On Moho Determination by the Vening Meinesz-Moritz Technique

This chapter describes a theory and application of satellite gravity and altimetry data for determining Moho constituents (i.e. Moho depth and density contrast) with support from a seismic Moho model in a least-squares adjustment. It presents and applies the Vening Meinesz-Moritz gravimetric-isostatic model in recovering the global Moho features. Internal and external uncertainty estimates are also determined. Special emphasis is devoted to presenting methods for eliminating the so-called non-isostatic effects, i.e. the gravimetric signals from the Earth both below the crust and from partly unknown density variations in the crust and effects due to delayed Glacial Isostatic Adjustment as well as for capturing Moho features not related with isostatic balance. The global means of the computed Moho depths and density contrasts are 23.8±0.05 km and 340.5 ± 0.37 kg/m3, respectively. The two Moho features vary between 7.6 and 70.3 km as well as between 21.0 and 650.0 kg/m3. Validation checks were performed for our modeled crustal depths using a recently published seismic model, yielding an RMS difference of 4 km.

Decorrelative Mollifier Gravimetry

<p>The lecture highlights arguments that, coming from multiscale mathematics, have fostered the advancement of gravimetry, as well as those that, generated by gravimetric problems, have contributed to the enhancement in constructive approximation and numerics. Inverse problems in gravimetry are delt with multiscale mollifier decorrelation strategies. Two examples are studied in more detail: (i) Vening Meinesz multiscale surface mollifier regularization to determine locally the Earth's disturbing potential from deflections of vertical, (ii) Newton multiscale volume mollifier regularization of the inverse gravimetry problem to derive locally the density contrast distribution from functionals of the Newton integral and to detect fine particulars of geological relevance. All in all, the Vening Meinesz medal  lecture is meant as an  \lq \lq appetizer'' served to enjoy the tasty meal "Mathematical Geoscience Today'' to be shared by geoscientists and mathematicians in the field of gravimetry. It provides innovative concepts and locally relevant applications presented in a monograph to be published by Birkhäuser in the book series “Geosystems Mathematics” (2021).</p>

Comparison among Gravimetric, Astrogravimetric and Astrogeodetic Geoids: Case Study for Austria

<p>It is used to state that all geoid determination techniques should yield to the same geoid if the indirect effect is properly taken into account (Heiskanen and Moritz, 1967). The current study compares different geoid determination techniques for Austria. The used techniques are the gravimetric, astrogravimetric and astrogeodetic geoid determination techniques. The available data sets (gravity, deflections of the vertical, height, GPS) are described. The window remove-restore technique (Abd-Elmotaal and Kuehtreiber, 2003) has been used. The available gravity anomalies and the deflections of the vertical have been topographically-isostatically reduced using the Airy isostatic hypothesis. The reduced deflections have been used to interpolate deflections on a relatively dense grid covering the data window. These gridded reduced deflections have been used to compute an astrogeodetic geoid for Austria using least-squares collocation technique within the remove-restore scheme. The Vening Meinesz formula has been used to compute an astrogravimetric geoid for Austria. Another gravimetric geoid for Austria has been determined in the framework of the window remove-restore technique using Stokes integral with modified Stokes kernel. All computed geoids have been validated using GNSS/levelling derived geoid. A wide comparison among the derived geoids computed within the current investigation has been carried out.</p>

The Deflection of the Vertical, from Bouguer to Vening-Meinesz, and Beyond – the unsung hero of geodesy and geophysics

<p>When thinking of gravity in geodesy and geophysics, one usually thinks of its magnitude, often referred to a reference field, the normal gravity.  It is, after all, the free-air gravity anomaly that plays the significant role in terrestrial data bases that lead to Earth Gravitational Models (such as EGM96 or EGM2008) for a multitude of geodetic and geophysical applications.  It is the Bouguer anomaly that geologists and exploration geophysicists use to infer deep crustal density anomalies.  Yet, it was also Pierre Bouguer (1698-1758) who, using the measured direction of gravity, was the first to endeavor a determination of Earth’s mean density (to “weigh the Earth”), that is, by observing the deflection of the vertical due to Mount Chimborazo in Ecuador.  Bouguer’s results, moreover, sowed initial seeds for the theories of isostasy.  With these auspicious beginnings, the deflection of the vertical has been an important, if not illustrious, player in geodetic history that continues to the present day.  Neglecting the vertical deflection in fundamental surveying campaigns in the mid to late 18<sup>th</sup> century (e.g., Lacaille in South Africa and Méchain and Delambre in France) led to errors in the perceived shape of the Earth, as well as its scale that influenced the definition of the length of a meter.  The vertical deflection, though generally excluded from modern EGM developments, nevertheless forms a valuable resource to validate such models.  It is also the vertical deflection that is indispensable for precision autonomous navigation (i.e., without external aids such as GPS) using inertial measurement units.  It is the deflection of the vertical that, measured solely along horizontal lines, would readily provide geoid undulation profiles, essential for the modernization of height systems (i.e., vertical geodetic control) without the laborious and traditional methods of spirit leveling.  But, measuring the deflection of the vertical is itself an arduous undertaking and this has essentially contributed to its neglect and/or underusage.  Even Vening-Meinesz’s formulas of convolution with gravity anomalies do not greatly facilitate its determination.  This presentation offers a review of the many roles the vertical deflection has, or could have, played over the centuries, how it has been measured or computed, and how gravity gradiometry might eventually awaken its full potential.</p>

Moho density contrast in Antarctica determined by satellite gravity and seismic models

SUMMARY As recovering the crust–mantle/Moho density contrast (MDC) significantly depends on the properties of the Earth's crust and upper mantle, varying from place to place, it is an oversimplification to define a constant standard value for it. It is especially challenging in Antarctica, where almost all the bedrock is covered with a thick layer of ice, and seismic data cannot provide a sufficient spatial resolution for geological and geophysical applications. As an alternative, we determine the MDC in Antarctica and its surrounding seas with a resolution of 1° × 1° by the Vening Meinesz-Moritz gravimetric-isostatic technique using the XGM2019e Earth Gravitational Model and Earth2014 topographic/bathymetric information along with CRUST1.0 and CRUST19 seismic crustal models. The numerical results show that our model, named HVMDC20, varies from 81 kg m−3 in the Pacific Antarctic mid-oceanic ridge to 579 kg m−3 in the Gamburtsev Mountain Range in the central continent with a general average of 403 kg m−3. To assess our computations, we compare our estimates with those of some other gravimetric as well as seismic models (KTH11, GEMMA12C, KTH15C and CRUST1.0), illustrating that our estimates agree fairly well with KTH15C and CRUST1.0 but rather poor with the other models. In addition, we compare the geological signatures with HVMDC20, showing how the main geological structures contribute to the MDC. Finally, we study the remaining glacial isostatic adjustment effect on gravity to figure out how much it affects the MDC recovery, yielding a correlation of the optimum spectral window (7≤ n ≤12) between XGM2019e and W12a GIA models of the order of ∼0.6 contributing within a negligible $ \pm 14$ kg m−3 to the MDC.

A New Moho Depth Model for Fennoscandia with Special Correction for the Glacial Isostatic Effect

In this study, we present a new Moho depth model in Fennoscandia and its surroundings. The model is tailored from data sets of XGM2019e gravitationl field, Earth2014 topography and seismic crustal model CRUST1.0 using the Vening Meinesz-Moritz model based on isostatic theory to a resolution of 1° × 1°. To that end, the refined Bouguer gravity disturbance is determined by reducing the observed field for gravity effect of topography, density heterogeneities related to bathymetry, ice, sediments, and other crustal components. Moreover, stripping of non-isostatic effects of gravity signals from mass anomalies below the crust due to crustal thickening/thinning, thermal expansion of the mantle, Delayed Glacial Isostatic Adjustment (DGIA), i.e., the effect of future GIA, and plate flexure has also been performed. As Fennoscandia is a key area for GIA research, we particularly investigate the DGIA effect on the gravity disturbance and gravimetric Moho depth determination in this area. One may ask whether the DGIA effect is sufficiently well removed in the application of the general non-isostatic effects in such an area, and to answer this question, the Moho depth is determined both with and without specific removal of the DGIA effect prior to non-isostatic effect and Moho depth determinations. The numerical results yield that the RMS difference of the Moho depth from our model HVMD19 vs. the seismic CRUST19 and GRAD09 models are 3.8/4.2 km and 3.7/4.0 km when the above strategy for removing the DGIA effect is/is not applied, respectively, and the mean value differences are 1.2/1.4 km and 0.98/1.4 km, respectively. Hence, our study shows that the specific correction for the DGIA effect on gravity disturbance is slightly significant, resulting in individual changes in the gravimetric Moho depth up to − 1.3 km towards the seismic results. On the other hand, our study shows large discrepancies between gravimetric and seismic Moho models along the Norwegian coastline, which might be due to uncompensated non-isostatic effects caused by tectonic motions.

A new Moho depth model for Fennoscandia and surroundings

<p>Seismic data are the preliminary information for investigating Earth’s interior structure. Since large parts of the world are not yet sufficiently covered by such data, products from Earth satellite gravity and altimetry missions can be used as complimentary for this purpose. This is particularly the case in most of the ocean areas, where seismic data are sparse. One important information of Earth’s interior is the crustal/Moho depth, which is widely mapped from seismic surveys. In this study, we aim at presenting a new Moho depth model from satellite altimetry derived gravity and seismic data in Fennoscandia and its surroundings using the Vening Meinesz-Moritz (VMM) model based on isostatic theory. To that end, the refined Bouguer gravity disturbance (reduced for gravity of topography, density heterogeneities related to bathymetry, ice, sediments, and other crustal components by applying so-called stripping gravity corrections) is corrected for so-called non-isostatic effects (NIEs) of nuisance gravity signals from mass anomalies below the crust due to crustal thickening/thinning, thermal expansion of the mantle, Delayed Glacial Isostatic Adjustment (DGIA) and plate flexure. As Fennoscandia is a key area for GIA research, we particularly investigate the DGIA effect on the gravity disturbance and Moho depth determination from gravity in this area. To do so, the DGIA effect is removed and restored from the NIEs prior to the application of the VMM formula. The numerical results show that the RMS difference of the Moho depth from the (mostly) seismic CRUST1.0 model is 3.6/4.3 km when the above strategy for removing the DGIA effect is/is not applied, respectively. Also, the mean value differences are 0.9 and 1.5 km, respectively. Hence, our study shows that our method of correcting for the DGIA effect on gravity disturbance is significant, resulting in individual changes in Moho depth up to several kilometres.</p>

Geodesy and Mathematics: Interactions in Multiscale Mollifier Regularization Methods

<p>The lecture highlights arguments that, coming from Mathematics, have fostered the advancement of Geodesy, as well as those that, generated by geodetic problems, have contributed to the enhancement in Mathematics.</p><p>We particularly deal with novel applications to Geodesy in the context of multiscale approximation (MA). In fact, multiscale reconstruction and decorrelation methods are a research field originated in geophysics for, e.g., earthquake modeling some decades ago, in which today's Geodesy and Mathematics show mutual influences, especially on the subject of spectral and space data sampling.</p><p>We particularly focus the attention on inverse problems of Geodesy and multiscale mollifier regularization strategies. Two examples are studied in more detail:</p><p>(i) Vening Meinesz multiscale surface mollifier regularization to determine locally the Earth's disturbing potential from deflections of vertical,</p><p>(ii) Newton multiscale volume mollifier regularization of the inverse gravimetry problem to derive locally the density contrast distribution from functionals of the Newton integral and to detect fine particulars of geological relevance.</p><p>Neither extreme depth to explain all facets of the geodetic observational situation nor penetrative handling of mathematical obligations and technicalities can be expected. The lecture is just an \lq \lq appetizer'' served to enjoy the tasty meal "Mathematical Geodesy Today'' to be shared by geodesists and mathematicians.</p>

Determination of Vertical Deflection Based on Terestrial Gravity Disturbance Data (A Case Study in Semarang City)

Vertical deflection can be determined by geometrical and physical measurement. In geometrical way, vertical deflection is obtained by comparing astronomical coordinate and geodetical coordinate. In physical way, vertical deflection can be computed from gravity measurement. In the past, vertical deflection was computed from gravity anomaly data. Gravity anomaly data measurement is difficult because it need reduction of gravity from surface of the earth to the geoid using orthometric height from spirit level measurement. In modern era, gravity anomaly data may be replaced by gravity disturbance data whose only required gravity and GNSS (Global Navigation Satellite System) measurement. This research aims to determine vertical deflection in Semarang City from terrestrial gravity disturbance data. The gravity data were measured in March of 2016. Formula of Vening Meinesz that usually used for vertical deflection was replaced by new formula that generated from derivation of function of Hotine. Applying gravity disturbance gave vertical deflection of east-west component that were vary from -1.2” to 12.2” while north-south component were vary from -4.2” to 4.2”. Comparing vertical deflection as computed from terrestrial data to as computed from EGM2008 coefficients showed conformity in shape and values. It was concluded that derivation of function of Hotine could be applied for vertical deflection determination from gravity disturbance.