GELE Photogrammetry & Cartography — Scale, Relief Displacement and ParallaxSummary

Photo scale is the ratio of photo distance to ground distance: scale = f/H, where f is the focal length and H is the flying height above the datum. On a vertical photograph, scale is uniform only if the terrain is perfectly level. When terrain varies in elevation, different points on the photo have different scales—areas at higher elevation appear larger (larger scale) and areas at lower elevation appear smaller (smaller scale). This variation is the basis for relief displacement. The average scale of a vertical photo is determined by the flying height and camera focal length, and understanding scale variation is critical for accurate measurements and cartographic applications.

Relief displacement is the radial outward shift of an image point on a vertical photograph caused by elevation above (or below) the datum plane. The formula is: d = (r × h) / H, where d is the displacement (mm), r is the radial distance of the image from the principal point (mm), h is the height of the object above the datum (m), and H is the flying height above the datum (m). Tall objects—buildings, towers, trees, mountains—appear shifted away from the principal point. The displacement is proportional to the object height and its radial distance from the principal point, and inversely proportional to flying height. At the principal point (r = 0), displacement is zero; at the image edges (large r), displacement is maximum. This radial outward shift is inherent to the perspective geometry of aerial photography.

By rearranging the relief displacement formula, the height of an object can be calculated: h = (d × H) / r. This allows measurement of object heights (buildings, trees, terrain features) from a single vertical photograph without requiring a stereo pair. The process involves identifying the image of the object's top on the photograph, measuring its radial distance r from the principal point, estimating the relief displacement d (the difference between where the top appears and where the base would appear), knowing the flying height H, and solving for h. This method is practical for isolated tall objects but less precise for distributed terrain features; it requires careful radial distance measurement and assumes accurate knowledge of flying height and datum.

Parallax is the apparent shift of a point's position between two overlapping aerial photographs taken from different positions along the flight line. Absolute parallax (P) is the horizontal distance between corresponding image points measured in the same direction on two stereo photos. For a point at ground level, absolute parallax is P = B × f / H, where B is the air base (ground distance between camera positions), f is focal length, and H is flying height. Parallax difference (Δp) is the difference in parallax between the top and base of an object: Δp = P_top − P_base. The relationship between parallax difference and height is: h ≈ (H × Δp) / P when Δp << P, or more precisely: h = (H × Δp) / (P + Δp). Stereoscopic parallax is the basis for automatic elevation measurement in digital photogrammetry and remains the most reliable method for three-dimensional mapping.

When two overlapping vertical photographs from different camera stations are viewed stereoscopically (either directly through a stereoscope or digitally), the human brain perceives a three-dimensional model. This stereo model is created because each eye sees a slightly different view—just as binocular vision works. The parallax difference between corresponding points in the left and right images creates the perception of depth. In a digital stereo workstation, an operator can move a floating mark (a small symbol) that appears to hover above or below the terrain surface; when the mark is positioned so it appears to touch a point on the terrain in both images simultaneously, the three-dimensional coordinates of that point can be calculated from the image coordinates and the parallax. This principle—turning parallax measurement into XYZ coordinates—is the core of digital photogrammetry and is used to generate orthophotos, digital terrain models (DTMs), digital elevation models (DEMs), and orthorectified maps.

Flying height H (measured above the datum) is the vertical distance of the aircraft camera above the reference surface (usually mean sea level or a local datum). Flying height directly determines photo scale: larger H produces smaller scale (more area covered in one photo but less detail); smaller H produces larger scale (more detail but smaller area covered). Flying height inversely affects relief displacement magnitude: larger H reduces displacement, making corrections less critical; smaller H increases displacement, requiring careful corrections. In the displacement formula d = (r × h) / H, a larger H (higher flying) results in smaller displacement for the same object height. This relationship is important: flying at higher altitudes produces more uniform scale across the photo and reduces relief displacement effects, making geometric corrections easier—but requires larger-format cameras or longer focal lengths to maintain ground detail. Flying height is specified in survey specifications and must be known precisely for accurate photogrammetric calculations.

Photo scale varies across a vertical photograph when terrain is not level. Higher elevation has larger scale (objects appear larger); lower elevation has smaller scale (objects appear smaller). This scale variation is the physical cause of relief displacement. A tall object at point A appears at position A on the photo (its base) but the top appears at position A' offset radially outward; this offset is relief displacement d.

Flying height H inversely affects both average photo scale and relief displacement magnitude. Higher flying (larger H) produces smaller scale and smaller displacement; lower flying (smaller H) produces larger scale and larger displacement. Expressed mathematically: scale ∝ 1/H and d ∝ 1/H. Survey planning must balance the need for ground detail (requiring lower flying, larger scale) against the need for reduced displacement errors (favoring higher flying).

Relief displacement is not merely an error to correct—it is a measurement tool. By measuring displacement d at radial distance r on a photograph of flying height H, one can calculate object height h = (d × H) / r. This single-photo method is practical for spot heights but less accurate and comprehensive than stereoscopic parallax, which measures height differences across an entire stereo model.

Parallax difference Δp between the top and base of an object directly yields height via h = (H × Δp) / (P + Δp). Unlike relief displacement measurement, parallax measurement is: (1) more accurate because stereoscopic viewing resolves height precisely, (2) applicable to all points in the stereo model (not just well-positioned features), (3) independent of radial distance from principal point, and (4) the basis for automated DEM generation in digital photogrammetry. This is why stereoscopic parallax is the preferred method in modern surveying.

In relief displacement height calculation h = (d × H) / r, measurement error in r directly appears in error in h (inverse relationship). For example, a 1 mm error in r when r = 80 mm is a 1.25% error; the same 1 mm error when r = 40 mm is a 2.5% error. This is why heights are measured away from the principal point (where r is large) for better accuracy. Near the principal point, where r is small, both displacement and accuracy are poor.

Orthorectification is the process of removing relief displacement and tilt distortion from aerial photographs to create orthophotos—images with uniform scale that can be used directly for mapping like maps. Digital orthophotos require accurate DEM (digital elevation model) data, which is generated from parallax measurement in stereo models. The DEM provides elevation h at each pixel; the image geometry is then corrected for relief displacement and tilt at each pixel. This process, now standard in digital mapping workflows, requires understanding of both relief displacement and parallax concepts.

Surveyors and assessors use aerial photographs to measure building heights and estimate roof areas for property assessment and urban planning. Relief displacement on a vertical photograph can yield a building height quickly (useful for spot checks), but stereoscopic parallax in a stereo model provides more reliable measurements for large datasets. This is common in Philippine urban projects and property assessment under real estate surveying practices.

Modern topographic surveys rely entirely on DEM generation from stereoscopic parallax measurement. Aerial photographs are acquired along parallel flight lines with 60% forward overlap (creating stereo pairs) and 30% lateral overlap. Parallax differences are measured automatically across the entire model to generate a dense DEM at 0.5–2 m resolution. This DEM is then used to create contour maps, orthophotos, and 3D terrain models for infrastructure planning, environmental monitoring, and flood modeling under WGS84/PRS92 standards. The Philippine government's National Mapping and Resource Information Authority (NAMRIA) and surveying firms routinely produce DEMs using this principle.

Aerial photographs have perspective distortion and relief displacement; they cannot be used directly for mapping without correction. Orthophotos are created by applying a DEM to remove relief displacement and tilt, producing images with uniform scale that align geometrically with maps. Orthophotos are used as backgrounds for topographic maps, cadastral maps, and land-use maps. Understanding relief displacement is essential for quality assurance—artifacts in orthophotos (ghosting, seams, distortion) often indicate DEM errors or improper displacement correction. This is a standard workflow in Philippine surveying and mapping projects.

Temporal series of aerial photographs can be used to monitor infrastructure (roads, buildings, dams) for changes or damage. Relief displacement must be understood when comparing features across different photo sets acquired at different times, flying heights, or seasons. Change detection algorithms must account for displacement-induced position shifts to avoid false positives. This is used in disaster response (after typhoons, earthquakes) and infrastructure maintenance under Philippine surveying and civil engineering practice.

Photogrammetric technicians use stereo plotters (analog or digital) to graphically locate spot heights, contour lines, and feature positions by stereoscopic parallax measurement. These photogrammetrically derived positions are then integrated with GPS/ground survey data to produce final survey maps and digital products. Understanding parallax enables technicians to identify and correct measurement errors, recognize elevation anomalies (which may indicate data processing errors), and ensure that photogrammetric products align with surveying standards (PRS92 datum, Philippine cadastral system under PD 1529, RA 4374 land registration requirements).

Philippine land surveys under the Land Registration Authority (LRA) often use orthophotos and aerial surveys to delineate property boundaries and prepare cadastral maps. Relief displacement in uncorrected photos would cause boundary positions to shift, potentially affecting legal land descriptions (as required by RA 4374 and CA 141). Modern cadastral surveys use orthophotographs (which have displacement corrected) and integrate aerial data with GPS ground surveys for legal accuracy. Understanding displacement and parallax ensures that photogrammetric contributions to cadastral mapping meet legal and technical standards.

Modern digital photogrammetry (Structure-from-Motion, SfM, with software like PhotoScan, Pix4D, Metashape) automates parallax measurement and DEM generation, but automated processing can fail or produce artifacts if input data is poor or parameters are incorrect. Skilled operators understand parallax principles well enough to: (1) detect processing failures by checking DEM reasonableness and contour patterns, (2) identify areas of poor parallax measurement (shadows, occlusions, water surfaces), (3) recognize relief displacement artifacts in orthophoto output, and (4) apply targeted corrections. This quality-control competency is essential in professional surveying firms and is tested in the PRC Geodetic Engineer Licensure Examination.