GeographicLib: Gravity models

GeographicLib 2.5

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Gravity models
Back to Geoid height. Forward to Normal gravity. Up to Contents.

GeographicLib can compute the earth's gravitational field with an earth gravity model using the GravityModel and GravityCircle classes and with the Gravity utility. These models expand the gravitational potential of the earth as sum of spherical harmonics. The models also specify a reference ellipsoid, relative to which geoid heights and gravity disturbances are measured. Underlying all these models is normal gravity which is the exact solution for an idealized rotating ellipsoidal body. This is implemented with the NormalGravity class and some notes on are provided in section Normal gravity

Go to

- Installing the gravity models
- The format of the gravity model files
- Comments on the NGA harmonic synthesis code
- Details of the geoid height and anomaly calculations
- The effect of the mass of the atmosphere
- Geoid heights on a multi-processor system
- Normal gravity (now on its own page)

The supported models are

- egm84, the Earth Gravity Model 1984, which includes terms up to degree 180.
- egm96, the Earth Gravity Model 1996, which includes terms up to degree 360.
- egm2008, the Earth Gravity Model 2008, which includes terms up to degree 2190. See also Pavlis et al. (2012)
- wgs84, the WGS84 Reference Ellipsoid. This is just reproduces the normal gravitational field for the reference ellipsoid. This includes the zonal coefficients up to order 20. Usually NormalGravity::WGS84() should be used instead.
- grs80, the GRS80 Reference Ellipsoid. This is just reproduces the normal gravitational field for the GRS80 ellipsoid. This includes the zonal coefficients up to order 20. Usually NormalGravity::GRS80() should be used instead.

See

- W. A. Heiskanen and H. Moritz, Physical Geodesy (Freeman, San Francisco, 1967). https://archive.org/details/HeiskanenMoritz1967PhysicalGeodesy

for more information.

Acknowledgment: I would like to thank Mathieu Peyréga for sharing EGM_Geoid_CalculatorClass from his Geo library with me. His implementation was the first I could easily understand and he and I together worked through some of the issues with overflow and underflow the occur while performing high-degree spherical harmonic sums.

# Installing the gravity models

These gravity models are available for download:

Available gravity models
name | max degree | size (kB) | Download Links (size, kB)
--- | --- | --- | ---
tar file | Windows installer | zip file
egm84 | 180 | 27 | link (26) | link (55) | link (26)
egm96 | 360 | 2100 | link (2100) | link (2300) | link (2100)
egm2008 | 2190 | 76000 | link (75000) | link (72000) | link (73000)
wgs84 | 20 | 1 | link (1) | link (30) | link (1)

The "size" column is the size of the uncompressed data.

For Linux and Unix systems, GeographicLib provides a shell script geographiclib-get-gravity (typically installed in /usr/local/sbin) which automates the process of downloading and installing the gravity models. For example

       geographiclib-get-gravity all  # to install egm84, egm96, egm2008, wgs84
       geographiclib-get-gravity -h   # for help

This script should be run as a user with write access to the installation directory, which is typically /usr/local/share/GeographicLib (this can be overridden with the -p flag), and the data will then be placed in the "gravity" subdirectory.

Windows users should download and run the Windows installers. These will prompt for an installation directory with the default being

       C:/ProgramData/GeographicLib

(which you probably should not change) and the data is installed in the "gravity" sub-directory. (The second directory name is an alternate name that Windows 7 uses for the "Application Data" directory.)

Otherwise download either the tar.bz2 file or the zip file (they have the same contents). To unpack these, run, for example

       mkdir -p /usr/local/share/GeographicLib
       tar xofjC egm96.tar.bz2 /usr/local/share/GeographicLib
       tar xofjC egm2008.tar.bz2 /usr/local/share/GeographicLib
       etc.

and, again, the data will be placed in the "gravity" subdirectory.

However you install the gravity models, all the datasets should be installed in the same directory. GravityModel and Gravity uses a compile time default to locate the datasets. This is

- /usr/local/share/GeographicLib/gravity, for non-Windows systems
- C:/ProgramData/GeographicLib/gravity, for Windows systems

consistent with the examples above. This may be overridden at run-time by defining the GEOGRAPHICLIB_GRAVITY_PATH or the GEOGRAPHICLIB_DATA environment variables; see GravityModel::DefaultGravityPath() for details. Finally, the path may be set using the optional second argument to the GravityModel constructor or with the "-d" flag to Gravity. Supplying the "-h" flag to Gravity reports the default path for gravity models for that utility. The "-v" flag causes Gravity to report the full path name of the data file it uses.

# The format of the gravity model files

The constructor for GravityModel reads a file called NAME.egm which specifies various properties for the gravity model. It then opens a binary file NAME.egm.cof to obtain the coefficients of the spherical harmonic sum.

The first line of the .egm file must consist of "EGMF-v" where EGMF stands for "Earth Gravity Model Format" and v is the version number of the format (currently "1").

The rest of the file is read a line at a time. A # character and everything after it are discarded. If the result is just white space it is discarded. The remaining lines are of the form "KEY WHITESPACE VALUE". In general, the KEY and the VALUE are case-sensitive.

GravityModel only pays attention to the following keywords

- keywords that affect the field calculation, namely:
  - ModelRadius (required), the normalizing radius for the model in meters.
  - ReferenceRadius (required), the equatorial radius a for the reference ellipsoid meters.
  - ModelMass (required), the mass constant GM for the model in meters3/seconds2.
  - ReferenceMass (required), the mass constant GM for the reference ellipsoid in meters3/seconds2.
  - AngularVelocity (required), the angular velocity ω for the model and the reference ellipsoid in rad seconds−1.
  - Flattening, the flattening of the reference ellipsoid; this can be given a fraction, e.g., 1/298.257223563. One of Flattening and DynamicalFormFactor is required.
  - DynamicalFormFactor, the dynamical form factor J2 for the reference ellipsoid. One of Flattening and DynamicalFormFactor is required.
  - HeightOffset (default 0), the constant height offset (meters) added to obtain the geoid height.
  - CorrectionMultiplier (default 1), the multiplier for the "zeta-to-N" correction terms for the geoid height to convert them to meters.
  - Normalization (default full), the normalization used for the associated Legendre functions (full or schmidt).
  - ID (required), 8 printable characters which serve as a signature for the .egm.cof file (they must appear as the first 8 bytes of this file).
  The parameters ModelRadius, ModelMass, and AngularVelocity apply to the gravity model, while ReferenceRadius, ReferenceMass, AngularVelocity, and either Flattening or DynamicalFormFactor characterize the reference ellipsoid. AngularVelocity (because it can be measured precisely) is the same for the gravity model and the reference ellipsoid. ModelRadius is merely a scaling parameter for the gravity model and there's no requirement that it be close to the equatorial radius of the earth (although that's typically how it is chosen). ModelMass and ReferenceMass need not be the same and, indeed, they are slightly difference for egm2008. As a result the disturbing potential T has a 1/r dependence at large distances.
- keywords that store data that the user can query:
  - Name, the name of the model.
  - Description, a more descriptive name of the model.
  - ReleaseDate, when the model was created.
- keywords that are examined to verify that their values are valid:
  - ByteOrder (default little), the order of bytes in the .egm.cof file. Only little endian is supported at present.

Other keywords are ignored.

The coefficient file NAME.egm.cof is a binary file in little endian order. The first 8 bytes of this file must match the ID given in NAME.egm. This is followed by 2 sets of spherical harmonic coefficients. The first of these gives the gravity potential and the second gives the zeta-to-N corrections to the geoid height. The format for each set of coefficients is:

- N, the maximum degree of the sum stored as a 4-byte signed integer. This must satisfy N ≥ −1.
- M, the maximum order of the sum stored as a 4-byte signed integer. This must satisfy N ≥ M ≥ −1.
- Cnm, the coefficients of the cosine coefficients of the sum in column (i.e., m) major order. There are (M + 1) (2N - M + 2) / 2 elements which are stored as IEEE doubles (8 bytes). For example for N = M = 3, there are 10 coefficients arranged as C00, C10, C20, C30, C11, C21, C31, C22, C32, C33.
- Snm, the coefficients of the sine coefficients of the sum in column (i.e., m) major order starting at m = 1. There are M (2N - M + 1) / 2 elements which are stored as IEEE doubles (8 bytes). For example for N = M = 3, there are 6 coefficients arranged as S11, S21, S31, S22, S32, S33.

Although the coefficient file is in little endian order, GeographicLib can read it on big endian machines. It can only be read on machines which store doubles in IEEE format.

As an illustration, here is egm2008.egm:

    EGMF-1
    # An Earth Gravity Model (Format 1) file.  For documentation on the
    # format of this file see
    # https://geographiclib.sourceforge.io/C++/doc/gravity.html#gravityformat
    Name            egm2008
    Publisher       National Geospatial Intelligence Agency
    Description     Earth Gravity Model 2008
    URL             https://earth-info.nga.mil/index.php?dir=wgs84&action=wgs84#tab_egm2008
    ReleaseDate     2008-06-01
    ConversionDate  2011-11-19
    DataVersion     1
    ModelRadius     6378136.3
    ModelMass       3986004.415e8
    AngularVelocity 7292115e-11
    ReferenceRadius 6378137
    ReferenceMass   3986004.418e8
    Flattening      1/298.257223563
    HeightOffset    -0.41

    # Gravitational and correction coefficients taken from
    # EGM2008_to2190_TideFree and Zeta-to-N_to2160_egm2008 from
    # the egm2008 distribution.
    ID              EGM2008A

The code to produce the coefficient files for the wgs84 and grs80 models is

// Write the coefficient files needed for approximating the normal gravity
// field with a GravityModel. WARNING: this creates files, wgs84.egm.cof and
// grs80.egm.cof, in the current directory.
#include <cmath>
#include <fstream>
#include <iostream>
#include <GeographicLib/NormalGravity.hpp>
#include <GeographicLib/Utility.hpp>
using namespace std;
using namespace GeographicLib;
int main() {
try {
Utility::set_digits();
const char* filenames[] = {"wgs84.egm.cof", "grs80.egm.cof"};
const char* ids[] = {"WGS1984A", "GRS1980A"};
for (int grs80 = 0; grs80 < 2; ++grs80) {
ofstream file(filenames[grs80], ios::binary);
Utility::writearray<char, char, false>(file, ids[grs80], 8);
const int N = 20, M = 0,
cnum = (M + 1) * (2 * N - M + 2) / 2; // cnum = N + 1
vector<int> num(2);
num[0] = N; num[1] = M;
Utility::writearray<int, int, false>(file, num);
vector<Math::real> c(cnum, 0);
const NormalGravity& earth(grs80 ? NormalGravity::GRS80() :
NormalGravity::WGS84());
for (int n = 2; n <= N; n += 2)
c[n] = - earth.DynamicalFormFactor(n) / sqrt(Math::real(2*n + 1));
Utility::writearray<double, Math::real, false>(file, c);
num[0] = num[1] = -1;
Utility::writearray<int, int, false>(file, num);
}
}
catch (const exception& e) {
cerr << "Caught exception: " << e.what() << "\n";
return 1;
}
catch (...) {
cerr << "Caught unknown exception\n";
return 1;
}
}
main
int main(int argc, const char *const argv[])
Definition CartConvert.cpp:28
NormalGravity.hpp
Header for GeographicLib::NormalGravity class.
Utility.hpp
Header for GeographicLib::Utility class.
GeographicLib::Math::real
double real
Definition Math.hpp:110
GeographicLib::NormalGravity
The normal gravity of the earth.
Definition NormalGravity.hpp:80
GeographicLib::NormalGravity::WGS84
static const NormalGravity & WGS84()
Definition NormalGravity.cpp:61
GeographicLib::NormalGravity::GRS80
static const NormalGravity & GRS80()
Definition NormalGravity.cpp:69
GeographicLib::Utility::writearray
static void writearray(std::ostream &str, const IntT array[], size_t num)
Definition Utility.hpp:455
GeographicLib::Utility::set_digits
static int set_digits(int ndigits=0)
Definition Utility.cpp:184
GeographicLib
Namespace for GeographicLib.
Definition Accumulator.cpp:12
std
Definition NearestNeighbor.hpp:806

# Comments on the NGA harmonic synthesis code

GravityModel attempts to reproduce the results of NGA's harmonic synthesis code for EGM2008, hsynth_WGS84.f. Listed here are issues that I encountered using the NGA code:

1. A compiler which allocates local variables on the stack produces an executable with just returns NaNs. The problem here is a missing `SAVE` statement in subroutine `latf`.
2. Because the model and references masses for egm2008 differ (by about 1 part in 109), there should be a 1/r contribution to the disturbing potential T. However, this term is set to zero in hsynth_WGS84 (effectively altering the normal potential). This shifts the surface W = U0 outward by about 5 mm. Note too that the reference ellipsoid is no longer a level surface of the effective normal potential.
3. Subroutine `radgrav` computes the ellipsoidal coordinate β incorrectly. This leads to small errors in the deflection of the vertical, ξ and η, when the height above the ellipsoid, h, is non-zero (about 10−7 arcsec at h = 400 km).
4. There are several expressions which will return inaccurate results due to cancellation. For example, subroutine `grs` computes the flattening using f = 1 − sqrt(1 − e2). Much better is to use f = e2/(1 + sqrt(1 − e2)). The expressions for q and q' in `grs` and `radgrav` suffer from similar problems. The resulting errors are tiny (about 50 pm in the geoid height); however, given that's there's essentially no cost to using more accurate expressions, it's preferable to do so.
5. hsynth_WGS84 returns an "undefined" value for xi and eta at the poles. Better would be to return the value obtained by taking the limit lat → ± 90°.

Issues 1–4 have been reported to the authors of hsynth_WGS84. Issue 1 is peculiar to Fortran and is not encountered in C++ code and GravityModel corrects issues 3–5. On issue 2, GravityModel neglects the 1/r term in T in GravityModel::GeoidHeight and GravityModel::SphericalAnomaly in order to produce results which match NGA's for these quantities. On the other hand, GravityModel::Disturbance and GravityModel::T do include this term.

# Details of the geoid height and anomaly calculations

Ideally, the geoid represents a surface of constant gravitational potential which approximates mean sea level. In reality some approximations are taken in determining this surface. The steps taking by GravityModel in computing the geoid height are described here (in the context of EGM2008). This mimics NGA's code hsynth_WGS84 closely because most users of EGM2008 use the gridded data generated by this code (e.g., Geoid) and it is desirable to use a consistent definition of the geoid height.

- The model potential is band limited; the minimum wavelength that is represented is 360°/2160 = 10' (i.e., about 10NM or 18.5km). The maximum degree for the spherical harmonic sum is 2190; however the model was derived using ellipsoidal harmonics of degree and order 2160 and the degree was increased to 2190 in order to capture the ellipsoidal harmonics faithfully with spherical harmonics.
- The 1/r term is omitted from the T (this is issue 2 in Comments on the NGA harmonic synthesis code). This moves the equipotential surface by about 5mm.
- The surface W = U0 is determined by Bruns' formula, which is roughly equivalent to a single iteration of Newton's method. The RMS error in this approximation is about 1.5mm with a maximum error of about 10 mm.
- The model potential is only valid above the earth's surface. A correction therefore needs to be included where the geoid lies beneath the terrain. This is NGA's "zeta-to-N" correction, which is represented by a spherical harmonic sum of degree and order 2160 (and so it misses short wavelength terrain variations). In addition, it entails estimating the isostatic equilibrium of the earth's crust. The correction lies in the range [−5.05 m, 0.05 m], however for 99.9% of the earth's surface the correction is less than 10 mm in magnitude.
- The resulting surface lies above the observed mean sea level, so −0.41m is added to the geoid height. (Better would be to change the potential used to define the geoid; but this would only change the result by about 2mm.)

A useful discussion of the problems with defining a geoid is given by Dru A. Smith in There is no such thing as "The" EGM96 geoid: Subtle points on the use of a global geopotential model, IGeS Bulletin No. 8, International Geoid Service, Milan, Italy, pp. 17–28 (1998).

GravityModel::GeoidHeight reproduces the results of the several NGA codes for harmonic synthesis with the following maximum discrepancies:

- egm84 = 1.1mm. This is probably due to inconsistent parameters for the reference ellipsoid in the NGA code. (In particular, the value of mass constant excludes the atmosphere; however, it's not clear whether the other parameters have been correspondingly adjusted.) Note that geoid heights predicted by egm84 differ from those of more recent gravity models by about 1 meter.
- egm96 = 23nm.
- egm2008 = 78pm. After addressing some of the issues alluded to in issue 4 in Comments on the NGA harmonic synthesis code, the maximum discrepancy becomes 23pm.

The formula for the gravity anomaly vector involves computing gravity and normal gravity at two different points (with the displacement between the points unknown ab initio). Since the gravity anomaly is already a small quantity it is sometimes acceptable to employ approximations that change the quantities by O(f). The NGA code uses the spherical approximation described by Heiskanen and Moritz, Sec. 2-14 and GravityModel::SphericalAnomaly uses the same approximation for compatibility. In this approximation, the gravity disturbance delta = grad T is calculated. Here, T once again excludes the 1/r term (this is issue 2 in Comments on the NGA harmonic synthesis code and is consistent with the computation of the geoid height). Note that delta compares the gravity and the normal gravity at the same point; the gravity anomaly vector is then found by estimating the gradient of the normal gravity in the limit that the earth is spherically symmetric. delta is expressed in spherical coordinates as deltax, deltay, deltaz where, for example, deltaz is the radial component of delta (not the component perpendicular to the ellipsoid) and deltay is similarly slightly different from the usual northerly component. The components of the anomaly are then given by

- gravity anomaly, Dg01 = deltaz − 2T/R, where R distance to the center of the earth;
- northerly component of the deflection of the vertical, xi = − deltay/gamma, where gamma is the magnitude of the normal gravity;
- easterly component of the deflection of the vertical, eta = − deltax/gamma.

NormalGravity computes the normal gravity accurately and avoids issue 3 of Comments on the NGA harmonic synthesis code. Thus while GravityModel::SphericalAnomaly reproduces the results for xi and eta at h = 0, there is a slight discrepancy if h is non-zero.

# The effect of the mass of the atmosphere

All of the supported models use WGS84 for the reference ellipsoid. This has (see TR8350.2, table 3.1)

- a = 6378137 m
- f = 1/298.257223563
- ω = 7292115 × 10−11 rad s−1
- GM = 3986004.418 × 108 m3 s−2.

The value of GM includes the mass of the atmosphere and so strictly only applies above the earth's atmosphere. Near the surface of the earth, the value of g will be less (in absolute value) than the value predicted by these models by about δg = (4πG/g) A = 8.552 × 10−11 A m2/kg, where G is the gravitational constant, g is the earth's gravity, and A is the pressure of the atmosphere. At sea level we have A = 101.3 kPa, and δg = 8.7 × 10−6m s−2, approximately. (In other words the effect is about 1 part in a million; by way of comparison, buoyancy effects are about 100 times larger.)

# Geoid heights on a multi-processor system

The egm2008 model includes many terms (over 2 million spherical harmonics). For that reason computations using this model may be slow; for example it takes about 78 ms to compute the geoid height at a single point. There are two ways to speed up this computation:

- Use a GravityCircle to compute the geoid height at several points on a circle of latitude. This reduces the cost per point to about 92 μs (a reduction by a factor of over 800).
- Compute the values on several circles of latitude in parallel. One of the simplest ways of doing this is with OpenMP; on an 8-processor system, this can speed up the computation by another factor of 8.

Both of these techniques are illustrated by the following code, which computes a table of geoid heights on a regular grid and writes on the result in a .gtx file. On an 8-processor Intel 3.0 GHz machine using OpenMP (-DHAVE_OPENMP=1), it takes about 10 minutes of elapsed time to compute the geoid height for EGM2008 on a 1' grid. (Without these optimizations, the computation would have taken about 50 days!)

// Write out a gtx file of geoid heights above the ellipsoid. For egm2008 at
// 1' resolution this takes about 10 mins on a 8-processor Intel 3.0 GHz
// machine using OpenMP.
//
// For the format of gtx files, see
// https://vdatum.noaa.gov/docs/gtx_info.html#dev_gtx_binary
//
// data is binary big-endian:
// south latitude edge (degrees double)
// west longitude edge (degrees double)
// delta latitude (degrees double)
// delta longitude (degrees double)
// nlat = number of latitude rows (integer)
// nlong = number of longitude columns (integer)
// nlat * nlong geoid heights (meters float)
#include <vector>
#include <iostream>
#include <fstream>
#include <string>
#include <algorithm>
#if defined(_OPENMP)
#define HAVE_OPENMP 1
#else
#define HAVE_OPENMP 0
#endif
#if HAVE_OPENMP
# include <omp.h>
#endif
#include <GeographicLib/GravityModel.hpp>
#include <GeographicLib/GravityCircle.hpp>
#include <GeographicLib/Utility.hpp>
using namespace std;
using namespace GeographicLib;
int main(int argc, const char* const argv[]) {
// Hardwired for 3 args:
// 1 = the gravity model (e.g., egm2008)
// 2 = intervals per degree
// 3 = output GTX file
if (argc != 4) {
cerr << "Usage: " << argv[0]
<< " gravity-model intervals-per-degree output.gtx\n";
return 1;
}
try {
// Will need to set the precision for each thread, so save return value
int ndigits = Utility::set_digits();
string model(argv[1]);
// Number of intervals per degree
int ndeg = Utility::val<int>(string(argv[2]));
string filename(argv[3]);
GravityModel g(model);
int
nlat = Math::hd * ndeg + 1,
nlon = Math::td * ndeg;
Math::real
delta = 1 / Math::real(ndeg), // Grid spacing
latorg = -Math::qd,
lonorg = -Math::hd;
// Write results as floats in binary mode
ofstream file(filename.c_str(), ios::binary);
// Write header
{
Math::real transform[] = {latorg, lonorg,
1/Math::real(ndeg), 1/Math::real(ndeg)};
unsigned sizes[] = {unsigned(nlat), unsigned(nlon)};
Utility::writearray<double, Math::real, true>(file, transform, 4);
Utility::writearray<unsigned, unsigned, true>(file, sizes, 2);
}
// Compute and store results for nbatch latitudes at a time
const int nbatch = 64;
vector< vector<float> > N(nbatch, vector<float>(nlon));
for (int ilat0 = 0; ilat0 < nlat; ilat0 += nbatch) { // Loop over batches
int nlat0 = min(nlat, ilat0 + nbatch);
#if HAVE_OPENMP
# pragma omp parallel for
#endif
for (int ilat = ilat0; ilat < nlat0; ++ilat) { // Loop over latitudes
Utility::set_digits(ndigits); // Set the precision
Math::real lat = (latorg * ndeg + ilat) / ndeg, h = 0;
GravityCircle c(g.Circle(lat, h, GravityModel::GEOID_HEIGHT));
for (int ilon = 0; ilon < nlon; ++ilon) { // Loop over longitudes
Math::real lon = (lonorg * ndeg + ilon) / ndeg;
N[ilat - ilat0][ilon] = float(c.GeoidHeight(lon));
} // longitude loop
} // latitude loop -- end of parallel section
for (int ilat = ilat0; ilat < nlat0; ++ilat) // write out data
Utility::writearray<float, float, true>(file, N[ilat - ilat0]);
} // batch loop
}
catch (const exception& e) {
cerr << "Caught exception: " << e.what() << "\n";
return 1;
}
catch (...) {
cerr << "Caught unknown exception\n";
return 1;
}
}
GravityCircle.hpp
Header for GeographicLib::GravityCircle class.
GravityModel.hpp
Header for GeographicLib::GravityModel class.
GeographicLib::GravityCircle
Gravity on a circle of latitude.
Definition GravityCircle.hpp:41
GeographicLib::GravityModel
Model of the earth's gravity field.
Definition GravityModel.hpp:83
GeographicLib::GravityModel::GEOID_HEIGHT
@ GEOID_HEIGHT
Definition GravityModel.hpp:155
GeographicLib::Math::qd
static constexpr int qd
degrees per quarter turn
Definition Math.hpp:145
GeographicLib::Math::td
static constexpr int td
degrees per turn
Definition Math.hpp:149
GeographicLib::Math::hd
static constexpr int hd
degrees per half turn
Definition Math.hpp:148
GeographicLib::Utility::val
static T val(const std::string &s)
Definition Utility.hpp:225

cmake will add in support for OpenMP for `examples/GeoidToGTX.cpp`, if it is available.

Back to Geoid height. Forward to Normal gravity. Up to Contents.