed as a unitless fraction (e.g., $1 : 10,000$). A scale denominator $m$ is defined such that $S = 1/m$, where $m = \frac{H - h}{f}$.

[!IMPORTANT] GELE Exam Rule: Higher elevation terrain (larger $h$) is closer to the camera lens, resulting in a larger scale (smaller scale denominator $m$). Lower elevation terrain (smaller $h$) is farther from the camera lens, resulting in a smaller scale (larger scale denominator $m$).

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3. Relief Displacement and Height Determination

Relief displacement ($d$) is the shift or displacement in the position of an image point on an aerial photograph caused by the elevation of the object above or below a chosen reference datum. Relief displacement radiates directly outward from the photograph's principal point (for true vertical photos) or nadir point.

Relief Displacement Formula

$$d = \frac{r \cdot h}{H}$$

Where:

• $d = \text{relief displacement of the point measured on the photograph (mm or m)}$
• $r = \text{radial distance from the principal point to the top (displaced position) of the object (mm or m)}$
• $h = \text{height of the object above reference datum (m)}$
• $H = \text{flying height of the aircraft above the same reference datum (m)}$

Alternatively, if the radial distance is measured to the base of the object ($r'$), the displacement is given by:

$$d = \frac{r' \cdot h}{H - h}$$

Key Characteristics of Relief Displacement

1. Radial Nature: Relief displacement occurs along radial lines passing through the principal point.
2. Direct Height Dependence: Relief displacement is directly proportional to the object height ($h$).
3. Direct Radial Distance Dependence: Relief displacement increases linearly with radial distance ($r$) from the center of the photo. An object located at the exact principal point ($r = 0$) has zero relief displacement ($d = 0$).
4. Inverse Flying Height Dependence: Relief displacement is inversely proportional to flying height ($H$). Higher altitude photography reduces relief displacement.
5. Direction of Shift: Points above the datum shift outward (away from the principal point); points below the datum shift inward (toward the principal point).

Height Determination from Relief Displacement

By measuring the length of the image displacement of a vertical structure (such as a building, telecommunication tower, or tree) from its base to its top, the object's physical height ($h$) can be computed:

$$h = \frac{d \cdot H}{r}$$

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4. Stereoscopy and Binocular Parallax

Stereoscopy is the science of viewing two overlapping photographs (a stereo pair) taken from different exposure stations to perceive a three-dimensional mental image (stereomodel).

Stereoscopic Parallax ($p$)

Parallax is the apparent displacement of the position of an object caused by a shift in the point of observation. In aerial photogrammetry, stereoscopic parallax ($p$) of a point is the algebraic difference of the distances of the point's images from their respective principal points measured parallel to the flight line ($x$-axis).

$$p = x - x'$$

Where:

• $x = \text{photo coordinate of point } P \text{ on the left photograph along the flight axis}$
• $x' = \text{photo coordinate of point } P \text{ on the right photograph along the flight axis (with sign)}$

Parallax Equations for Terrain Elevation

$$p = \frac{B \cdot f}{H - h}$$

Rearranging to solve for terrain elevation ($h$) or flying height above ground ($H - h$):

$$H - h = \frac{B \cdot f}{p} \implies h = H - \frac{B \cdot f}{p}$$

Where:

• $B = \text{air base (horizontal distance between exposure stations in meters)}$
• $f = \text{focal length of camera (mm or m)}$
• $H = \text{flying height above datum (m)}$
• $h = \text{elevation of ground point } P \text{ above datum (m)}$
• $p = \text{absolute stereoscopic parallax of point } P \text{ (mm or m)}$

Height Difference Calculation Using Parallax Difference ($\Delta p$)

When calculating the elevation difference ($\Delta h$) between two points (Point 1 and Point 2) in a stereo pair:

$$\Delta h = \frac{\Delta p \cdot (H - h_1)}{p_2} = \frac{\Delta p \cdot (H - h_1)}{p_1 + \Delta p}$$

Where $\Delta p = p_2 - p_1$ is the parallax difference between the top and base of the feature.

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5. Flight Planning Parameters and Formulas

Flight planning establishes camera parameters, aircraft speed, altitude, line spacing, and exposure intervals required to achieve complete stereo coverage at a specified ground sample size and precision.

Standard Overlap Requirements

• End Lap (Forward Overlap): Overlap between consecutive photographs along the same flight line. Standard requirement is 60% (acceptable range: $55% - 65%$). This ensures every ground point appears in at least two photos for 3D stereoscopic viewing.
• Side Lap (Lateral Overlap): Overlap between adjacent parallel flight strips. Standard requirement is 30% (acceptable range: $20% - 40%$). Side lap prevents coverage gaps caused by aircraft drift.

+---------------------------------------+
|                                       |
|       Photo 1         Photo 2         |
|    +-----------+   +-----------+      |
|    |   | 60%   |   |   |       |      |
|    |   | End   |   |   |       |      |
|    |   | Lap   |   |   |       |      |
|    +-----------+   +-----------+      |
|          <---- Air Base B ---->       |
+---------------------------------------+

Core Flight Planning Formulas

1. Ground Coverage of Single Photo ($W_G \times L_G$): Given square image format size $s \times s$ (e.g., $230\text{ mm} \times 230\text{ mm}$): $$W_G = s \cdot m = s \cdot \frac{H - h_{avg}}{f}$$
2. Air Base Distance ($B$) (Distance between exposures along strip): $$B = W_G \cdot (1 - \text{EndLap}) = s \cdot m \cdot (1 - PE)$$ Where $PE = 0.60$ for 60% end lap.
3. Flight Line Spacing ($W$) (Distance between parallel strips): $$W = W_G \cdot (1 - \text{SideLap}) = s \cdot m \cdot (1 - PS)$$ Where $PS = 0.30$ for 30% side lap.
4. Number of Photos per Strip ($N_{strip}$): $$N_{strip} = \frac{L_{strip}}{B} + 1 + 1 \quad \text{(plus safety photos at ends)}$$
5. Number of Flight Lines ($N_{strips}$): $$N_{strips} = \frac{W_{total}}{W} + 1$$
6. Exposure Interval ($T_{exp}$): Given aircraft ground speed $V$ (m/s): $$T_{exp} = \frac{B}{V}$$

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6. Ground Control Points (GCPs)

Ground Control Points (GCPs) are points of known horizontal coordinates ($X, Y$) and elevation ($Z$) determined via high-precision GNSS or terrestrial surveying. GCPs anchor the photogrammetric block model to the geodetic reference frame (e.g., PRS92 / PGD2020).

GCP Distribution Guidelines

• Minimum Density: For traditional aerial block bundle adjustment, a minimum of 4 corner GCPs plus internal control points per block is required.
• Signalization (Pre-Marking): Targets must be placed prior to flight execution using high-contrast geometric shapes (Maltese cross, L-target, or central dot with crossarms) sized such that the target width on the photograph is at least 3 to 5 times the pixel size/GSD.
• LMB & NAMRIA Requirements: Land Management Bureau Technical Bulletins require cadastral photogrammetric surveys to tie all block adjustments to primary and secondary network control monuments of the Philippine Reference System of 1992 (PRS92).

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7. Step-by-Step Worked Numerical Examples

Example 1: Photo Scale & Ground Distance Calculation

Problem: An aerial camera with focal length $f = 152.4\text{ mm}$ (6 inches) takes a vertical photograph from an aircraft flying at an altitude $H = 3,200\text{ m}$ above MSL. Calculate:

1. The photographic scale at a terrain elevation $h = 450\text{ m}$ above MSL.
2. The ground distance represented by a line segment $d = 45.0\text{ mm}$ measured on the photograph at this elevation.

Solution:

Step 1: Compute flying height above ground level ($H'$)

$$H' = H - h = 3,200\text{ m} - 450\text{ m} = 2,750\text{ m}$$

Step 2: Convert focal length to meters

$$f = 152.4\text{ mm} = 0.1524\text{ m}$$

Step 3: Compute photo scale ($S$)

$$S = \frac{f}{H - h} = \frac{0.1524\text{ m}}{2,750\text{ m}} = \frac{1}{18,044.62}$$

Thus, the scale representative fraction is 1 : 18,045.

Step 4: Compute ground distance ($D$)

$$D = \frac{d}{S} = d \cdot m = 0.0450\text{ m} \times 18,044.62 = 812.01\text{ m}$$

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Example 2: Building Height Determination from Relief Displacement

Problem: On a vertical aerial photograph taken with $f = 152.4\text{ mm}$ from a flying height $H = 1,800\text{ m}$ above the base of a building, the image of the top of a communication tower lies at a radial distance $r = 92.50\text{ mm}$ from the principal point. The relief displacement of the tower (distance from base image to top image) is measured as $d = 6.40\text{ mm}$. Calculate the height of the tower ($h$).

Solution:

Step 1: Identify given parameters

• Flying height above base $H = 1,800\text{ m}$
• Radial distance to top image $r = 92.50\text{ mm}$
• Relief displacement $d = 6.40\text{ mm}$

Step 2: Apply relief displacement height formula

$$h = \frac{d \cdot H}{r}$$ $$h = \frac{6.40\text{ mm} \times 1,800\text{ m}}{92.50\text{ mm}} = \frac{11,520}{92.50} = 124.54\text{ m}$$

Answer: The height of the communication tower is 124.54 meters.

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Example 3: Parallax and Elevation Calculation

Problem: A stereo pair is taken with a camera focal length $f = 152.4\text{ mm}$ from an altitude $H = 2,500\text{ m}$ above MSL. The air base $B = 750\text{ m}$. The absolute stereoscopic parallax of ground Point A is measured as $p_A = 91.44\text{ mm}$. Ground Point B has a measured parallax $p_B = 95.00\text{ mm}$. Compute:

1. The elevation of Point A ($h_A$).
2. The elevation of Point B ($h_B$).
3. The height difference $\Delta h = h_B - h_A$.

Solution:

Step 1: Compute elevation of Point A ($h_A$) Convert $f$ and $p_A$ to meters ($f = 0.1524\text{ m}$, $p_A = 0.09144\text{ m}$):

$$H - h_A = \frac{B \cdot f}{p_A} = \frac{750\text{ m} \times 0.1524\text{ m}}{0.09144\text{ m}} = \frac{114.3}{0.09144} = 1,250\text{ m}$$ $$h_A = H - 1,250\text{ m} = 2,500\text{ m} - 1,250\text{ m} = 1,250.00\text{ m}$$

Step 2: Compute elevation of Point B ($h_B$) Convert $p_B$ to meters ($p_B = 0.09500\text{ m}$):

$$H - h_B = \frac{B \cdot f}{p_B} = \frac{750\text{ m} \times 0.1524\text{ m}}{0.09500\text{ m}} = \frac{114.3}{0.09500} = 1,203.16\text{ m}$$ $$h_B = H - 1,203.16\text{ m} = 2,500\text{ m} - 1,203.16\text{ m} = 1,296.84\text{ m}$$

Step 3: Compute height difference ($\Delta h$)

$$\Delta h = h_B - h_A = 1,296.84\text{ m} - 1,250.00\text{ m} = 46.84\text{ m}$$

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Example 4: Flight Planning Block Calculation

Problem: A project area measuring $12\text{ km} \times 18\text{ km}$ is to be mapped using an aerial camera with $f = 152.4\text{ mm}$ and standard photo format size $230\text{ mm} \times 230\text{ mm}$. The desired nominal photo scale is $1 : 10,000$ over average terrain elevation $h_{avg} = 200\text{ m}$. Forward overlap (end lap) is set to 60%, and lateral overlap (side lap) is set to 30%. The flight lines are parallel to the $18\text{ km}$ side. Calculate:

1. The required flying altitude above MSL ($H$).
2. The air base distance ($B$).
3. The flight line spacing ($W$).
4. The total number of photographs required to cover the area (including 1 extra photo at each end of every flight line).

Solution:

Step 1: Compute flying height above MSL ($H$)

$$m = 10,000 \implies H - h_{avg} = m \cdot f = 10,000 \times 0.1524\text{ m} = 1,524\text{ m}$$ $$H = 1,524\text{ m} + 200\text{ m} = 1,724\text{ m}$$

Step 2: Compute single photo ground coverage ($W_G$)

$$W_G = s \cdot m = 0.230\text{ m} \times 10,000 = 2,300\text{ m}$$

Step 3: Compute air base distance ($B$)

$$B = W_G \cdot (1 - PE) = 2,300\text{ m} \times (1 - 0.60) = 2,300\text{ m} \times 0.40 = 920\text{ m}$$

Step 4: Compute flight line spacing ($W$)

$$W = W_G \cdot (1 - PS) = 2,300\text{ m} \times (1 - 0.30) = 2,300\text{ m} \times 0.70 = 1,610\text{ m}$$

Step 5: Compute number of flight lines ($N_{strips}$) The width across flight lines is $12\text{ km} = 12,000\text{ m}$.

$$N_{strips} = \frac{12,000\text{ m}}{1,610\text{ m}} + 1 = 7.45 + 1 \approx 8.45 \implies 9 \text{ flight lines}$$

Step 6: Compute photos per line ($N_{photo}$) The length along flight lines is $18\text{ km} = 18,000\text{ m}$.

$$N_{line} = \frac{18,000\text{ m}}{920\text{ m}} + 1 + 2 \text{ (safety margin)} = 19.57 + 3 = 22.57 \implies 23 \text{ photos/line}$$

Step 7: Total photos

$$\text{Total Photos} = 9 \text{ strips} \times 23 \text{ photos/strip} = 207 \text{ photographs}$$