Orthometric height
Orthometric height, denoted as H, is the vertical distance from the geoid—an equipotential surface approximating mean sea level—to a point on the Earth's surface, measured along the local plumb line and taken as positive upward from the geoid.[1][2] This height system provides a physically meaningful measure of elevation that accounts for gravitational variations, distinguishing it from purely geometric heights and enabling applications in hydrology, surveying, and vertical datum establishment where water flow and potential energy are relevant.[2] In geodesy, orthometric height relates to ellipsoidal height (h), which is the geometric distance above a reference ellipsoid such as that used in NAD 83, through the equation H = h - N, where N is the geoid height or undulation representing the separation between the ellipsoid and the geoid.[1][3] Unlike ellipsoidal heights directly obtained from Global Navigation Satellite Systems (GNSS) like GPS, orthometric heights cannot be measured directly and are instead determined via spirit leveling combined with gravity observations to compute geopotential numbers, often using corrections such as the Helmert orthometric method that incorporates average gravity along the plumb line and an assumed crustal density.[1][2] The practical implementation of orthometric heights underpins national vertical datums, such as the North American Vertical Datum of 1988 (NAVD 88, to be replaced by the North American-Pacific Geodetic Datum of 2022 as part of the National Spatial Reference System modernization in 2025 or 2026) in the United States, which was established through over 625,000 kilometers of leveling tied to a single benchmark at Pointe au Père, Quebec, with typical local leveling precisions of about ±3 cm between benchmarks.[1][2] In modern GNSS surveying, orthometric heights are derived by transforming GPS-measured ellipsoidal heights using geoid models like GEOID18 or later iterations from the National Geodetic Survey, with guidelines recommending baselines under 10 km for local accuracies of 2–5 cm at 95% confidence and geoid differences below 1 cm over 20 km.[1][3] This integration enhances efficiency in engineering, mapping, and flood modeling by bridging geometric satellite data with gravity-informed elevations.[3]Basic Concepts
Definition
Orthometric height, denoted as H, is the vertical distance from the geoid to a point on the Earth's surface, measured along the local plumb line in the direction of gravity.[4] This measurement provides a gravity-related reference that aligns with the physical direction of downward force at any location.[4] The geoid serves as the reference surface for orthometric heights and is defined as the equipotential surface of the Earth's gravity field that best approximates global mean sea level in a least-squares sense.[5] As an equipotential surface, the geoid represents a level where gravitational potential is constant, closely coinciding with the undisturbed ocean surface extended under landmasses.[5] This makes orthometric height particularly meaningful for applications involving gravitational potential, as it accounts for local variations in gravity strength due to the Earth's irregular mass distribution.[5] For instance, at points on the geoid itself—such as mean sea level—orthometric height is zero (H = 0), while on mountaintops, it indicates the elevation above this reference, reflecting the accumulated gravitational effect along the plumb line.[4] In contrast to geometric measures like ellipsoidal height, orthometric height integrates the influence of the gravity field for a more physically intuitive vertical coordinate.[4]Comparison to Other Height Types
Orthometric height, denoted as H, represents the distance above the geoid measured along the plumb line, providing a physically meaningful elevation tied to Earth's gravity field.[1] In contrast, ellipsoidal height, denoted as h, is the geometric distance from a reference ellipsoid—such as the GRS 80 ellipsoid used in NAD 83—to the point of interest, measured perpendicular to the ellipsoid surface.[2] This height is directly obtainable from Global Navigation Satellite Systems (GNSS) like GPS, offering high precision in three-dimensional positioning without requiring gravity data.[6] The fundamental relationship linking these heights is h \approx H + N, where N is the geoid undulation, representing the separation between the geoid and the reference ellipsoid, which can vary by tens of meters globally.[7] One key advantage of orthometric height over ellipsoidal height lies in its alignment with intuitive concepts of "elevation above sea level," as it follows the direction of gravity and references the equipotential surface approximated by mean sea level, making it essential for applications like flood modeling and construction.[1] Ellipsoidal heights, while geometrically consistent and globally applicable, lack this physical basis and often require correction via geoid models to yield practical elevations; for instance, in New York City, where the geoid undulation N \approx -32 m, an uncorrected ellipsoidal height would underestimate true orthometric elevation by about 30 meters.[8][2] This geometric nature renders ellipsoidal heights unsuitable as direct substitutes for orthometric ones in gravity-dependent contexts, though they enable efficient hybrid determination when combined with gravimetric data.[6] Normal height, denoted as H^*, closely resembles orthometric height but is defined using the normal gravity field of a reference ellipsoid rather than the actual, irregular gravity field.[9] Specifically, H^* = C / \gamma, where C is the geopotential number and \gamma is the average normal gravity along the ellipsoidal normal, avoiding the need for direct measurements of terrain-induced gravity variations.[1] This makes normal heights simpler to compute in regions with sparse gravity data, as they rely on theoretical models like those from the GRS 80 ellipsoid, but they introduce approximations that can lead to discrepancies of up to several centimeters compared to orthometric heights, particularly in areas with significant topography.[2] Orthometric heights, by incorporating observed gravity (e.g., via Helmert's method), achieve higher fidelity to the true geopotential, with accuracies around ±3 cm in modern systems like NAVD 88.[6] Dynamic height, denoted as H_d, is another potential-based system, computed as H_d = C / \gamma_{45}, where \gamma_{45} is the normal gravity at 45° latitude, scaling the geopotential number to define equipotential surfaces suitable for fluid dynamics.[10] Unlike orthometric height, which uses local actual gravity for a more general elevation reference, dynamic height is optimized for hydrology and oceanography, such as maintaining consistent water levels across the Great Lakes in the International Great Lakes Datum of 1985 (IGLD 85).[1] The differences arise from the fixed gravity scaling in dynamic heights, which can introduce corrections of up to 0.2 meters in non-equatorial regions like Albuquerque, emphasizing its specialized role over the broader applicability of orthometric heights.[2]| Height Type | Reference Surface | Measurement Basis | Key Relation to Orthometric H | Primary Advantage |
|---|---|---|---|---|
| Ellipsoidal (h) | Reference ellipsoid | Geometric (GNSS) | h \approx H + N | Direct GNSS compatibility |
| Normal (H^*) | Telluroid | Normal gravity | H^* \approx H (with corrections) | Simpler computation, no local gravity needed |
| Dynamic (H_d) | Equipotential (scaled) | Geopotential | H_d = C / \gamma_{45}, related via C | Suited for hydraulic and oceanographic uses |