Earth figure and terrain

workingupdated 2026-09-15earth-figureellipsoidelevationdemgeoidsvs

The Besselian formulation places the observer on an oblate spheroid, through $\rho \sin\phi' = S \sin\phi$ and $\rho \cos\phi' = C \cos\phi$ with $\tan\phi' = (1-f)^2 \tan\phi$ . The 1961 Explanatory Supplement used Hayford's spheroid ( $a = 6378.388$ km, $f = 1/297$ ), the 1992 edition the IAU 1976 figure ( $a = 6378140$ m, $f = 1/298.257$ ), and NASA SVS uses WGS 84 ( $a = 6378137$ m, $f = 1/298.257223563$ ) ¹ ² ³.
Choosing among modern ellipsoids is irrelevant; choosing a sphere is not. IAU 1976, GRS 80 and WGS 84 differ by 3 m in equatorial radius and 16 mm in polar radius. A sphere with geodetic latitude misplaces the observer by up to 21 km in latitude and 21 km in radius, which is tens of seconds in contact times and the full path width at the limits.
Observer height enters through $\rho$ and $\zeta$ . A limit line shifts perpendicular to the path by $h \cot A \sin D$ : 577 m per 1000 m of elevation when the Sun is at 60°, 1.7 km at 30°, 5.7 km at 10° ⁴. The umbra widens by only $2 h \tan f_2$ , 82 m at the height of Everest ⁵.
NASA SVS is the only predictor that puts every observer on the terrain. For 2017 and 2024 it used SRTM elevations on the WGS 84 ellipsoid with the EGM96 geoid, LRO LOLA and SLDEM2015 lunar topography, and JPL DE421 on the product pages, with DE440 in the paper's appendix. It reports up to 3 km of umbra shift toward the Sun over the western United States in 2017 ³ ⁶ ⁷.
Everyone else draws the path at sea level. Espenak publishes an elevation factor to shift the limits by hand, and timeanddate states sea level explicitly. The local-circumstance calculators take an altitude for one site only. They are Jubier's calculator, The Photographer's Ephemeris (Photo Ephemeris), the Swiss Ephemeris and Occult ⁸ ⁵ ⁹ ¹⁰ ¹¹ ¹².
Every free global DEM gives heights above a geoid, not above the ellipsoid. SRTM, NASADEM, ASTER GDEM and GMTED2010 are on EGM96, Copernicus DEM on EGM2008 ¹³ ¹⁴ ¹⁵ ¹⁶. The geoid sits between 106 m below and 85 m above WGS 84 ¹⁷. Only SVS documents a geoid model in its constants. Ignoring the separation costs up to about 170 m of path at a 30° Sun.

The question. What figure of the Earth does the Besselian-element formulation assume, how is an observer with a geodetic latitude, longitude and height mapped into the fundamental plane, what does it cost to get the figure wrong, how does terrain height move the path limits and the contact times, and what did the major predictors actually do about terrain, with which elevation datasets, and with or without a geoid correction?

The ellipsoid the formulas assume

The formulation treats the Earth as an ellipsoid of revolution, described by an equatorial radius $a$ and a flattening $f$ , and it measures every Besselian length in units of $a$ . The 1961 Explanatory Supplement states the figure used for parallax and eclipse work: "the dimensions of the Earth are usually taken to be those of Hayford's spheroid ... defined by the equatorial radius ( $a$ ) and the flattening ( $f$ ), for which an exact value of 1/297 is adopted. The adopted value for $a$ is 6378.388 km, from which the polar radius $b = a(1-f)$ is derived as 6356.912 km", with a footnote "From 1968: $f = 1/298.25$ , $a = 6\,378\,160$ m, $b = 6\,356\,775$ m" ¹. The 1992 edition gives, for terrestrial coordinates, the MERIT 1983 values " $a = 6378.137$ km and $f = 1/298.257$ " and, in the reduction of observations, "The IAU (1976) constants are $a = 6378\,140$ m and $f = 1/298.257$ " ². The reference ellipsoid of GPS and of every modern DEM is WGS 84, whose defining parameters are "Semi-major Axis $a$ 6378137.0 meters" and "Flattening Factor of the Earth $1/f$ 298.257223563" ¹⁸.

Implementations pick one of these. NASA SVS lists "Earth radius 6378.137 km, Earth flattening 1/298.257 (the WGS 84 ellipsoid)" for its first 2017 umbra product and "Earth radius 6378.137 km, Ellipsoid WGS84, Geoid EGM96" for the terrain-aware 2017 animation ¹⁹ ³. The Swiss Ephemeris defines EARTH_RADIUS 6378136.6 and EARTH_OBLATENESS (1.0/298.25642), both attributed to page K6 of the 2006 Astronomical Almanac, and its eclipse routine sets the scale with de = 6378140.0 / AUNIT ²⁰ ¹¹. Espenak's pages do not state the figure they use. His Besselian elements are in units of the Earth's equatorial radius, and the NASA 2001 bulletin, which describes his method, gives $\tan f_1$ and $\tan f_2$ without naming an ellipsoid ⁴ ²¹.

The observer's coordinates

The 1961 Supplement gives the observer's rectangular coordinates in the fundamental plane, in units of $a$ , for an observer at ephemeris longitude $\lambda$ (west positive in that edition), geocentric latitude $\phi'$ and geocentric distance $\rho$ :

\xi = \rho \cos\phi' \sin\theta

\eta = \rho \sin\phi' \cos d - \rho \cos\phi' \sin d \cos\theta

\zeta = \rho \sin\phi' \sin d + \rho \cos\phi' \cos d \cos\theta

with $\theta = \mu - \lambda$ the local hour angle of the shadow axis and $d$ the declination of the axis ¹. The hourly variations are $\xi' = \mu'(-\eta \sin d + \zeta \cos d)$ , $\eta' = \mu' \xi \sin d - d' \zeta$ , $\zeta' = -\mu' \xi \cos d + d' \eta$ , with $\mu'$ and $d'$ in radians per hour ¹. The quantities that carry the Earth's shape are $\rho \sin\phi'$ and $\rho \cos\phi'$ . For a point on the spheroid:

\rho \sin\phi' = \frac{(1-e^2)\sin\phi}{\sqrt{1 - e^2 \sin^2\phi}} = S \sin\phi, \qquad \rho \cos\phi' = \frac{\cos\phi}{\sqrt{1 - e^2 \sin^2\phi}} = C \cos\phi

"in which $e$ is the ellipticity of the Earth's spheroid. For Hayford's spheroid, $e^2$ is equal to 0.00672267", and " $e^2 = 0.00669454$ for 1968 onwards" ¹. The geodetic to geocentric conversion and the auxiliary functions follow from the ellipse:

\tan\phi' = \frac{b^2}{a^2}\tan\phi = (1-f)^2 \tan\phi, \qquad S = (1-f)^2 C, \qquad C = \left\{\cos^2\phi + (1-f)^2 \sin^2\phi\right\}^{-1/2}

\rho^2 = C^2\left\{\cos^2\phi + (1-f)^4 \sin^2\phi\right\}

where the Supplement notes that these "assume that the observer is at sea level" ¹. The 1992 edition writes the same relations with the height $h$ above the spheroid included:

x = \rho \cos\phi' \cos\lambda = (aC + h)\cos\phi \cos\lambda, \quad y = \rho \cos\phi' \sin\lambda = (aC + h)\cos\phi \sin\lambda, \quad z = \rho \sin\phi' = (aS + h)\sin\phi

with " $C = (\cos^2\phi + (1-f)^2 \sin^2\phi)^{-1/2}$ , $S = (1-f)^2 C$ " ². Dividing by $a$ gives $\rho \sin\phi' = (S + h/a)\sin\phi$ and $\rho \cos\phi' = (C + h/a)\cos\phi$ , which is the form a developer needs. At $\phi = 40^{\circ}$ on WGS 84 these evaluate to $\rho \sin\phi' = 0.63937$ , $\rho \cos\phi' = 0.76711$ and $\rho = 0.99862$ (computed here from the formulas above).

Bessel's device for the inverse problem, finding the geodetic point under a given $(\xi, \eta)$ without iterating on $\rho$ , introduces the parametric latitude $\phi_1$ and the auxiliaries $\rho_1$ , $\rho_2$ , $d_1$ , $d_2$ :

\rho_1 = (1 - e^2 \cos^2 d)^{1/2}, \quad \rho_2 = (1 - e^2 \sin^2 d)^{1/2}, \quad \rho_1 \sin d_1 = \sin d, \quad \rho_1 \cos d_1 = (1-e^2)^{1/2} \cos d

\rho_1 \rho_2 \sin(d_1 - d_2) = e^2 \sin d \cos d, \quad \rho_1 \rho_2 \cos(d_1 - d_2) = (1 - e^2)^{1/2}

after which $\eta_1 = \eta / \rho_1$ , $\zeta_1 = (1 - \xi^2 - \eta_1^2)^{1/2}$ , and the geodetic latitude follows from $\cos\phi_1 \sin\theta = \xi$ , $\cos\phi_1 \cos\theta = -\eta_1 \sin d_1 + \zeta_1 \cos d_1$ , $\sin\phi_1 = \eta_1 \cos d_1 + \zeta_1 \sin d_1$ , $\lambda = \mu - \theta$ , $\tan\phi = (1-e^2)^{-1/2}\tan\phi_1$ , where "for Hayford's spheroid (flattening 1/297) the coefficient $(1-e^2)^{-1/2}$ is equal to 1.003378" ¹. The Supplement is explicit that this is not optional: "Precise eclipse calculations must take into account the flattening of the Earth" ¹. The Swiss Ephemeris avoids the auxiliaries by a different device: "Instead of flattening the earth, we apply the correction to the z coordinate of the moon and the sun", dividing the equatorial $z$ of both bodies by $(1-f)$ so that a sphere of radius $a$ can be used for the intersection ¹¹.

What a wrong figure costs

The three modern ellipsoids are interchangeable at eclipse precision. With $a = 6378137$ m, the polar radius on WGS 84 is 6356752.314 m, on GRS 80 ( $1/f = 298.257222101$ ) it is 0.1 mm larger, and on the IAU 1976 rounding $1/f = 298.257$ it is 16 mm larger (computed here). The 3 m between the IAU 1976 equatorial radius of 6378140 m and the WGS 84 value of 6378137 m is a scale change of $5 \times 10^{-7}$ , which is 3 m at the limb of the Earth and far below the 40 m the Swiss Ephemeris attributes to the JPL ephemeris uncertainty alone ¹¹. A developer can therefore use WGS 84 throughout, which is also the datum of GPS and of the DEMs, without reconciling it to the almanac constants.

A sphere is a different matter. The polar radius is $af = 21.385$ km shorter than the equatorial radius, and the difference $\phi - \phi'$ between geodetic and geocentric latitude reaches $11.5'$ at $\phi = 45^{\circ}$ , which is 21.4 km of arc on the meridian (computed from $\tan\phi' = (1-f)^2\tan\phi$ ). A program that feeds a geodetic latitude into spherical formulas therefore puts the observer up to 21 km north or south of where they stand and up to 21 km too far from the centre. The penumbral edge crosses an observer at roughly 0.5 to 1 km per second, so the first and last contact times are wrong by tens of seconds, and a path limit computed on a sphere is displaced by a distance comparable to the whole width of a narrow total path. The 1961 Supplement allows the flattening to be neglected only for the outline curves drawn on small-scale maps, and repeats after each such derivation that "the flattening of the Earth may be introduced in the same manner" when precision is needed ¹.

Observer height in the fundamental plane

Height enters twice. First, it increases $\rho$ : the 1992 Supplement says of the second fundamental relation for the limits that " $\rho$ is usually not unity; it contains flattening and height above the geoid", and in the section on differential corrections, "For calculation of circumstances at elevations above the spheroid, the assumed radius of the Earth is increased accordingly and the calculations repeated" ². Second, the radius of the shadow at the observer depends on $\zeta$ , the observer's distance above the fundamental plane: "The radius ( $L$ ) of the shadow at height $\zeta$ above the fundamental plane is $L = l - \zeta \tan f$ , subscripted for umbra or penumbra, and $L$ may be positive or negative" ². Raising the observer by $h$ raises $\zeta$ by about $h \sin A$ , where $A$ is the Sun's altitude, so the umbral radius grows by $h \sin A \tan f_2$ , at most $h \tan f_2$ . With $\tan f_2 = 0.0046450$ for 2024 ²², the umbra at the height of Everest (8848 m) is $2 h \tan f_2 = 82$ m wider than at sea level, which is the "about 80 meters" that timeanddate quotes ⁵.

The large effect is not the widening but the shift. A raised observer sees the Sun at altitude $A$ , and the shadow boundary, which is a surface inclined at that altitude, intersects a plane $h$ higher at a point displaced horizontally by $h \cot A$ toward the Sun's azimuth. Only the component perpendicular to the path limit moves the observer into or out of the umbra. Espenak's bulletins give this as a multiplicative elevation factor: "The elevation factor is the product, $\tan(90 - A) \cdot \sin(D)$ , where $A$ is the altitude of the Sun and $D$ is the difference between the azimuth of the Sun and the azimuth of the limit line, with the sign selected to be positive if the path should be shifted north with positive elevations above sea level", with the worked example "if one's elevation is 1000 meters above sea level and the Elev Fact value is +0.20, then the shift is +200m" ⁴. Writing this out:

\Delta_\perp = h \cot A \sin D

gives, per 1000 m of elevation with the Sun square to the path ( $D = 90^{\circ}$ ): 577 m at $A = 60^{\circ}$ , 1000 m at $45^{\circ}$ , 1732 m at $30^{\circ}$ , 2747 m at $20^{\circ}$ and 5671 m at $10^{\circ}$ (computed here). The Photo Ephemeris technical note summarises the same physics as "typically of the order of ~500 m per 1,000 m of elevation", which corresponds to a Sun near 60° ¹⁰. timeanddate states the consequence for its sea-level map: "if the Sun appears in the eastern sky, the shadow's actual position will be slightly farther east. The higher your altitude and the lower the Sun's position above the horizon, the larger the difference", and bounds it at "up to 10 kilometers (about 6 miles) in extreme cases, when the observer is at a high altitude and the Sun is low in the sky" ⁵. Espenak's 2001 bulletin makes the weaker statement that "the elevation does not play a significant role in the predictions unless the location is near the umbral path limits and the Sun's altitude is relatively small (<10°)", which is true for the timing of mid-eclipse at a site well inside the path and not true for the limits ⁴.

What NASA SVS did with terrain in 2017 and 2024

The full SVS constants and dataset lists are in SVS products and data; this section keeps the rows this topic owns, which are the ellipsoid, the geoid and the terrain model.

SVS computed the 2017 umbra twice. The first product ignored terrain and stated so: "Both the elevations of locations on the Earth and the irregular limb of the Moon were ignored. The resulting small errors mostly affect the totality duration calculation, but they tend to cancel out: elevations above sea level slightly lengthen totality, while valleys along the lunar limb slightly shorten it" ¹⁹. The later animation and map replaced that with a full treatment: "In keeping with their paper and pencil origins, traditional eclipse calculations pretend that all observers are at sea level and that the Moon is a smooth sphere centered on its center of mass ... This animation shows the umbra and its path in a new way. Elevations on the Earth's surface and the irregular lunar limb (the silhouette edge of the Moon's disk) are both fully accounted for" ³. The entries in that page's constants block that this topic owns are "Earth radius 6378.137 km, Ellipsoid WGS84, Geoid EGM96", and the dataset list names "DEM [SRTM: SIR-C]", "DEM [LRO: LOLA]" and "SLDEM2015 [LRO/SELENE: LOLA/TC]" ³. The rest of the block, with the lunar and solar radii, the ephemeris, the Earth-orientation kernel and the ΔT, is transcribed in SVS products and data.

The method page states where the elevations come from and what they do: "Observer elevations are taken from SRTM, a digital elevation map of the Earth based on radar data collected during the February, 2000 flight of the Space Shuttle Endeavor. As illustrated by the following cartoon, higher elevations can lift the observer either into or out of the umbra cone. The overall effect is to shift the umbra toward the Sun." The quantitative result for 2017: "As these edges pass over mountain ranges (for the 2017 eclipse, the Cascades, Rockies, and Appalachians), they are scalloped by the peaks and valleys of the landscape. The higher elevations in the western states in 2017 also shift the umbra toward the southeast (in the direction of the Sun's azimuth) by as much as 3 kilometers" ⁶. The 2017 shapefile page confirms "Elevations are from SRTM ... Lunar topography, used for precise shadow calculations, is from NASA LRO laser altimetry and JAXA Kaguya stereo imaging. Planetary positions are from the JPL DE421 ephemeris. The lunar limb profile and eclipse calculations are by the visualizer" ²³.

For 2023 and 2024 the method statement is the same: "The eclipse data were calculated by visualizer Ernie Wright using elevation information from SRTM, lunar topography from LRO, and planetary positions from the JPL DE421 ephemeris", and the 2024 path page says the umbra "were calculated in a way that accounts for both elevations on the Earth's surface and the irregular lunar limb" ⁷ ²⁴ ²⁵. Every SVS product page names DE421, while the paper's appendix uses DE440, as set out in Wright and Young 2024. The difference is under a metre at the Moon. The 2024 pages do not print a constants block, so the ΔT used, the SRTM resolution and the geoid model for 2024 are not documented there. The listed dataset is SRTM from the SIR-C instrument, which does not distinguish the 1 arc-second from the 3 arc-second product. A 3 km shift cannot be resolved by SRTM's own errors: the 1 arc-second product has a true resolution of "50 to 80 m" and its heights are on the EGM96 geoid ¹³. The presence of "Geoid EGM96" in the 2017 constants shows that SVS converted SRTM orthometric heights to ellipsoidal heights before placing observers on WGS 84, which is the correct treatment and the only documented one among the predictors surveyed.

What the other predictors do

Espenak's NASA and EclipseWise products are sea-level products. The 2001 bulletin states that local circumstances use "the location's elevation (meters) above sea-level, if known. If the elevation is unknown (i.e., not in the data base), then the local circumstances for that location are calculated at sea-level", and that the graze-zone tables are "at sea level" with the elevation factor supplied for manual correction ⁴. The NASA and EclipseWise pages for 2017 and 2024 say nothing about elevation ²⁶ ²⁷ ²².

Jubier's local-circumstances calculator takes "Altitude: meters" as an input next to latitude and longitude and draws the Sun and Moon "(astronomical horizon, no atmospheric refraction)" ⁹. The archived HTML of his 2024 interactive map is JavaScript only and carries no method text, but the calculator's change log records "Added elevation profile at click location", so the map does assign an elevation to a clicked point. The Earth model in that code is WGS 84 with an equatorial radius of 6378137.0 m. Both are read from the source in commercial and institutional tools ²⁸. The Photo Ephemeris note describes the practice of a comparable application: observer elevation "is obtained from the elevation data source in the app (e.g. SRTM3, AsterGDEM, Google Elevation), which may optionally be adjusted by a user-provided offset", while "Eclipse paths assume sea-level observers" ¹⁰.

timeanddate: "Our calculations do not take elevation into account; they are based on sea level for each location" ⁵.

Occult refers site coordinates to the GPS datum and takes a site altitude. A graze listing in the tutorial states: "Path coordinates are referred to WGS84 (as used by GPS), with the nominal site altitude being referenced to Mean Sea Level" ¹². Whether its solar-eclipse limit maps are drawn at sea level or with a DEM is not documented on the pages read ²⁹.

The Swiss Ephemeris eclipse functions take a height in metres as the third element of geopos and reject values outside "-500 and 25000 m above sea", while the global functions are "referred to sea level / the mean ellipsoid" ¹¹ ³⁰. Its own error budget for the central line lists "from deviation of the geoid from the ellipsoid: a few meters" ¹¹. That estimate is for the intersection of the shadow axis with the surface, where a 100 m radial error moves the point by $100 \cot A$ m, and it is not an estimate for an observer at 2000 m near a limit.

Elevation datasets

Dataset	Posting	Coverage	Vertical datum	Stated vertical accuracy	Licence and access
SRTM GL1 v003 (SRTMGL1)	1" (30 m), true resolution 50 to 80 m	60°N to 56°S	WGS84/EGM96 geoid	Not stated in the catalogue; the guide gives no global figure	Open, DOI 10.5067/MEASURES/SRTM/SRTMGL1.003, LP DAAC ³¹ ¹³
SRTM GL3 v003 (SRTMGL3)	3" (90 m), 3×3 average of GL1	60°N to 56°S	WGS84/EGM96 geoid	as above	Open, DOI 10.5067/MEaSUREs/SRTM/SRTMGL3.003 ³²
NASADEM HGT v001	1" (30 m)	60°N to 56°S	Geoid (SRTM reprocessed, ICESat GLAS control)	Not stated on the catalogue page	Open, DOI 10.5067/MEASURES/NASADEM/NASADEM_HGT.001 ³³
ASTER GDEM v003 (ASTGTM)	1" (30 m)	83°N to 83°S	WGS84/EGM96 geoid	Not stated; "may contain anomalies and artifacts"	Open, DOI 10.5067/ASTER/ASTGTM.003 ¹⁴
Copernicus DEM GLO-30, GLO-90	30 m, 90 m	Global	EGM2008	"< 4m (90% linear error)"	Free with attribution, DOI 10.5270/ESA-c5d3d65; also s3://copernicus-dem-30m ¹⁶ ³⁴
GMTED2010	30", 15", 7.5"	84°N to 56°S	EGM96 geoid	Aggregated global RMSE 25 to 41 m at 30", about 10 to 16 m at 7.5"	Public domain, USGS ¹⁵
ETOPO 2022	15", 30", 60"	Global, with bathymetry	Ice-surface and bedrock versions; a geoid-height GeoTIFF is supplied	Not stated on the product page	NOAA NCEI, DOI 10.25921/fd45-gt74 ³⁵

The SRTM guide is the authority for the datum: "The unit of elevation is meters as referenced to the WGS84/EGM96 geoid" ¹³. GMTED2010 was built on the same surface, and its report states that source datasets on the WGS 84 ellipsoid "were converted to the EGM96" geoid before merging ¹⁵. The Copernicus DEM uses "EGM2008" as its vertical reference and was acquired "by the TanDEM-X mission between 2011 and 2015" ¹⁶. For an eclipse product the Copernicus GLO-30 is the best-documented free choice: global, 30 m, under 4 m at 90 percent, with a stated licence. SRTM 1" is what SVS used and is adequate where its 50 to 80 m resolution and unstated global accuracy are acceptable, which at the path limits means an elevation error of roughly 10 m producing under 20 m of limit shift at a 30° Sun.

The geoid question

The formulas want the height above the ellipsoid, because $\rho$ is the distance from the geocentre in units of $a$ and the ellipsoid is the reference for $S$ and $C$ . Every dataset in the table gives the orthometric height $H$ above a geoid. The ellipsoidal height is $h = H + N$ , where $N$ is the geoid undulation. The undulation "ranges from +85 m (Iceland) to −106 m (southern India)" ¹⁷. EGM96 is "provided as a set of normalized, geopotential coefficients to degree and order 360" with "a 15-minute worldwide geoid height file", and EGM2008 "complete to degree and order 2159" with "a 2.5-minute worldwide geoid height file" ¹⁸; the difference between the two models is far below a metre for this purpose, and EGM2008 has "an accuracy approaching 10 cm" ³⁶.

The 1992 Supplement itself blurs the two heights, writing "height above the geoid" for the quantity inside $\rho$ ², and Espenak's tables list "elevation (meters) above sea-level" ⁴. Treating $H$ as $h$ introduces an error of $N$ in the observer's radial position, which by the elevation-factor relation moves the observer relative to a limit by $N \cot A \sin D$ : 173 m for $N = 100$ m at $A = 30^{\circ}$ , and 58 m at $A = 60^{\circ}$ . Over the 2024 path in North America $N$ is about $-30$ m, so the omission is of the order of 50 m there. It is a bias, not noise, and it is trivially removed by adding the EGM96 or EGM2008 undulation from the NGA grids. SVS's constants list "Geoid EGM96" and are the only evidence found of a predictor doing this ³. No statement about a geoid correction was found on the Espenak, Jubier, timeanddate, Occult or Photo Ephemeris pages read, and the Swiss Ephemeris treats the geoid only as a source of error in its comment block ¹¹.

Sources compared

Source	Figure of the Earth	Height treatment	Terrain data	Geoid	What it uniquely provides
Explanatory Supplement 1961 ¹	Hayford $a = 6378.388$ km, $f = 1/297$ ; $1/298.25$ from 1968	Sea level in $S$ , $C$ ; flattening "must" be included	none	none	The complete Bessel parametric-latitude algorithm with constants
Explanatory Supplement 1992 ²	IAU 1976 $a = 6378140$ m, $f = 1/298.257$ ; MERIT $a = 6378.137$ km	$(aC + h)$ , $(aS + h)$ ; $L = l - \zeta \tan f$	none	Calls $h$ "height above the geoid"	Height in the coordinate formulas and in the shadow radius
Espenak NASA bulletins and pages ⁴ ⁸	Not stated	Sea level; elevation factor $\tan(90-A)\sin D$ for manual shifts	City elevation tables	none	The elevation factor and its sign convention
NASA SVS 2017 and 2024 ³ ⁶ ⁷	WGS 84	Every observer at terrain height	SRTM (SIR-C)	EGM96	Terrain-aware umbra polygons; 3 km shift measured for 2017
Jubier calculator ⁹	WGS 84, read from the calculator's JavaScript, see commercial and institutional tools	Altitude input per site, with an elevation profile at the clicked point	Not named	Not documented	Limb correction LC at contacts
timeanddate ⁵	"slightly squashed" ellipsoid	Sea level, stated	none	none	Honest bound: up to 10 km at high altitude and low Sun
Occult ¹²	WGS 84	Site altitude above mean sea level	Not documented	none stated	GPS-datum path coordinates
Swiss Ephemeris ¹¹ ²⁰	$a = 6378136.6$ m, $f = 1/298.25642$ ; eclipse scale 6378140 m	Height input, −500 to 25000 m; global paths at sea level	none	Listed as "a few meters" of error	z-scaling device instead of Bessel's auxiliaries; an explicit error budget
Photo Ephemeris ¹⁰	Not stated	Site height from DEM; paths at sea level	SRTM3, ASTER GDEM, Google Elevation	Not stated	Practical rule of about 500 m per 1000 m

What a developer should do

Use WGS 84 ( $a = 6378137$ m, $1/f = 298.257223563$ ) for the observer and express all Besselian lengths in units of that $a$ . Do not spend effort reconciling it to the IAU 1976 figure: the differences are millimetres to metres ¹⁸ ².
Place the observer with $\rho \sin\phi' = (S + h/a)\sin\phi$ , $\rho \cos\phi' = (C + h/a)\cos\phi$ , $C = (\cos^2\phi + (1-f)^2\sin^2\phi)^{-1/2}$ , $S = (1-f)^2 C$ , and never with spherical formulas ² ¹.
Take $h$ as ellipsoidal height. Look up $H$ in the Copernicus DEM GLO-30 (EGM2008) or SRTM GL1 (EGM96) and add the undulation $N$ from the matching NGA geoid grid ¹⁶ ¹³ ¹⁸.
For the umbral limits, compute the intersection of the shadow surface with the terrain, as SVS does, rather than shifting a sea-level limit by $h \cot A \sin D$ . Keep the elevation factor as a check: at 2000 m and a 45° Sun it predicts 2 km, which is the order SVS found ⁶ ⁴.
Use $L = l - \zeta \tan f$ with the observer's actual $\zeta$ so that the duration includes the small widening from height ².

Read first: Section 9B of the 1961 Supplement for the algorithm, Sections 3.244 and 8.34 to 8.36 of the 1992 Supplement for the height formulas, and SVS 4517 for the only public description of a terrain-aware umbra.

What this changes

The pipeline needs a terrain stage between the global Besselian solution and the published path polygons: a DEM lookup with a geoid conversion, and an intersection of the umbral cone with the terrain surface rather than with the ellipsoid. It also needs a stated datum for every published coordinate. Nothing changes in the Besselian-element stage itself, since the elements are independent of the observer.

Open questions

Obtain the SRTM product version (GL1 or GL3) and the sampling used by SVS for 2024, and the ΔT it used. Neither appears on SVS 5073, 5123 or 5219; a constants block like the one on SVS 4515 is needed, possibly in the shapefile metadata of SVS 5073 ⁷.
Obtain the elevation service behind Jubier's interactive maps. That a clicked point is assigned an elevation is settled by the change log quoted in commercial and institutional tools. Which DEM supplies it is not ²⁸.
Obtain Occult's help topic on solar-eclipse limit mapping to learn whether its path limits are at sea level and whether a DEM is used ²⁹.
Obtain the eclipse chapter of the 2013 Explanatory Supplement to confirm that the height formulas are unchanged from 1992 and to see whether it now distinguishes ellipsoidal from orthometric height ³⁷.
Obtain the Copernicus DEM product handbook for the per-continent accuracy tables, to replace the single "< 4 m" figure quoted here ¹⁶.

References

1peer-reviewed Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac (HMSO, 1961) Section 6 (figure of the Earth: Hayford spheroid, the S and C functions) and Section 9B (eclipses: observer coordinates, Bessel's parametric-latitude device, rising and setting curves without refraction). Read from the archive.org OCR text kept in var/downloads/es1961_djvu.txt.
2peer-reviewed Explanatory Supplement to the Astronomical Almanac, P. K. Seidelmann ed. (University Science Books, 1992) Sections 2.553 (Stephenson and Morrison 1984 parabolas), 3.244 (terrestrial coordinates, MERIT 1983 ellipsoid), 3.283 (low-precision refraction, 34 arcmin horizontal refraction), 3.351 and 3.352 (IAU 1976 ellipsoid, GMST), 7.3 (apparent places include aberration), 8.12, 8.342, 8.353, 8.362 and 8.363 (eclipses: apparent places, shadow radius at height, height above the geoid, refraction as a refinement, the delta-T longitude correction). Read from the archive.org OCR text in var/downloads/es1992_djvu.txt.
3primary NASA SVS 4515: 2017 Total Solar Eclipse in the U.S., umbra animation with terrain and limb (E. Wright) Read from the Wayback Machine snapshot of 2026-01-14. Lists Earth radius 6378.137 km, Ellipsoid WGS84, Geoid EGM96, DE421, SPICE earth orientation kernel, Delta UTC 69.184 s and delta-T 68.917 s, DEM SRTM (SIR-C), lunar DEMs LOLA and SLDEM2015.
4primary Total Solar Eclipse of 2001 June 21, F. Espenak and J. Anderson, NASA/TP-1999-209484 Read from the PDF text kept in var/downloads/TP209484_2001.txt. States that predictions use centre-of-mass positions with no refraction or limb corrections, that local circumstances are at sea level unless the elevation is known, and defines the elevation factor tan(90-A) sin(D) for shifting the path limits.
5company How Accurate Are Eclipse Predictions? (Accuracy of Eclipse Times), K. Bikos, timeanddate.com Read from the Wayback Machine snapshot of 2025-12-24 (the live page returned HTTP 403). Calculations are at sea level, the umbra at Everest is only about 80 m wider, the path position may be off by up to 10 km at high altitude with a low Sun, rise and set times assume a flat horizon.
6primary NASA SVS 4517: Umbra Shapes (E. Wright) Read from the Wayback Machine snapshot of 2025-12-10. Explains the limb-profile point-cloud method, states that observer elevations come from SRTM, and that the western-US elevations in 2017 shift the umbra toward the Sun's azimuth by as much as 3 km.
7primary NASA SVS 5073: The 2023 and 2024 Solar Eclipses, Map and Data (E. Wright, M. Garrison) Read from the Wayback Machine snapshot of 2026-01-02. Eclipse data calculated with elevation from SRTM, lunar topography from LRO, DE421. Shapefile list (umbra polygons at 1 s and 10 s). No delta-T value is printed on the page.
8primary Explanation of Solar Eclipse Predictions (NASA Eclipse Web Site, F. Espenak) Read. Predictions use the best delta-T available at preparation and centre-of-mass positions with no refraction; limb corrections to central durations are given separately.
9company Solar Eclipse Calculator and Diagram, X. Jubier Read from the Wayback Machine snapshot of 2026-02-08 (xjubier.free.fr refused connections). Takes latitude, longitude and altitude in metres, shows the Sun-Moon diagram for the astronomical horizon with no refraction, prints the delta-T used per eclipse, and applies a lunar-limb correction LC to second and third contact.
10company Technical Note: Solar Eclipse Functionality, The Photographer's Ephemeris Read. Based on Meeus (Astronomical Algorithms and Elements of Solar Eclipses), the Astronomical Almanac 2023 and the 2013 Explanatory Supplement; delta-T from USNO data and predictions (2017 changed from 70.3 s to 68.8373 s); elevation from SRTM3, ASTER GDEM or Google Elevation; paths assume sea level with about 500 m of shift per 1000 m of elevation; simulator refraction from the US Standard Atmosphere.
11company Swiss Ephemeris source, swecl.c (eclipse functions), Astrodienst Read from the local copy var/downloads/swecl.c. Central-line algorithm from Montenbruck, refraction considered only for maxima of partial and non-central eclipses, positions referred to sea level and the mean ellipsoid, a listed error budget (JPL 40 m, refraction, geoid and polar motion a few metres each, under 100 m except near the horizon), the z-scaling device for oblateness, observer altitude limits -500 to 25000 m, a horizon visibility criterion with 34.4556 arcmin of refraction plus dip.
12company Occult tutorial (grazing occultation examples), PDF distributed with Occult Read from the local copy var/downloads/OccultTutorial.pdf. A graze prediction listing states that path coordinates are referred to WGS84 with the site altitude referenced to mean sea level and that the path is adjusted for refraction at low Moon altitudes.
13primary SRTM Version 3.0 User Guide (NASA JPL and LP DAAC) Read from the PDF (var/downloads/srtm_guide.txt). Heights are metres referenced to the WGS84/EGM96 geoid; the true spatial resolution of the 1 arc-second data is 50 to 80 m; the LP DAAC 3 arc-second product averages 3x3 posts; voids are filled with ASTER GDEM2, GMTED2010 and NED.
14primary ASTER Global Digital Elevation Model V003 (ASTGTM), NASA Earthdata catalog Read. 30 m posts, 83 N to 83 S, WGS84/EGM96 geoid, DOI 10.5067/ASTER/ASTGTM.003, open.
15primary Global Multi-resolution Terrain Elevation Data 2010 (GMTED2010), J. J. Danielson and D. B. Gesch, USGS Open-File Report 2011-1073 Read from the PDF (var/downloads/gmted2010.txt). 30, 15 and 7.5 arc-second products, heights on the EGM96 geoid, aggregated global RMSE 25 to 41 m at 30 arc-seconds and about 10 to 16 m at 7.5 arc-seconds against 1.5 million NGA control points, public domain.
16primary Copernicus DEM (COP-DEM) collection description, Copernicus Data Space Ecosystem Read. EEA-10, GLO-30 and GLO-90 from TanDEM-X 2011 to 2015 (WorldDEM), vertical reference EGM2008, absolute vertical accuracy under 4 m at 90 percent linear error, GLO-30 and GLO-90 free with attribution, DOI 10.5270/ESA-c5d3d65.
17unsourced Geoid (Wikipedia) Read. Used only for the range of geoid undulation, +85 m (Iceland) to -106 m (southern India). Encyclopaedia, so graded unsourced.
18primary World Geodetic System 1984 (NGA Office of Geomatics) Read. Defining parameters a = 6378137.0 m, 1/f = 298.257223563, omega = 7292115 x 10^-11 rad/s, GM = 3.986004418 x 10^14 m^3/s2; EGM2008 to degree 2159 with a 2.5-minute geoid grid, EGM96 to degree 360 with a 15-minute grid.
19primary NASA SVS 4314: 2017 eclipse shadow cones and umbra, visualisation constants (E. Wright) Read from the Wayback Machine snapshot of 2025-12-19 (svs.gsfc.nasa.gov refused connections). Lists Earth radius 6378.137 km, flattening 1/298.257, Moon radius 1737.4 km (k = 0.2723993), Sun radius 696,000 km, DE421, SPICE earth_070425_370426_predict.bpc, Delta UTC 68.184 s, and states that elevations and the lunar limb were ignored in that product.
20company Swiss Ephemeris header, swephexp.h (SE_TIDAL constants), Astrodienst Read (downloaded 2026-09-15). Tidal accelerations per ephemeris: DE200 -23.8946, DE403 and DE404 -25.580, DE405 and DE406 -25.826, DE421 and DE422 -25.85, DE430 -25.82, DE431 -25.80, DE441 -25.936, SE_TIDAL_26 = -26.0, default DE431.
21primary Besselian Elements for the Total Solar Eclipse of 2024 Apr 08 (NASA Eclipse Web Site) Read. Delta-T = 70.6 s, VSOP87/ELP2000-85, k1 = 0.272488, k2 = 0.272281, centre-of-mass lunar coordinates.
22company Total Solar Eclipse of 2024 Apr 08, EclipseWise prime page (F. Espenak) Read. Delta-T = 71.5 s, JPL DE405, k penumbra 0.2725076, k umbra 0.2722810, tan f1 = 0.0046683, tan f2 = 0.0046450.
23primary NASA SVS 4518: 2017 Total Solar Eclipse Map and Shapefiles (E. Wright) Read from the Wayback Machine snapshot of 2025-12-20. Elevations from SRTM, lunar topography from LRO LOLA and Kaguya, DE421, limb profile and eclipse calculations by the visualiser.
24primary NASA SVS 5123: The 2024 Total Solar Eclipse (E. Wright, M. Garrison) Read from the Wayback Machine snapshot of 2025-12-09. Same method statement as SVS 5073: SRTM, LRO, DE421. No delta-T value printed.
25primary NASA SVS 5219: 2024 Path of Totality (E. Wright) Read from the Wayback Machine snapshot of 2025-12-12. States that the 2024 umbra and path account for Earth elevations and the irregular lunar limb, linking to SVS 4517 for the method.
26primary Total Solar Eclipse of 2017 Aug 21, Google Maps page (NASA Eclipse Web Site, F. Espenak) Read. States JPL DE405, delta-T = 68.4 s, no lunar limb corrections, and the 1 to 3 km and 1 to 3 s limb effect.
27primary Total Solar Eclipse of 2024 Apr 08, Google Maps page (NASA Eclipse Web Site, F. Espenak) Read. States VSOP87/ELP2000-85 ephemerides, delta-T = 70.6 s, no lunar limb corrections, zoom limited to about 0.7 km per cm.
28company Mexico - USA - 2024 April 8 Total Solar Eclipse - Interactive Google Map, X. Jubier Wayback Machine snapshot of 2025-12-30 fetched; the page is JavaScript only and the archived HTML carries no method text, so nothing about its delta-T, elevation source or refraction could be read.
29company Occult v4 occultation prediction software, D. Herald Read from the Wayback Machine snapshot of 2025-12-23 (the live site failed TLS). Feature list only: solar and lunar eclipses, 42 downloadable data files. No method statement on delta-T, elevation or refraction.
30company Swiss Ephemeris programming interface (swephprg.htm), Astrodienst Read through a fetch summary. Eclipse functions take UT and a geopos triple (longitude, latitude, height in metres); the search for the next eclipse anywhere on Earth is independent of delta-T.
31primary NASA Shuttle Radar Topography Mission Global 1 arc second V003 (SRTMGL1), NASA Earthdata catalog Read. 1 arc-second (30 m) posts, 60 N to 56 S, HGT files, DOI 10.5067/MEASURES/SRTM/SRTMGL1.003, openly shared without restriction under the EOSDIS data use guidance.
32primary NASA Shuttle Radar Topography Mission Global 3 arc second V003 (SRTMGL3), NASA Earthdata catalog Read. 3 arc-second (90 m) product derived from the 1 arc-second data by averaging, DOI 10.5067/MEaSUREs/SRTM/SRTMGL3.003, open.
33primary NASADEM Merged DEM Global 1 arc second V001, NASA Earthdata catalog Read. SRTM reprocessed with ICESat GLAS control, ASTER GDEM2 and AW3D30 void fill, 60 N to 56 S, geoid-referenced, DOI 10.5067/MEASURES/NASADEM/NASADEM_HGT.001, open.
34primary Copernicus DEM on the Registry of Open Data on AWS Read. Buckets s3://copernicus-dem-30m and s3://copernicus-dem-90m (eu-central-1), cloud-optimised GeoTIFFs of the 2021 release, no-sign-request access.
35primary ETOPO Global Relief Model (ETOPO 2022), NOAA NCEI Read. 15, 30 and 60 arc-second grids, ice-surface and bedrock versions, a companion geoid-height GeoTIFF, DOI 10.25921/fd45-gt74.
36unsourced Earth Gravitational Model (Wikipedia) Read. EGM96 degree 360 with a 15-minute grid; EGM2008 degree 2159 with a 2.5-minute grid and accuracy approaching 10 cm. Encyclopaedia, so graded unsourced.
37peer-reviewed Explanatory Supplement to the Astronomical Almanac, 3rd edition, S. E. Urban and P. K. Seidelmann eds. (University Science Books, 2013) Not read for this note. Cited by the Photo Ephemeris technical note as one basis of its eclipse paths. Its eclipse chapter should be checked for changes to the 1992 formulas.