charm_gfm#

Module for spectral gravity forward modelling. Offers:

global and spatially limited spectral gravity forward modelling of band-limited topographic masses having constant, lateral and 3D varying mass density.

Note

This documentation is written for double precision version of CHarm.

Note

Spatially limited integration radius is available only if --enable-mpfr is specified when calling ./configure during the installation. See Installing for further details.

Symbolic constants representing types of truncation coefficients

Note

The symbolic constants that follow are available only if --enable-mpfr is specified when calling ./configure during the installation. See Installing for further details.

CHARM_GFM_Q00 0U#: Truncation coefficients \(Q^{0,0,j}_{npi}(r, \psi_0, R)\) defined by Eq. (88) of Bucha (2025).

CHARM_GFM_Q10 10U#: Truncation coefficients \(Q^{1, 0, j}_{npi}(r, \psi_0, R)\) defined by Eq. (88) of Bucha (2025).

CHARM_GFM_Q11 11U#: Truncation coefficients \(Q^{1, 1, j}_{npi}(r, \psi_0, R)\) defined by Eq. (88) of Bucha (2025).

CHARM_GFM_Q20 20U#: Truncation coefficients \(Q^{2, 0, j}_{npi}(r, \psi_0, R)\) defined by Eq. (88) of Bucha (2025).

CHARM_GFM_Q21 21U#: Truncation coefficients \(Q^{2, 1, j}_{npi}(r, \psi_0, R)\) defined by Eq. (88) of Bucha (2025).

CHARM_GFM_Q22 22U#: Truncation coefficients \(Q^{2, 2, j}_{npi}(r, \psi_0, R)\) defined by Eq. (88) of Bucha (2025).

CHARM_GFM_NEAR_ZONE 1#: Symbolic constant that can be used to forward model near-zone masses in the charm_gfm_cap_density_* functions. Near-zone masses are defined as masses up to some spherical distance \(\psi_0\) from evaluation points.

CHARM_GFM_FAR_ZONE (-1)#: Symbolic constant that can be used to forward model far-zone masses in the charm_gfm_cap_density_* functions. Far-zone masses are defined as masses beyond some spherical distance \(\psi_0\) from evaluation points.

Global spectral gravity forward modelling

These functions perform spectral gravity forward modelling using full integration radius, that is, masses all around the body are forward modelled. The density may be constant, laterally variable or 3D variable.

void charm_gfm_global_density_3d(const charm_shc *shape_shcs, unsigned long shape_nmax, double shape_radius_ref, charm_shc **density_shcs, unsigned long *density_nmax, unsigned density_order, double grav_const, double mass, unsigned shape_power_min, unsigned shape_power_max, unsigned long potential_shcs_nmax, const char *shape_density_shcs_path, const char *potential_shcs_path, const char *shcs_file_format, charm_shc *potential_shcs, charm_err *err)#

Performs global spectral gravity forward modelling of 3D mass densities.

Assume a body having a star-shaped surface expandable into surface spherical harmonics. Let the body’s density vary in spherical coordinates \(r', \varphi', \lambda'\) as a finite-degree polynomial in the radial direction as

\[\rho(r', \varphi', \lambda') = \sum_{i = 0}^I \rho_i(\varphi', \lambda') \, \left( r' \right)^i{,}\]

where \(\rho_i(\varphi', \lambda')\) are laterally varying polynomial coefficients of the density that are expandable into surface spherical harmonics. From now on, we assume only those masses that are found between some internal sphere and the surface of the body. This function computes spherical harmonic coefficients of the gravitational potential implied by such mass distributions.

The routine has several input parameters, so it is useful to group them into a few categories:

parameters defining the shape of the gravitating body (shape_shcs, shape_nmax, shape_radius_ref),
parameters defining the density of the gravitating body (density_shcs, density_nmax, density_order),
parameters related to outputs (shape_density_shcs_path, potential_shcs_path, shcs_file_format, potential_shcs, err),
parameters related to the method (shape_power_min, shape_power_max, grav_const, mass, potential_shcs_nmax).

Each parameter is described in detail below.

References:

Bucha B (2025) Spectral gravity forward modelling of 3D variable densities using an arbitrary integration radius with application to lunar topographic masses. Journal of Geodesy, doi: 10.1007/s00190-025-01951-9.

Parameters:

shape_shcs – [in] Spherical harmonic coefficients defining the shape of the gravitating body (e.g., planetary topography), that is, the upper integration limit in the radial direction. It must hold that (i) surface spherical harmonic synthesis of coefficients from shape_shcs gives the full spherical radius of surface points in metres, (ii) shape_shcs->mu == 1.0 and (iii)shape_shcs->r == 1.0.
shape_nmax – [in] Maximum harmonic degree to take when defining the shape of the gravitating body by the spherical harmonic coefficients in shape_shcs. The value cannot be larger than shape_shcs->nmax.
shape_radius_ref – [in] Radius in metres defining the lower integration limit in the radial direction. Forward modelled will be the masses between the sphere of radius shape_radius_ref and the body’s surface defined by shape_shcs. The value must be positive.
density_shcs – [in] Pointer to an array of pointers to charm_shc structures defining spherical harmonic coefficients of the polynomial density coefficients. For instance, density_shcs[0] is a pointer to charm_shc structure defining the spherical harmonic coefficients of the zero-order laterally varying polynomial density coefficients \(\rho_0(\varphi', \lambda')\); density_shcs[1] is a pointer to charm_shc structure holding the spherical harmonic coefficients of the first-order laterally varying polynomial density coefficients \(\rho_1(\varphi', \lambda')\); etc.
density_nmax – [in] Pointer to an array of unsigned long integers defining the maximum harmonic degrees to be used with the respective charm_shc structures in density_shcs when defining the polynomial density coefficients. More specifically, density_nmax[i] is the maximum harmonic degree that will be used when defining the polynomial density coefficients of order \(i\) by density_shcs[i]; it must hold that density_nmax[i] <= density_shcs[i]->nmax.
density_order – [in] Maximum order of the density polynomial. Both density_shcs and density_nmax must have an access to at least density_order + 1 array elements.
grav_const – [in] Newton’s gravitational constant.
mass – [in] Mass of the body. The value affects the output potential coefficients potential_shcs (see below) but do not affect the gravitational effects synthesized from the coefficients. This is because the value is also used to compute potential_shcs->mu, so it cancels out during the synthesis. Only non-zero values are accepted.
shape_power_min – [in] Minimum topography power. Usually, the value is 1. Values higher than 1 do not produce a complete gravity field model, but are useful in specific situations. At any rate, the value must be positive.
shape_power_max – [in] Maximum topography power. The value cannot be smaller than shape_power_min. The larger is the value, the more complete is the spectral gravity forward modelling.
potential_shcs_nmax – [in] Maximum harmonic degree of the output spherical harmonic coefficients of the implied gravitational potential in potential_shcs (see below). The value cannot be larger than potential_shcs->nmax.
shape_density_shcs_path – [in] Path and prefix of output files containing spherical harmonic coefficients of shape-density functions. For instance, the "/tmp/shape-density" string will cause that several files (depending on shape_power_max and density_order) having the prefix shape-density will be created in the /tmp folder. If NULL, coefficients of shape-density functions are not exported. The topographic-density function is defined as (see Bucha 2025) \(\left( \frac{r_{\mathrm{S}}(\varphi', \lambda') - R}{R} \right)^p \, \rho_i(\varphi', \lambda') \, R^i\), where \(r_{\mathrm{S}}\) is the radius of a surface point synthesized from shape_shcs up to degree shape_nmax, \(R\) is shape_radius_ref and \(\rho_i(\varphi', \lambda')\) are the polynomial density coefficients of the \(i\)-th order synthesized from density_shcs[i] up to degree density_nmax[i].
potential_shcs_path – [in] Path and prefix of output files containing spherical harmonic coefficients of gravitational potential contributions due to the shape-density functions. If NULL, the coefficients are not exported.
shcs_file_format – [in] String defining the format of output files that are specified by shape_density_shcs_path and/or potential_shcs_path. Accepted strings are "tbl", "mtx", "dov", "bin". If both shape_density_shcs_path and potential_shcs_path are NULL, shcs_file_format can be NULL, too.
potential_shcs – [out] Spherical harmonic coefficients of the output gravitational potential. In a special case, when potential_shcs is NULL, computed and exported are only partial contributions due to the individual shape and density powers to the potential coefficients, that is, they are not summed to yield potential coefficients. The partial outputs are exported through shape_density_shcs_path and potential_shcs_path. The output potential coefficients can be reproduced by summing all output files with the prefix potential_shcs_path. This option is useful if you need to save some memory.
err – [out] Error reported by the function (if any).

void charm_gfm_global_density_lateral(const charm_shc *shape_shcs, unsigned long shape_nmax, double shape_radius_ref, charm_shc *density_shcs, unsigned long density_nmax, double grav_const, double mass, unsigned shape_power_min, unsigned shape_power_max, unsigned long potential_shcs_nmax, const char *shape_density_shcs_path, const char *potential_shcs_path, const char *shcs_file_format, charm_shc *potential_shcs, charm_err *err)#

Performs global spectral gravity forward modelling with laterally varying mass density.

The function input parameters are the same as in charm_gfm_global_density_3d() with two exceptions:

density_shcs is now a pointer to a charm_shc structure that holds spherical harmonic coefficients of the laterally varying mass density function,
density_nmax is an unsigned long integer indicating the maximum harmonic degree, up to which the coefficients from density_shcs are used in forward modelling.

void charm_gfm_global_density_const(const charm_shc *shape_shcs, unsigned long shape_nmax, double shape_radius_ref, double density, double grav_const, double mass, unsigned shape_power_min, unsigned shape_power_max, unsigned long potential_shcs_nmax, const char *shape_shcs_path, const char *potential_shcs_path, const char *shcs_file_format, charm_shc *potential_shcs, charm_err *err)#

Performs global spectral gravity forward modelling with a constant mass density after Hirt and Kuhn (2014).

The function input parameters are the same as in charm_gfm_global_density_3d() with two exceptions.

A single constant mass density replaced the density_shcs and density_nmax parameters.
The shape_shcs_path variable replaced shape_density_shcs_path and is used to export spherical harmonic coefficients of the powers of the topographic height function \(\left( \frac{r_{\mathrm{S}}(\varphi', \lambda') - R}{R} \right)^p\), where \(r_{\mathrm{S}}\) is defined by shape_shcs and shape_nmax and \(R\) is shape_radius_ref. This time, the coefficients of the exported powers of the topographic height function are therefore not multiplied by the constant density, hence the new variable name.

Note

The function is optimized in terms of memory usage. It therefore requires smaller amount of working RAM than charm_gfm_global_density_3d() and charm_gfm_global_density_lateral() would require if used with a constant mass density as their special case.

Cap-modified spectral gravity forward modelling

These functions perform spectral gravity forward modelling using spatially restricted integration radius. They can be used to compute the gravitational contribution of masses up to (near-zone) or beyond (far-zone) some distance from evaluation points as opposed to the global integration from the previous routines. The density may be constant, laterally variable or 3D variable.

Note

These functions are compiled only if --enable-mpfr is specified when calling ./configure during the installation. See Installing for further details.

void charm_gfm_cap_density_3d(const charm_shc *shape_shcs, unsigned long shape_nmax, double shape_radius_ref, charm_shc **density_shcs, unsigned long *density_nmax, unsigned density_order, double grav_const, double mass, unsigned shape_power_min, unsigned shape_power_max, unsigned radial_derivative_min, unsigned radial_derivative_max, double evaluation_spherical_radius, double integration_radius, unsigned u, unsigned v, int zone, mpfr_prec_t nbits, unsigned long potential_shcs_nmax, const char *shape_density_shcs_path, const char *potential_shcs_path, const char *shcs_file_format, charm_shc **potential_shcs, charm_err *err)#

Performs cap-modified spectral gravity forward modelling of 3D mass densities.

Similarly as charm_gfm_global_density_3d(), this function forward models 3D mass distributions. The difference is that it considers only those topographic masses that are located inside or outside a spherical cap centred at evaluation points. The output spherical harmonic coefficients can thus be used to compute near-zone or far-zone effects on some gravitational field quantity.

Many of the function parameters have exactly the same meaning as in charm_gfm_global_density_3d(), so explained will be the new ones only.

Tip

Use charm_gfm_cap_q_check_prec() to determine the appropriate value of the input parameter nbits.

References:

Bucha B (2025) Spectral gravity forward modelling of 3D variable densities using an arbitrary integration radius with application to lunar topographic masses. Journal of Geodesy, doi: 10.1007/s00190-025-01951-9.

Note

This function is compiled only if --enable-mpfr is specified when calling ./configure during the installation. See Installing for further details.

Warning

The function parameters u and v specify the output quantity, of which spherical harmonic coefficients potential_shcs are computed. However, the output quantity is not necessarily some derivative of the gravitational potential in the local north-oriented reference frame (LNOF). Yet, you can get the directional potential derivatives in LNOF if you do some rather simple manipulations on your own (see Bucha, 2025). The exception is the case u = 0 and v = 0, which leads directly to the spherical harmonic coefficients of the gravitational potential.

Parameters:

radial_derivative_min – [in] Minimum radial derivative of the quantity specified by u and v (see below). Use 0 for no radial derivative.
radial_derivative_max – [in] The same as radial_derivative_min but for the maximum radial derivative.
evaluation_spherical_radius – [in] Spherical radius in metres of evaluation points. The output coefficients potential_shcs refer to this sphere. If your goal is to compute your u,v-quantity on an irregular surface (e.g., on the Earth’s surface), use analytical continuation through radial derivatives. These can be specified by radial_derivative_min and radial_derivative_max.
integration_radius – [in] Cap radius in radians defining the masses to be integrated. Masses within the cap are here called near-zone masses and masses beyond the cap are here called far-zone masses.
u – [in] Order of the potential derivative, of which spherical harmonic coefficients potential_shcs are computed. Use 0 to get the gravitational potential, use 1 to get quantities related to the first-order potential derivatives or use 2 for quantities related to the second-order potential derivatives. Values larger than 2 are not allowed.
v – [in] Order of the potential derivative with respect to the spherical distance. Valid values are 0, ..., u.
zone – [in] Non-negative integer to integrate near-zone masses (e.g., CHARM_GFM_NEAR_ZONE) or negative integer to integrate far-zone masses (e.g., CHARM_GFM_FAR_ZONE).
nbits – [in] Number of bits to represent the significand of all floating point numbers used to compute the truncation coefficients.
potential_shcs – [out] Double pointer to the charm_shc structure. potential_shcs must have access to radial_derivative_max - radial_derivative_min + 1 charm_shc structures, each of which must have allocated memory for coefficients at least up to degree potential_shcs_nmax. In a special case, when potential_shcs is NULL, computed and exported are only partial contributions of individual shape and density powers to the potential coefficients, that is, they are not summed. The partial outputs are exported through shape_density_shcs_path and potential_shcs_path. The output potential coefficients can be reproduced using the output files with the prefix potential_shcs_path. This option is useful if you need to save some memory.

void charm_gfm_cap_density_lateral(const charm_shc *shape_shcs, unsigned long shape_nmax, double shape_radius_ref, charm_shc *density_shcs, unsigned long density_nmax, double grav_const, double mass, unsigned shape_power_min, unsigned shape_power_max, unsigned radial_derivative_min, unsigned radial_derivative_max, double evaluation_spherical_radius, double integration_radius, unsigned u, unsigned v, int zone, mpfr_prec_t nbits, unsigned long potential_shcs_nmax, const char *shape_density_shcs_path, const char *potential_shcs_path, const char *shcs_file_format, charm_shc **potential_shcs, charm_err *err)#

Performs cap-modified spectral gravity forward modelling with laterally varying mass density.

The function input parameters are the same as in charm_gfm_cap_density_3d() with two exceptions:

density_shcs is now a pointer to a charm_shc structure that holds spherical harmonic coefficients of the laterally varying mass density function,
density_nmax is an unsigned long integer indicating the maximum harmonic degree, up to which the coefficients from density_shcs are used in forward modelling.

void charm_gfm_cap_density_const(const charm_shc *shape_shcs, unsigned long shape_nmax, double shape_radius_ref, double density, double grav_const, double mass, unsigned shape_power_min, unsigned shape_power_max, unsigned radial_derivative_min, unsigned radial_derivative_max, double evaluation_spherical_radius, double integration_radius, unsigned u, unsigned v, int zone, mpfr_prec_t nbits, unsigned long potential_shcs_nmax, const char *shape_shcs_path, const char *potential_shcs_path, const char *shcs_file_format, charm_shc **potential_shcs, charm_err *err)#

Performs cap-modified spectral gravity forward modelling with a constant mass density after Bucha et al (2019).

The function input parameters are the same as in charm_gfm_cap_density_3d() with two exceptions.

A single constant mass density replaced the density_shcs and density_nmax parameters.
The shape_shcs_path variable replaced shape_density_shcs_path and is used to export spherical harmonic coefficients of the powers of the topographic height function \(\left( \frac{r_{\mathrm{S}}(\varphi', \lambda') - R}{R} \right)^p\), where \(r_{\mathrm{S}}\) is defined by shape_shcs and shape_nmax and \(R\) is shape_radius_ref. This time, the coefficients of the exported powers of the topographic height function are therefore not multiplied by the constant density, hence the new variable name.

Note

The function is optimized in terms of memory usage. It therefore requires smaller amount of working RAM than charm_gfm_cap_density_3d() and charm_gfm_cap_density_lateral() would require if used with a constant mass density as their special case.

void charm_gfm_cap_q(const mpfr_t shape_radius_ref, const mpfr_t evaluation_spherical_radius, const mpfr_t integration_radius, unsigned long potential_shcs_nmax, unsigned shape_power_max, unsigned radial_derivative_min, unsigned radial_derivative_max, unsigned density_order, int zone, unsigned type, mpfr_prec_t nbits, mpfr_t *qkpin, charm_err *err)#

Computes trunction coefficients CHARM_GFM_Q00, CHARM_GFM_Q10, CHARM_GFM_Q11, CHARM_GFM_Q20, CHARM_GFM_Q21, CHARM_GFM_Q22 that are used in spectral gravity forward modelling with spatially limited integration radius.

Tip

Use charm_gfm_cap_q_check_prec() to determine the appropriate value of the input parameter nbits.

Note

This function is compiled only if --enable-mpfr is specified when calling ./configure during the installation. See Installing for further details.

Parameters:

shape_radius_ref – [in] Radius of the reference sphere in metres.
evaluation_spherical_radius – [in] Spherical radius of evaluation points in metres.
integration_radius – [in] Integration radius in radians.
potential_shcs_nmax – [in] Maximum spherical harmonic degree of the truncation coefficients.
shape_power_max – [in] Maximum topography power of the truncation coefficients (must be larger than zero).
radial_derivative_min – [in] Minimum radial derivative of the truncation coefficients.
radial_derivative_max – [in] Maximum radial derivative of the truncation coefficients.
density_order – [in] Maximum order of the polynomial density coefficients.
zone – [in] Non-negative integer for near-zone truncation coefficients (e.g., CHARM_GFM_NEAR_ZONE) or negative integer for far-zone truncation coefficients (e.g., CHARM_GFM_FAR_ZONE).
type – [in] Type of the truncation coefficients (symbolic constant CHARM_GFM_Q00, CHARM_GFM_Q10, CHARM_GFM_Q11, CHARM_GFM_Q20, CHARM_GFM_Q21 or CHARM_GFM_Q22).
nbits – [in] Number of bits to represent the significand of all floating point numbers used to compute the truncation coefficients.
qkpin – [out] Pointer to an array of nq elements of the mpfr_t data type. nq can be obtained by charm_gfm_cap_nq() or it can be computed as follows. Let nmax = potential_shcs_nmax, pmax = shape_power_max, kmin = radial_derivative_min, kmax = radial_derivative_max, imax = density_order. The total number of elements is then nq = (kmax - kmin + 1) * pmax * (nmax + 1) * (imax + 1). The value of \(\partial^k Q^{uvj}_{npi}(r, \psi_0, R) / \partial r^k\) can be accessed as qkpin[(((k - kmin) * pmax + (p - 1)) * (imax + 1) + i) * (nmax + 1) + n]. qkpin can thus be considered as a 4D-array, in which harmonic degree n varies fastest and the order of the radial derivative varies slowest, or, more precisely, the order is n, i, p, and k.
err – [out] Error reported by the function (if any).

size_t charm_gfm_cap_nq(unsigned long potential_shcs_nmax, unsigned shape_power_max, unsigned radial_derivative_min, unsigned radial_derivative_max, unsigned density_order, charm_err *err)#

Returns the number of truncation coefficients qkpin computed by charm_gfm_cap_q().

Parameters:

potential_shcs_nmax – [in] Maximum spherical harmonic degree of the truncation coefficients.
shape_power_max – [in] Maximum topography power of the truncation coefficients (must be larger than zero).
radial_derivative_min – [in] Minimum radial derivative of the truncation coefficients.
radial_derivative_max – [in] Maximum radial derivative of the truncation coefficients.
density_order – [in] Maximum order of the polynomial density coefficients.
err – [out] Error reported by the function (if any).

Returns:

On success, returned is the number of truncation coefficients. On error, 0 is returned in addition to the error reporting through err.

long charm_gfm_cap_q_check_prec(const mpfr_t shape_radius_ref, const mpfr_t evaluation_spherical_radius, const mpfr_t integration_radius, unsigned long potential_shcs_nmax, unsigned shape_power_max, unsigned radial_derivative_min, unsigned radial_derivative_max, unsigned density_order, unsigned type, mpfr_prec_t nbits, mpfr_prec_t nbits_ref, charm_err *err)#

Estimates the lowest number of correct digits of the sum of near- and far-zone truncation coefficients obtained by charm_gfm_cap_q() when using the same input parameters. The uncertainty of the estimate is 1 digit. In case of an error, returned is a negative integer.

First, near- and far-zone truncation coefficients are computed and summed using the nbits precision. Then, reference coefficients from global spectral gravity forward modelling are evaluated using the nbits_ref precision. Finally, the sums are compared with respect to the reference values.

Tip

Use this function before calling charm_gfm_cap_density_*() to check whether nbits that you enter to charm_gfm_cap_density_*() is large enough to ensure accurate computation of truncation coefficients. The number returned by this function should be at least 7, 16 or 34 for gravity forward modelling in single, double or quadruple precision, respectively. Otherwise, increase nbits and nbits_ref until the desired accuracy is met. Once the value returned by this function is large enough, that particular (or higher) nbits can safely enter charm_gfm_cap_density_*().

Note

If all sums of truncation coefficients are computed accurately to the last bit, returned is a large integer (LONG_MAX from limits.h). This may happen, for instance, if nmax is so small that all truncation coefficients of a given type are zero (e.g., nmax = 0 for type = CHARM_GFM_Q20).

Note

The number returned is related to the sum of near- and far-zone truncation coefficients. Usually, all near- and far-zone coefficients are more accurate than indicated by the returned value. Therefore, even a smaller value of nbits may suffice to get accurate outputs from charm_gfm_cap_density_*().

Note

This function is compiled only if --enable-mpfr is specified when calling ./configure during the installation. See Installing for further details.

Parameters:

shape_radius_ref – [in] Radius of the reference sphere in metres.
evaluation_spherical_radius – [in] Spherical radius of the evaluation point in metres.
integration_radius – [in] Integration radius in radians.
potential_shcs_nmax – [in] Maximum spherical harmonic degree of the truncation coefficients.
shape_power_max – [in] Maximum topography power of the truncation coefficients (must be larger than zero).
radial_derivative_min – [in] Minimum radial derivative of the truncation coefficients.
radial_derivative_max – [in] Maximum radial derivative of the truncation coefficients.
density_order – [in] Maximum order of the density polynomial.
type – [in] Type of the truncation coefficients (symbolic constant CHARM_GFM_Q00, CHARM_GFM_Q10, CHARM_GFM_Q11, CHARM_GFM_Q20, CHARM_GFM_Q21 or CHARM_GFM_Q22).
nbits – [in] Number of bits to represent the significand of all floating point numbers used to compute the truncation coefficients.
nbits_ref – [in] Number of bits to represent the significand of all floating point numbers used to compute the reference values (cannot be smaller thannbits).
err – [out] Error reported by the function (if any).

Returns:

An estimate of the lowest number of correct digits across all harmonic degrees up to potential_shcs_nmax, all topography powers up to shape_power_max and radial derivatives from radial_derivative_min to radial_derivative_max, respectively.

charm_gfm

Contents

charm_gfm#