The geodetic and geophysical literature shows an abundance of mascon approaches for modelling the gravity field of the Moon or Earth on global or regional scales. This article illustrates the differences and similarities between the methods, which are labelled as mascon approaches by their authors.

Point mass mascons and planar disc mascons were developed for modelling the lunar gravity field from Doppler tracking data. These early models had to consider restrictions in observation geometry, computational resources or geographical pre-knowledge, which influenced the implementation. Mascon approaches were later adapted and applied for the analysis of GRACE observations of the Earth's gravity field, with the most recent methods based on the simple layer potential.

Differences among the methods relate to the geometry of the mascon patches and to the implementation of the gradient and potential for field analysis and synthesis. Most mascon approaches provide a direct link between observation and mascon parameters – usually the surface density or the mass of an element – while some methods serve as a post-processing tool of spherical harmonic solutions. This article provides a historical overview of the different mascon approaches and sketches their properties from a theoretical perspective.

The gravity field of the Earth influences daily life in many ways. Local plumb lines define the upward direction, and several scientific instruments must be levelled before usage. Gravity measurements provide corrections for geophysical height systems and allow the exploration of mineral deposits or caves. A gravity field model is also required for inertial navigation systems within aeroplanes, ships or submarines. On a regional or global scale, mass redistributions – like the melting of glaciers or the changes in ground water – are reflected in the temporal variations of the gravity field.

The gravity field of the Earth – or another celestial body – can be analysed on a global scale when enough orbiting satellites are tracked, even without special onboard instruments or ground-based measurements. Satellite data provide a fast and homogeneous sampling in contrast to ground-based observations. The gravity field analysis establishes a connection between the tracking data and a set of base functions to model the gravity field. An analysis by spherical harmonic functions is preferable for spherical bodies, as these base functions are the natural solution of the Laplace equation. This set of base functions is also a complete and orthogonal system, which simplifies the analysis. However, a reasonable spherical harmonic analysis requires orbit observations with global and homogeneous data distribution, and the model will have the same resolution everywhere.

Alternative localising base functions – e.g. point masses, spherical radial base functions, wavelets, Slepian function – are investigated and applied when the data distribution is irregular or when more details in a region of interest shall be detected. This article will summarise the localising base functions, which are labelled as

Studies of the early lunar orbiters demonstrated significant orbit disturbances, which were traced back to an irregular lunar gravity field.
The term “

A closer inspection of the publications, however, shows a variety of approaches under the label of mascons. This article will give a historical overview of the most prominent representatives
and an adequate definition of the mascon base functions. All different meanings of the investigated mascon approaches can be covered by the following definition:
the term

This publication is focused on the mascons' definitions and will ignore other processing steps, like background models, regional constraints or regularisation techniques. Each mascon approach is presented by the associated gravitational potential of a single element and its gradient in the notation of representative literature. The properties of each approach are deduced from the theoretical perspective only, but without treating programming experiments or numerical aspects. Such a detailed and comprehensive review of the different mascon approaches cannot be found in literature to our knowledge.

In several previous articles, the authors quote only the original publication

A point mass model and planar discs are applied for modelling the lunar gravity field in

mascons that have an analytical expression for the gravitational potential and explicit partial derivatives for the gradient;

mascons that are represented by a finite series of spherical harmonic functions and with partial derivatives derived via the chain rule;

mascons that serve as a post-processing tool to obtain regional mass changes from monthly spherical harmonic solutions.

Many recent publications are related to the mascon solutions of either the NASA Goddard Space Flight Center (GSFC), the Jet Propulsion Laboratory (JPL) or the Center for Space Research (CSR). The current JPL solutions are spherical cap mascons with analytical partial derivatives – i.e. category A in the classification – which are presented in Sect.

The origin of the mascon concept is closely related to early models of the lunar gravity field.

In the space race between the Soviet Union (USSR) and the United States of America (USA), both nations wanted to send their representatives to the Moon first. The possible landing sites were investigated by spacecrafts, starting with Luna 1 (USSR) in 1959, which missed the Moon due to navigation issues. The first man-made object on the Moon was the space probe Luna 2 (USSR) in a design impact in 1959, followed by several missions by both nations. The spacecraft Luna 10 (USSR) and Lunar Orbiter 1 (USA) were the first artificial orbiters around the Moon in 1966

In both orbiter missions, the observed orbits differed after a short time from the predicted ones, which indicated either an incorrect or an incomplete model. As other error sources could be excluded soon, the orbit disturbances were explained by significant mass anomalies below the Moon's surface. For these anomalies, the term “

Lunar maria and selected tracks of the Lunar Orbiter given in

All identified mascons on the nearside of the Moon cause relatively large and positive effects up to 200 mGal, and their locations are one-to-one correlated to the major lunar maria, including Imbrium, Serenitatis, Crisium, Humorum and Nectaris, which are visualised in Fig.

In particular for the Moon it is still common to call a large area with a significant positive mass anomaly a mascon independent of its mathematical representation

A quick modelling of the mass anomalies was important for the preparation of the latter space missions and the landing on the Moon. The chosen representation should

consider the geographical pre-knowledge, i.e. the lunar maria as expected locations of the mass anomalies;

consider the observation geometry, i.e. the fact that only the near side of the Moon allows observations from terrestrial ground stations;

enable a direct relation between observables – Doppler tracking data in the case of the early lunar missions – and the estimated mascon parameters;

remain simple due to limited computer resources.

The original papers

Please note that, for consistency, all mascon quantities and their geometries are labelled in this article by means of an index (here:

A standard observation technique for space probes is the Doppler tracking, i.e. the change in frequency of a (re)-transmitted signal due to the relative motion of the spacecraft and the ground station. The American missions use a few globally distributed stations, which meanwhile form the Deep Space Network of the NASA and which are operated by JPL today

The equivalent system of the USSR is not discussed in the investigated material.

. The Doppler signal does not provide complete information on the position or velocity; rather, it only projects the relative velocity between station and space probe onto the line of sightThe relationship between observation and mascon parameters requires a description of the change in the velocity – i.e. the acceleration – of the spacecraft caused by the gravitational potential. Hence, it is sufficient to derive the gradient by

To emphasise the special requirements and restrictions for the early lunar modelling, some details will be sketched here as well: according to

Modelling the gravity field of the asteroid 216 a.k.a. Kleopatra by point mass mascons according to

Point mass mascons are also used in a different way to determine the gravity field of irregular celestial bodies. An example can be found in

The point mass mascons have closed formulas for potential and explicit partial derivatives, which identify them as type A mascons in the threefold scheme.

The method is very easy to implement and requires only a few computational resources.

The gradient and all other field quantities are found without quadrature.

The model is singular for the potential and the gradient at the location of the point masses.

In the case of the lunar gravity field, assumptions are required for the location and depth below ground, as the Doppler tracking data and the observation geometry do not allow a detection of this information from the measurement.

It should be pointed out that the modelling by point masses is applied, for example, in

As a response to

The obvious issues of point masses are discussed in

the singularities of the model at the centres;

bad fitting of the residual tracking data in the equatorial zone of the Moon due to the observation geometry;

and combination issues with the spherical harmonic models (of very low degree and order at the time).

For a simple and efficient solution, the finite mass elements are chosen to be oblique rotational ellipsoids, also known as spheroids

These planar disc mascons must be rotated and translated on the surface or close to it onto different locations, which is only implicitly indicated due to the definition of the coordinates

The planar disc mascons have closed formulas for explicit partial derivatives of the potential, which identifies them as type A mascons.

The closed formulas do not require any integration for the gradient.

The surface elements all have the same shape, size and area for each mascon.

The potential of a mascon requires a series expansion.

The model is singular for the potential at the centre of the disc.

The surface elements do not cover the complete surface, even in a global analysis.

Most points within a disc are either above or below the spherical surface.

Modelling the gravitational potential by a simple layer was well known in geodesy and became popular around 1970.

The method can be applied to the complete potential or to a residual field after subtracting a reference field. The basic idea is to condensate the (remaining) in-homogeneous mass distribution onto the surface

The gravitational potential of the layer is given by

The unusual arguments of the distance expressions are introduced here for highlighting the dependency on two distinct point sets.

between calculation pointIn the mascon version of the simple layer potential, the surface

A linear combination Eq. (

Solving the Laplace equation in spherical coordinates leads to the spherical harmonic functions as a natural basis for gravity field modelling. An adequate linear combination of spherical harmonic functions can also be used to define localising base functions like the mascons in the spectral domain. Due to this combination over all degrees and orders, the result is sometimes labelled as the “lumped spherical harmonic approach”

Firstly, the gravity field is decomposed into a static field and its temporal variations:

The approach arose at GSFC when analysing the data of the GRACE mission, and it is presented in a sequence of articles

The mascons are generated in the spectral domain by (time-dependent)

The mascon solutions of the JPL are published online
(

The lumped spherical harmonic approach can be used for any (almost spherical) body, but the approach is introduced for analysing the temporal variations of Earth's gravity field due to the variable water storage in particular. Taking into account that a uniform layer of 1 cm fresh water within an area of 1 m

If the maximum degree

A straightforward sub-division of a sphere is given by a longitude–latitude grid, i.e. all boundaries are either part of parallel circles or of meridians. In this case, the integrals Eq. (

Sub-division in a longitude–latitude grid and a differential volume element of the sphere

The size and shape of the surface elements vary between the publications:

Equal areas within a longitude–latitude grid can be obtained by stretching or shrinking one of the angles dependent on latitude, which is discussed already in

In

In the CSR solution, the equal area per mascon is considered to be more relevant than a simple sub-division or a complete coverage of the sphere

Definition of mascon surface elements in the Mississippi basin. The figure originates from

The mascons were introduced for analysing the Earth's gravity field in the mission GRACE (Gravity Recovery And Climate Experiment). The mission consisted of two identical satellites, which were launched in 2002 in a cooperation between NASA/JPL and the German DLR. The satellites fell around the Earth in one common and almost circular orbit, with a low altitude of originally 500 km in height. The positions were quasi-permanently observed by GPS receivers with three antennas, and onboard accelerometers with three axes were measuring the combined influence of all non-gravitational effects. The main observable was the variation of the distance between the two GRACE satellites, measured by microwaves in the K-band and Ka-band via a range-rate measurement system. The distance of

The orbit observations and the gravity field parameters can be linked in different ways – e.g. the variational equation, the energy balance approach, the short arc approach or the acceleration approach – which are sketched, for example, in

For the range-rate

The lumped spherical harmonic approach is a representative of the type B mascons.

The method is very easy to implement after a previous analysis of the GRACE observations by spherical harmonic functions.

After the determination of all

The required integration Eq. (

A high degree

The planar disc mascon approach (in Sect.

To reduce the effort of quadrature for the gradient expression, a

The gradient of the potential

In the local mascon coordinate system, the gradient operator is of the form

The mascon potential is calculated by quadrature, and analytical derivatives have been derived, which leads to a class A mascon in the threefold scheme.

The two-dimensional quadrature for the gradients are reduced to a one-dimensional integration.

The calculation takes place only in the spatial domain and avoids the truncation error of spherical harmonic synthesis.

The surface elements have all the same shape, size and area for each mascon.

The surface elements do not cover the complete surface, even in a global analysis.

The model is singular for the potential and the gradient at the location of the centre of the spherical cap.

A straightforward implementation of the formulas Eq. (

To avoid truncation errors and aliasing into coefficients of lower degree and order via the spherical harmonic expansion Eq. (

The derivatives of the range-rate

The dynamical components are determined by the variational equation

The approach does not fit into the threefold scheme.

The method avoids truncation errors and aliasing by integration in the spatial domain.

The surface elements cover the complete surface in a global analysis.

The potential and the gradient require numerical quadrature.

The variational equations lead to a high computational burden, which is already admitted in

Since the successful GRACE mission, it is also possible to observe the temporal variations of the gravity field. The standard output of these investigations are monthly solutions of spherical harmonic coefficients, which are meanwhile complemented by mascon solutions in the same time span by several research centres.

The question arises whether it is possible to estimate local variations from the spherical harmonic solutions by post processing. This is of particular interest for the ice masses and glaciers in Greenland, Antarctica and Alaska as well as for the highly variable water masses in the large water basins, which dominate the time-variable part of the gravity field.

The spherical harmonic functions have a global support, which contradicts a regional analysis. Another problem is the noise in the coefficients, which is overcome by filtering and de-striping techniques at the cost of the spatial resolution. To estimate regional mass changes, it can be helpful to determine an adequate field quantity by spherical harmonic synthesis and to analyse this newly generated signal by another base function with local support

A long-term mean value

Original GRACE data are not required, as the method is applied to the previous solution by spherical harmonics, which leads to type C mascons.

The estimation of the weights is a straightforward process via least-squares estimation.

The surface elements all have the same shape, size and area.

The required integration Eq. (

The effect of Gauß filtering and the temporal average on the solution's quality are difficult to predict; also, the chosen sampling in the spherical harmonic synthesis might have an effect on the estimated masses.

The surface elements do not cover the complete surface, even in a global analysis.

This signal is analysed – by least squares estimation of the surface densities

The method is applied to the previous solution by spherical harmonics, which leads to type C mascons.

The method is very easy to implement.

Integration per mascon element is replaced by a weighted sum of point masses located on a grid.

The model is singular for potential and radial derivatives at the location of the nodes.

Finding the weighting

Point mass models are an important tool for gravity field modelling due to their simplicity and efficiency. The point mass representation is used for celestial bodies with irregular shapes but also for Earth or the Moon on regional and global scales. Point mass mascons are also a key aspect of converting spherical harmonic solutions into regional mass variations, which supports the interpretation of geophysical processes.

Mascons represented by finite surface elements are based on the simple layer potential. These models form a subset of localising base functions for gravity field modelling. Without neighbourhood conditions, a solution close to the ground generates a discontinuous field. The discontinuity problem is damped for higher evaluation altitude or small patches. The constant density per mascon simplifies the interpretation of mass variations in comparison to other localising base functions (e.g. wavelets or Slepian functions), which vary within their region of interest. Planar disc and spherical cap mascons are radial symmetric base functions, while the other mascon concepts allow for patches with arbitrary shapes. In particular, the shape can consider the geometry of water basins, which reduces the leakage of signals in hydro-geodesy.

No data sets were used in this article.

The author has declared that there are no competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The author is grateful to Nico Sneeuw at the Institute of Geodesy (University of Stuttgart) for encouraging and supporting the idea of a historical review.

This paper was edited by Hans Volkert and reviewed by two anonymous referees.