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Vajda et al.: A new view of the concept of anomalous gravity in the GIP resubmission version of ms for SGG
A new view of the concept of anomalous gravity in the context of the gravimetric inverse problem
Vajda Peter1, Petr Vaníček, Bruno Meurers
keywords disturbing potential, gravity, anomaly, disturbance, geoidal height, density, inversion, interpretation, modeling, geophysical indirect effect, topographical correction, NETC space
The rigorous formulation of the gravimetric inverse problem in terms of sought subsurface anomalous density distribution and in terms of the gravity anomaly, the gravity disturbance, and the geoidal height is newly reviewed. The emphasis is put on the proper treatment of the respective effects of topographical masses. The aim is to prove, that the problem calls for using such topographical corrections that adopt the reference ellipsoid as the lower boundary of the topography, and a reference topographical density. We base our derivations on a reference topographical density that is chosen to be globally constant. Such topographical corrections are then referred to as “No Ellipsoidal Topography of Constant Density” corrections, abbreviated as “NETC” corrections, as they differ from the commonly used Bouguer topographical corrections. We derive the NETC corrections to the disturbing potential, the gravity disturbance, the gravity anomaly, and the geoidal height. Equations are derived, that link the anomalous density, via Newton integrals with specific kernels, with the NETC gravity disturbance, the NETC gravity anomaly, and the NETC geoidal height. Prescriptions for compiling the NETC gravity disturbance, anomaly, and geoidal height from observable data are derived. It is proved, that the NETC gravity disturbance is rigorously equal to the gravitational effect (attraction) of the anomalous density distribution inside the entire earth. That is the fundamental point of the paper. The spherical complete Bouguer anomaly (SCBA) widely used in geophysical studies is then compared with the NETC gravity disturbance. The SCBA is also demonstrated to be a hybrid quantity, neither an anomaly, nor a disturbance. The systematic deviation between the SCBA and the NETC gravity disturbance is shown to be the most general form of the so called geophysical indirect effect (GIE). The GIE is computed for the area of Eastern Alps.
One of the goals of geophysics is to acquire some knowledge on the underground geological structure, or at least on some of its elements. In gravimetry the subject of the study is the distribution of mass. In practice gravity or gravitation are observed, which are physically associated with (actual) gravity or gravitation potential. Other parameters of the field derived from actual potential are observable, as well. Given a mass (density) distribution, the actual potential, and parameters derived thereof, are uniquely determinable. Such a task is known as the direct (forward) problem, and the solution is given by means of the Newton integral (for potential or for parameters derived thereof). The determination of the density distribution from the actual potential, or from parameters derived thereof, is referred to as the inverse problem, which is known to be non–unique.
However, most often the gravimetric inverse problem (GIP) is formulated in terms of anomalous (disturbing) potential (or parameters of the anomalous field derived thereof), and anomalous density distribution, as such formulation offers several advantages. In this paper we shall deal with the formulation of the GIP in terms of the disturbing potential, the gravity disturbance, the gravity anomaly, and the geoidal (or quasigeoidal) height.
The anomalous field results from the subtraction of a mathematically defined normal field from the actual field. The matter is complicated by the existence of topographical masses. The effect of the topography must be taken into account when solving the GIP. A rigorous formulation of the GIP, considering also the topography, shall yield rigorous definitions of the parameters of the anomalous field that are to be inverted (or eventually modeled or interpreted). Our main motivation for this paper is to point out some deviations from this rigorous definitions of the data that enter the GIP as observables, that occur in the geophysical practice, and to assess their size and impact. Particular attention will be paid to the widely used complete Bouguer gravity anomaly. Extensive debate has already taken place regarding the Bouguer anomaly. Although called anomaly, many geophysicists use it in the sense that is rigorously known as disturbance. Much debated was also the issue of the lower boundary of the topographical masses used for constructing the complete Bouguer anomaly, which has lead to introducing an effect named “geophysical indirect effect”. We refer here to the issues discussed in (Chapman and Bodine, 1979; Vogel, 1982; Jung and Rabinowitz, 1988; Meurers, 1992; Talwani, 1998; Hackney and Featherstone, 2003). Our aim here is to take such discussions one step further.
Historically the separation between the reference ellipsoid and the geoid – either of them serving as the lower boundary of the topographical masses, as well as the vertical datum – has been commonly neglected in geophysical applications. We will attempt to indicate when the separation may be ignored, and when it should be taken into account.
Hackney and Featherstone (2003) conclude in their Summary and Recommendations: “… it is suggested that there should be an integrated examination of the ‘gravity anomaly’ from both the geophysical and geodetic perspectives…” and “… answers to questions in geophysics such as ‘should we be computing gravity anomalies or gravity disturbances and at what point?’…”, as well as “…the ‘gravity anomaly’ is an ambiguous quantity for many reasons and the terminology needs to be re–examined using some unified geodetic and geophysical approach…” Here we would like to take part in the above indicated challenge.
2 Theoretical background
Our investigations shall be based on the commonly known concepts of the theory of the gravity field (e.g., Kellogg, 1929; MacMillan, 1930; Molodenskij et al., 1960; Grant and West, 1965; Heiskanen and Moritz, 1967; Bomford, 1971; Pick et al., 1973; Moritz, 1980, Vaníček and Krakiwsky, 1986; Blakely, 1995; Torge, 2001). We were drawing especially from (Vaníček et al., 1999; 2004; Hackney and Featherstone, 2003). In the below subsections we wish to explicitly point out only those concepts, that are either of particular importance in the context of our developments, or often are, as we find, not fully understood or implemented in geophysical practice. Our approach shall be that of applying the methodology of model gravity field spaces, such as the “Helmert space” used in (Vaníček and Martinec, 1994; Vaníček et al., 1999;) or the “NT space” used by Vaníček et al. (2004), to the geophysical problems, particularly to the GIP.
Hereafter all the discussed quantities will be considered as already properly corrected for the effects of the atmosphere, tides, and all the other smaller temporal effects. Systematically throughout the paper we shall describe the discussed quantities in geocentric geodetic (Gauss ellipsoidal) coordinates (e.g., Heiskanen and Moritz, 1967, Sec. 5–3; Vaníček and Krakiwsky, 1986, Sec. 15.4), where is ellipsoidal (geodetic) height , is geodetic latitude , and is geodetic longitude. For brevity the horizontal position will often be denoted by . Sometimes the height of a point above “sea level” – orthometric height (above the geoid), or normal height (above the quasigeoid) – shall be used in addition to the ellipsoidal height. In the sequel, disturbing potential will be denoted by , actual gravity by , normal gravity by , and geoidal height by . Note, that the Bruns equation, the fundamental gravimetric equation, and even the Newton volume integrals will be all expressed in geodetic coordinates. Once getting used to it, one finds it extremely convenient, when dealing with data that are typically positioned using geographical coordinates. Also the spherical approximation (e.g., Moritz, 1980, p. 349) will be dealt by in terms of geodetic coordinates. Hence, for instance, if the Newton integral is evaluated in spherical approximation over the volume of the reference ellipsoid, the lower limit of the integration in is “” and the upper limit is “0”, where is the mean earth’s radius. Note also, that in the spherical approximation the geocentric distance of a point becomes .
We will make use of a generalization of the Bruns formula and of the fundamental gravimetric equation as applied to the pairs of actual and normal equipotential surfaces – separated by the vertical displacement – elsewhere, not only at the geoid/ellipsoid level (e.g., Heiskanen and Moritz, 1967; Vaníček et al., 1999). The physics of the gravity field will be applied also to the model gravity fields. The model gravity fields and the gravity field spaces are introduced by manipulating (removing, replacing, condensing) the topographical masses, and by applying a topographical correction to the actual, and thus disturbing, potentials. In this way Vaníček and Martinec (1994) and Vaníček et al. (1999) use the “Helmert space” and Vaníček et al. (2004) use the “NT space”. Vaníček et al. (2004) deal extensively with the NT–gravity anomaly, and relate it to the commonly known Bouguer anomaly. The gravity field in the NT space is produced by real density inside the geoid (earth with no real masses between the geoid and the topo–surface, hence the “no topography” shortly “NT” terminology). It will turn out in the sequel, that we shall need to work with a model gravity field produced by real density inside the reference ellipsoid and anomalous density between the ellipsoid and the topo–surface. Hence our earth model will be that of “no reference density between the ellipsoid and the topo–surface”.
Originally we have performed all our derivations, including evaluation of the Newton volume integrals, in (Gauss) ellipsoidal coordinates, followed by applying spherical approximations. For the sake of brevity and simplicity we start our developments here already using the spherical approximations. For the exact formulation of the Newton volume integrals in Gauss ellipsoidal (geodetic) coordinates we refer the reader to e.g. (Vajda et al., 2004a; Novák and Grafarend, 2005). Novák and Grafarend (2005) showed, that the ellipsoidal correction to the spherical approximation of the Newton integral is by three orders of magnitude smaller than the spherical term.
We shall use the term “reference ellipsoid” for both the body and the surface, trusting that the meaning will be always clear from the context.
2.1 Remarks on normal field as related to coordinates
The normal gravity potential is selected as a known, mathematically defined model of the earth’s actual gravity potential (e.g., Heiskanen and Moritz, 1967; Vaníček and Krakiwsky, 1986), that closely approximates the actual potential. The normal gravity potential defines a normal earth in terms of its gravity field and its shape. In general, the normal field represents a spheroidal reference field, but most commonly used, due to simplicity, is the ellipsoidal normal field of the “mean earth ellipsoid” (ibid). The mean earth ellipsoid is geocentric, properly oriented, and “level” (“equipotential”, cf. Heiskanen and Moritz, 1967, Sec. 2–7). That assures, that the (reference) ellipsoid as a reference coordinate surface (as a 3D datum for geodetic coordinates) is, at the same time, also the equipotential surface of the normal gravity field, such, that the value of the normal potential on the ellipsoid is equal to the value of the actual potential on the geoid. In this way a physically meaningful tie between (geodetic) coordinates and normal gravity is established. Therefore, rigorously, the anomalous gravity data will have an unbiased physical interpretation only if their positions are referred to a “mean earth ellipsoid”. We say this because in geodesy, and in practice in some countries, often locally best fitting (“relative”) reference ellipsoids, that are not geocentric and/or not properly oriented, and/or of improper size (e.g., Vaníček and Krakiwsky, 1986) are in use as datums for geodetic coordinates. For instance Hackney and Featherstone (2003, Sec. 2.3) discuss the impact of the use of a local ellipsoid, as a datum, on normal gravity values (which in turn systematically biases the gravity anomaly/disturbance data). Below we shall call the mean earth ellipsoid simply reference ellipsoid.
2.2 Remarks on normal field as related to normal density inside reference ellipsoid
According to Somigliana and Pizzetti (Somigliana, 1929) the normal potential makes sense only outside the reference ellipsoid. It needs and it provides no unique knowledge about the interior density distribution of the reference ellipsoid. Consequently, the disturbing potential as well as the normal gravity inside the reference ellipsoid remain unspecified. However, if we are to formulate the GIP in terms of parameters of anomalous field and in terms of anomalous density distribution, we need some reference density inside the reference ellipsoid. How to overcome this obstacle? What we can do is to find a particular normal density distribution inside the ellipsoid, such that satisfies the external normal potential. The proof of the existence of such a solution is considered outside the scope of this paper. Here we shall assume that it can be done to a satisfactory approximation. With respect to this topic we refer the reader to (Tscherning and Sünkel, 1980; Moritz, 1990).
2.3 Remarks on gravity disturbance and gravity anomaly
Having the pairs actual potential and actual gravity, normal potential and normal gravity, we would anticipate to encounter the pair disturbing potential and anomalous (disturbing) gravity. In fact, two such anomalous quantities have been used, the gravity anomaly and the gravity disturbance. Historically the gravity anomaly emerged as a practical alternative to the gravity disturbance, as the disturbance was not “observable” (realizable) while the ellipsoidal height was not “commonly” (readily, widely) “observable” (realizable). Both the disturbance and the anomaly can be defined either using actual gravity (e.g., Heiskanen and Moritz, 1967; Vaníček et al., 1999; 2004) – left hand sides of Eqs. (e1) and (e2) respectively – we refer to such definition as “point definition” (e.g., Vajda et al., 2004b, Sec. 2.5), or using the disturbing potential (e.g., Heiskanen and Moritz, 1967; Vaníček et al., 1999; 2004) – right hand sides of Eqs. (e1) and (e2) respectively
In Eq. (e2) is the vertical displacement at given by the generalised Bruns equation (ibid). The two sets of definitions are not rigorously compatible, they differ by the effect of the deflection of the vertical, which is of the order of 10 Gal (e.g., Vaníček et al., 1999; 2004). Both sets define the disturbance or anomaly anywhere, not only on the geoid, or on the topographical surface. The “point definition” becomes handy in the space of observables, while the definition using disturbing potential becomes useful in the model space (on the side of working with density). The realization of point gravity disturbance requires the knowledge of ellipsoidal height of the point of evaluation, while the realization of point gravity anomaly requires the knowledge of the vertical displacement at the point of evaluation, which for gravity anomalies on the topographical surface reduces to the knowledge of the normal or orthometric height of the evaluation point above the geoid (quasigeoid). No “reductions” or “altitude corrections” based on various vertical gradients of gravity are needed for defining the gravity disturbance or anomaly. Topographical corrections, if applied to gravity disturbance or anomaly, define specific “kinds” of gravity disturbance or anomaly. Upward or downward continuation of the gravity anomaly or disturbance is an altogether separate issue, which should not be mixed up with reducing the value of actual gravity upward or downward using simply some kind of vertical gradient of gravity.
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