GELE Photogrammetry & Cartography — Scale, Relief Displacement and ParallaxStudy Notes

Photo scale is the ratio of a distance on a photograph to the corresponding distance on the ground. It is expressed as a representative fraction (RF) or ratio, such as 1:5000, meaning 1 unit on the photo represents 5000 units on the ground. **Principal Scale Formula:** For a vertical photograph taken with a camera of focal length f at flying height H above a reference datum: Scale = f / H or expressed as a ratio: 1 : (H/f) Example: If f = 152 mm and H = 1500 m, then Scale = 152 / 1,500,000 = 1:9868 (approximately 1:10,000). **Important Scale Concepts:** 1. **Mean Scale** — the average scale across a photo, typically at the principal point. 2. **Scale at Height h** — when terrain rises to height h above the datum plane, the scale changes: Scale at height h = f / (H − h) Higher terrain appears at a larger scale (closer to the camera). 3. **Scale Variation** — across a single photo, scale varies with elevation. In mountainous terrain, this variation can be significant and must be accounted for in precise measurements. 4. **Nominal vs. Actual Scale** — the nominal scale is calculated from f and planned flight height; actual scale depends on actual flying height, which may differ due to terrain, wind, or pilot adjustment. **Practical Consideration for Philippine Surveys:** When working in areas with significant relief (e.g., Cordillera region or island surveys under RA 4374 land classification), always verify the actual flying height above the local terrain reference, not just mean sea level (MSL). PRS92 and PPCS/UTM grid systems require consistent datum interpretation.

**Definition:** Relief displacement is the radial outward shift of the image of a point on an elevated object from its position if that point were on the reference datum plane. On a vertical photograph, any tall object leans outward from the principal point, creating a characteristic radial displacement. **Physical Cause:** When an object rises above the datum, it appears at a different horizontal position on the photo because the camera views it from a slightly different angle. The top of a tower, for instance, is imaged farther from the principal point than the tower's base (which rests on the datum). **Relief Displacement Formula:** d = (r × h) / H where: - d = relief displacement (mm or other unit) - r = radial distance of the image (top of the object) from the principal point (mm) - h = height of the object above the datum (m) - H = flying height above the datum (m) **Key Properties:** 1. Displacement is zero at the principal point (center of photo). 2. Displacement increases linearly with object height h. 3. Displacement increases with radial distance r from the principal point. 4. Displacement is purely radial (along a line from the principal point outward). 5. Displacement is independent of camera focal length (it depends only on geometry). **Object Height from Displacement (Inverse Formula):** h = (d × H) / r This is the formula used to determine object heights when relief displacement is measured. **Measurement Procedure:** 1. Locate the principal point (intersection of fiducial marks on the photo). 2. Identify the object of interest and locate its image (typically the top of the object). 3. Measure the radial distance r from the principal point to the object's image using a ruler or comparator. 4. Measure the relief displacement d by comparing the position of the top of the object to where its base would be if extended radially (or by identifying the true base image nearby). 5. Apply the inverse formula to calculate height. **Why Measure Near the Edges?** Heights are measured far from the principal point (near photo edges) because the radial distance r is larger, making the relief displacement d also larger relative to measurement uncertainty. A larger d means better precision in the calculated height. Conversely, near the principal point, both r and d are very small, leading to large relative errors. **Practical Considerations for Philippine Surveys:** When applying relief displacement measurement to cadastral surveys (RA 4374, RA 8560), always: - Clearly identify the datum plane (mean sea level or local reference). - Account for sea-level vs. ellipsoidal heights in WGS84/PRS92 conversions. - Document the flying height and focal length used. - Use images with good contrast and sharp focus on object tops.

**Definition:** Parallax is the apparent shift in position of an object when viewed from two different locations. In photogrammetry, stereoscopic parallax is the apparent shift of a point between two overlapping aerial photographs taken from successive camera positions along the flight line. **Physical Basis:** When two photos are taken in sequence along a flight line, the same ground point appears at different positions on the two photos because the camera has moved. This shift, measured parallel to the flight line direction, is the parallax. The magnitude of parallax is inversely related to object distance (height)—closer objects show larger parallax shifts. **Absolute Parallax:** The absolute parallax P of a point is the lateral separation (parallel to the flight line) of the point's images on the left and right photos: P = x_L − x_R where x_L and x_R are measured from the same reference (typically the left photo's principal point or the conjugate principal point on the right photo). **Parallax Difference (Height Parallax):** For two points at the same horizontal location but different elevations (e.g., the top and base of an object), the parallax difference is: Δp = p_top − p_base where p_top and p_base are the absolute parallaxes at the top and base. **Height from Parallax Difference:** The fundamental parallax-to-height formula is: h = (H × Δp) / (P + Δp) For most practical cases where Δp is small compared to P, this simplifies to: h ≈ (H × Δp) / P where: - h = height of the object above the base datum (m) - H = flying height above the datum (m) - Δp = parallax difference (mm or other unit) - P = absolute parallax of the base point (mm or other unit) **Derivation Insight:** The parallax difference arises because the top of an object is closer to the camera than its base. Using similar triangles and the geometry of the stereo pair: At base (datum level): distance from camera = H At top (height h): distance from camera = H − h The parallax is proportional to the reciprocal of distance. The difference in parallax is related directly to the height difference. **Stereoscopic Model Elevation Profile:** Across a stereo model (the overlapping area of two photos), if the absolute parallax P is known, any point's elevation can be calculated by measuring its parallax difference from a reference base. **Measurement in the Stereo Viewer:** 1. Set up a stereoscope and place the stereo pair in correct orientation (left photo on left, right on right). 2. View the model stereoscopically to see the three-dimensional terrain. 3. Using a parallax bar (or digital measurement tool), align the floating dot with the top of the object. 4. Read the parallax value p_top. 5. Shift the parallax bar to the base (datum level or a reference surface) and read p_base. 6. Calculate Δp = p_top − p_base. 7. Apply the height formula to get h. **Advantages Over Relief Displacement:** - Heights can be measured anywhere in the stereo model, not just from principal points. - Typically provides better accuracy because both stereo photos are used. - Modern digital photogrammetry automates parallax measurement across entire stereo pairs. - Enables generation of digital elevation models (DEMs) and orthophotos. **Practical Considerations for Philippine Surveys:** When working with stereo pairs in Philippine cadastral surveys (RA 4374, PD 1529) or land classification (CA 141): - Verify that the stereo pair overlap is at least 50% (typically 55–65% for aerial surveys). - Use consistent datum reference (MSL for orthometric heights, or ellipsoidal heights if using WGS84). - For PPCS/UTM grid integration, ensure flying height is referenced to the correct datum. - Document the camera calibration certificate (focal length, principal distance, distortion parameters). - Cross-check parallax measurements with ground control points (GCPs) surveyed using conventional or GNSS methods.

**When to Use Each Method:** **Relief Displacement Method:** - Used when only a single vertical photo is available. - Measurements are made radially from the principal point. - Better for isolated objects far from the principal point. - No stereo pair required. - Heights can be measured even when stereo overlap is not available. **Stereoscopic Parallax Method:** - Requires a pair of overlapping photos. - Measurements are made parallel to the flight line within the stereo model. - Can measure heights anywhere in the overlapping area. - Typically more accurate (both photos are used for convergent geometry). - Modern digital photogrammetry standard; enables automation. - Allows generation of elevation profiles and DEMs. **Accuracy Considerations:** 1. **Relief Displacement Accuracy:** - Depends on accurate measurement of radial distance r and displacement d. - Sensitivity: d = (r × h) / H, so error in d directly propagates to h. - Precision improves with larger r (farther from principal point). - Uncertainty grows near the principal point (small r). 2. **Parallax Height Accuracy:** - Depends on accurate measurement of parallax difference Δp and absolute parallax P. - Formula h = (H × Δp) / (P + Δp) shows that error in either parallax component affects h. - Relative accuracy is usually better because both measurements are similar in scale. - Modern parallax bars and digital tools can achieve mm-level precision. **Practical Workflow:** In comprehensive photogrammetric surveys: 1. Establish ground control points (GCPs) using conventional or GNSS survey methods. 2. Perform interior orientation (determine camera calibration). 3. Perform relative orientation (align stereo pair for proper parallax measurement). 4. Perform absolute orientation (tie the stereo model to ground coordinates and datums). 5. Measure heights using parallax differences across the entire stereo model. 6. Validate with GCP check points. 7. Generate DTM or DEM for the survey area. Relief displacement may be used as a secondary check or when a single photo must be analyzed independently.

**Modern Context:** While classical photogrammetry relied on manual measurement of relief displacement and parallax using optical instruments (stereoscopes, comparators, parallax bars), contemporary surveys employ digital photogrammetry and automated image matching. **Digital Parallax Measurement:** 1. **Image Correlation** — software automatically matches corresponding pixels between stereo photos. 2. **Dense Point Clouds** — millions of 3D points are generated by computing parallax for each pixel or grid cell. 3. **Elevation Determination** — the z-coordinate of each point is derived from its parallax value using the formula: z = z_base + (H × Δp) / P 4. **Digital Elevation Models (DEMs)** — interpolated surfaces representing terrain elevation. 5. **Orthophotos** — photos corrected for tilt and relief displacement, registered to a map projection (e.g., PPCS or UTM). **Advantages of Digital Methods:** - Rapid processing of large areas. - Reduced operator fatigue and subjective error. - Consistent application of photogrammetric equations across entire stereo models. - Integration with GIS and land information systems. - Easy update and revision. **Integration with PRS92 and PPCS/UTM:** For Philippine surveys, digital photogrammetric processing typically includes: - Transformation of image coordinates to WGS84 geographic coordinates (latitude/longitude). - Further transformation to PRS92 (Philippine Reference System 1992) if needed for older projects or legacy data. - Projection to PPCS (Philippine Plane Coordinate System) or UTM (Universal Transverse Mercator) for mapping and cadastral purposes. - Proper datum conversion (ellipsoidal to orthometric heights if required). **Quality Control:** Modern digital photogrammetry projects include: - Ground control point (GCP) measurement and verification. - Accuracy assessment against independent check points. - Statistical analysis of residuals. - Compliance with accuracy standards (e.g., ±50 cm vertical for 1:5000 scale photography). **Philippine Regulatory Context:** Under RA 4374 (Law on the Classification of Public Lands), RA 8560 (Real Property Information System), PD 1529 (Property Registration Decree), and CA 141 (Public Land Act): - Aerial surveys for cadastral purposes must be performed by licensed geodetic engineers. - Survey methods and accuracy must be documented and approved by appropriate authorities (BIR, DENR, etc.). - Height measurements from aerial photos are acceptable when proper control and accuracy standards are met. - Digital products (orthophotos, DEMs) must be properly georeferenced to recognized datums (WGS84, PRS92, PPCS).

**Common Exam Question Types:** 1. **Basic Scale Calculation** - Given: f, H - Find: Scale ratio or RF - Strategy: Use Scale = f / H; convert units consistently. 2. **Scale at Variable Heights** - Given: f, H, elevation h - Find: Scale at height h - Strategy: Use Scale(h) = f / (H − h); note that higher elevations have larger scales. 3. **Relief Displacement Calculation** - Given: h, r, H - Find: d - Strategy: Use d = (r × h) / H; ensure units match (typically mm, m, m). 4. **Height from Relief Displacement** - Given: d, r, H - Find: h - Strategy: Use h = (d × H) / r; watch for unit conversions. 5. **Parallax Height Determination** - Given: H, P, Δp - Find: h - Strategy: Use h = (H × Δp) / (P + Δp) for exact; or h ≈ (H × Δp) / P for approximate when Δp << P. 6. **Mixed Problems** - Involve scale, relief displacement, and parallax in sequence. - Require logical connection between concepts. **Step-by-Step Exam Approach:** 1. **Read Carefully** - Identify what is given (f, H, h, r, d, P, Δp). - Identify what is asked. - Check units and note any conversions needed. 2. **Sketch a Diagram** - For relief displacement: show principal point, radial direction, object top, and base. - For parallax: show flight line, left and right photos, point positions. - Diagrams clarify the geometry and reduce sign errors. 3. **Select the Correct Formula** - Relief displacement: d = (r × h) / H or h = (d × H) / r - Parallax: h = (H × Δp) / (P + Δp) - Scale: Scale = f / H or Scale(h) = f / (H − h) 4. **Convert Units Consistently** - Typically: f in mm, H in m, h in m, r in mm, d in mm, P in mm, Δp in mm. - Example: if f = 152 mm and H = 1500 m, Scale = 0.152 m / 1500 m = 1:9868 (unitless ratio). 5. **Perform Calculation** - Show all steps clearly. - Carry appropriate significant figures (typically 3–4 for engineering). 6. **Verify the Result** - Is the answer reasonable? (A 50 m building on a 1500 m elevation photo should have a few mm of displacement.) - Check units in the final answer. - If possible, use an alternative method or inverse formula to cross-check. 7. **State the Answer Clearly** - Include units. - Round appropriately. - If asked, note any assumptions or limitations. **Common Mistakes to Avoid:** 1. **Unit Confusion** - Formula uses r in mm and h in m; mixing units causes factor-of-1000 errors. - Always state your unit choice upfront. 2. **Confusing H with h** - H = flying height (camera altitude), h = object height. - Easy to swap accidentally; double-check the problem statement. 3. **Principal Point Error** - Relief displacement is measured from the principal point (intersection of fiducial marks), not from the photo corner. - Parallax is measured from corresponding image points, not from the principal point. 4. **Direction and Sign** - Relief displacement is always radial outward (positive). - Parallax is measured parallel to the flight line; sign depends on convention (left to right or right to left). - For height calculation, use absolute values or be consistent with sign convention. 5. **Scale Interpretation** - A scale of 1:5000 means 1 unit on photo = 5000 units on ground. - Larger denominators = smaller scale (less detail, farther away). - Smaller denominators = larger scale (more detail, closer). 6. **Approximation Validity** - The approximation h ≈ (H × Δp) / P is valid only when Δp << P (typically Δp < 0.1 × P). - Always use the exact formula h = (H × Δp) / (P + Δp) if in doubt, especially for tall objects. **Practice Exam Questions:** Question 1: A vertical photo is taken with f = 152 mm from H = 2000 m. A tall building is located at r = 110 mm with relief displacement d = 3.3 mm. Calculate: (a) The photo scale. (b) The building height. (c) The ground distance from the nadir point to the building. Question 2: A stereo pair with H = 1800 m covers a mountainous region. The absolute parallax at sea level is P = 96 mm. Three points are measured: - Mountain peak A: Δp = 4.0 mm - Building B: Δp = 2.5 mm - Ground point C: Δp = 0 mm (reference datum) Calculate the heights of all three points above sea level. Question 3: Compare the heights calculated by relief displacement and parallax for the same object. Discuss why they might differ and which method is more reliable.