Comparison of geosat and gravimetric geoid profiles in the North Sea

Sea surface height profiles derived from 2‐year, repeat track, Geosat altimeter data have been compared with a regional gravimetric geoid in the western North Sea, computed using a geopotential model and terrestrial gravity data. The comparison encompasses 18 Geosat profiles covering a 750 × 850 km area of the North Sea. After a second‐order polynomial was used to model the long‐wavelength differences which cannot be clearly separated over an area of this size, results show agreement to better than ±3 cm for wavelengths between approximately 20 and 750 km. In regions where terrestrial gravity data were not available to improve the geoid, similar comparisons with the OSU91A geopotential model alone show differences of up to ±6 cm. This illustrates the importance of incorporating local gravity data in regional geoid computations, and partly validates the regional gravimetric geoid solution and Geosat sea surface profiles in the western North Sea. It is concluded that, in marine areas where the sea surface topography is known to be small in magnitude, Geosat sea surface profiles can act as an independent control on gravimetric geoids in the medium‐wavelength range.     

Transformation of Lambert Conic Conformal Coordinates from a Global Datum to a Local Datum

A detailed gravimetric geoid around Japan has been computed on the basis of 30’ × 30’ block mean free‐air gravity anomalies and GSFC GEM‐8 geopotential coefficient set. The 30’ × 30’ block means were read from various gravity maps around Japan, and the block means have been compiled into the JHDGF‐1 gravity file. Since the gravity file is restricted around Japan (see Figure 1), additional gravity data are needed to perform the Stokes’ integration in the cap with radius ψ₀ = 20°. The 1° × 1° block gravity means have been used outside the JHDGF‐1 region. The remarkable features of the gravimetric geoid occur over the trench areas. The geoidal dents over the trenches amount to ‐20~ ‐25 m in comparison with the geoidal heights in the land areas of Japan. The mean error of the 30’ × 30’ detailed gravimetric geoid obtained is estimated to be around 1.4 m, and the relative undulation of the geoid between the distance of a few hundred kilometers may be more accurate.     

High resolution gravimetric geoid heights and gravimetric vertical deflections of Europe including marine areas

Gravimetric geoid heights and gravimetric vertical deflections have been detemined for Europe including the Mediterranean Sea, North Sea, Norwegian Sea, Baltic Sea and parts of the North Atlantic Ocean in a 12′×20′ grid. The computation has been carried out by least squares spectral combination using closed integral formulas, combining 104 000 mean free air gravity anomalies in 6′×10′ blocks, 12 000 mean free air gravity anomalies in 1⁰×1⁰ blocks and the sherical harmonic model GEM9. The precision of the computed geoid heights has been estimated to ±1 m, the precision of the computed vertical deflections has been estimated to ±2″. Comparisons of the gravimetric geoid heights and vertical deflections with a number of other solutions have been carried out, confirming the precision estimation.     

Global ocean circulation patterns based on seasat altimeter data and the GEML2 gravity field

The Seasat altimeter data has been completely adjusted by a crossing arc technique to reduce the crossover discrepancies to approximately ±30 cm in five regional adjustments. This data was then used to create sea surface heights at 1° intersections in the ocean areas with respect to the GRS80 ellipsoid. These heights excluded the direct tidal effects but included the induced permanent deformation. A geoid corresponding to these sea surface heights was computed, based on the potential coefficients of the GEML2 gravity field up to degree 6, augmented by Rapp's coefficients up to degree 180. The differences between sea surface heights and the geoid were computed to give approximate estimates of sea surface topography. These estimates are dominated by errors in both sea surface heights and geoid undulations. To optimally determine sea surface topography a spherical harmonic analysis of raw estimates was carried out and the series was further truncated at degree 6, giving estimates with minimum wavelengths on the order of 6000 km. The direction of current flow can be computed on a global basis using the spherical harmonic expansion of the sea surface topography. Ths has been done, not only for Seasat/GEML2 estimates, but also using the recent dynamic topography estimates of Levitus. The results of the two solutions are very similar and agree well with the major circulation features of the oceans.     

The Role of Multi-Mission ERS Altimetry in the Determination of the Marine Geoid in the Azores

This study concerns the determination of a regional geoid model in the North Atlantic area surrounding the Azores islands by combining multi-mission altimetry from the ERS (European Remote Sensing) satellites and surface gravity data. A high resolution mean sea surface, named AZOMSS99, has been derived using altimeter data from ERS-1 and ERS-2 35-day cycles, spanning a period of about four years, and from ERS-1 geodetic mission. Special attention has been paid to data processing of points around the islands due to land contamination on some of the geophysical corrections. A gravimetric geoid has been computed from all available surface gravity, including land and sea observations acquired during an observation campaign that took place in the Azores in October 1997 in the scope of a European and a Portuguese project. Free air gravity anomalies were derived by altimetric inversion of the mean sea surface heights. These were used to fill the large gaps in the surface gravity and combined solutions were computed using both types of data. The gravimetric and combined solutions have been compared with the mean sea surface and GPS (Global Positioning System)-levelling derived geoid undulations in five islands. It is shown that the inclusion of altimeter data improves geoid accuracy by about one order of magnitude. Combined geoid solutions have been obtained with an accuracy of better than one decimetre.     

Global detailed gravimetric geoid

A 1 ° × 1 ° global detailed gravimetric geoid has been computed, using a combination of the Goddard Space Flight Center (GSFC) GEM‐8 potential field model and a set of 38,406 1° × 1° mean surface free air anomalies. Numerous short wavelength features are shown in the geoid contour map, e.g., the steep gradients associated with oceanic trenches. Comparison of this geoid with geoceiver derived and astrogeodetic geoid heights in the United States resulted in an r.m.s. difference of about 1.7 m. Comparisons with three GEOS‐3 altimeter derived geoidal profiles revealed that for areas with good surface data coverage, the relative agreement is generally better than 5 m.     

Identification of mantle and lithospheric components of the gravity field by isostatic gravity anomalies

All anomalous masses of the Earth are reflected in the free air gravity anomalies and the geoidal undulations. The low viscosity of the asthenosphere significantly reduces the possibility of existence of density inhomogeneities in the layer. This fact provides some physical basis for the separation of the gravity field anomalies. It has been shown by power spectrum analysis of the free air anomalies and gravity field of isostatically compensated model of the lithosphere for the North Atlantic and adjacent areas of America, Europe and Mediterranean, that the attraction of isostatically compensated model is significant for any wave length of the field. It causes significant error in the interpretation if long wavelength constituents of the free air gravity anomalies are considered as a field of deep anomalous masses. The isostatic anomalies und isostatic geoid are free from the influences of isostatically compensated lithosphere. The characteristic feature of the isostatic anomalies power spectrum is a pronounced minimum at the wavelength of about 1000 km. The relative homogeneity of the asthenosphere may explain this minimum. It means that principal density inhomogeneities of the Earth's interior are separated by the asthenospheric layer. Such a minimum has not been observed at the power spectrum of free air anomalies being masked by corresponding wavelength of the field of isostatically compensated lithosphere. Isostatic anomalies that reflect the differences between the real structure of the lithosphere and its isostatically compensated model have wavelengths less than 1000 km. Isostatic anomalies with the wavelength more than 1000 km reflect the attraction of density inhomogeneities situated under the level of isostatic compensation. The basic features of power spectrum of isostatic anomalies are the same for oceanic and continental areas. The method based on Kolmogorov-Wiener filtration which consideres statistical characteristics of the field has been developed to divide the isostatic gravity anomalies into lithosphere and mantle components. For the North Atlantic and adjacent areas the field of mantle inhomogeneities has been determined.     

The Role of Multi-Mission ERS Altimetry in the Determination of the Marine Geoid in the Azores

This study concerns the determination of a regional geoid model in the North Atlantic area surrounding the Azores islands by combining multi-mission altimetry from the ERS (European Remote Sensing) satellites and surface gravity data. A high resolution mean sea surface, named AZOMSS99, has been derived using altimeter data from ERS-1 and ERS-2 35-day cycles, spanning a period of about four years, and from ERS-1 geodetic mission. Special attention has been paid to data processing of points around the islands due to land contamination on some of the geophysical corrections. A gravimetric geoid has been computed from all available surface gravity, including land and sea observations acquired during an observation campaign that took place in the Azores in October 1997 in the scope of a European and a Portuguese project. Free air gravity anomalies were derived by altimetric inversion of the mean sea surface heights. These were used to fill the large gaps in the surface gravity and combined solutions were computed using both types of data. The gravimetric and combined solutions have been compared with the mean sea surface and GPS (Global Positioning System)-levelling derived geoid undulations in five islands. It is shown that the inclusion of altimeter data improves geoid accuracy by about one order of magnitude. Combined geoid solutions have been obtained with an accuracy of better than one decimetre.     

Combined satellite altimetry and shipborne gravimetry data processing

The recovery of quantities related to the gravity field (i.e., geoid heights and gravity anomalies) is carried out in a test area of the central Mediterranean Sea using 5' × 5' marine gravity data and satellite altimeter data from the Geodetic Mission (GM) of ERS‐J. The optimal combination of the two heterogeneous data sources is performed using (1) the space‐domain least‐squares collocation (LSC) method, and (2) the frequency‐domain input‐output system theory (IOST). The results derived by these methods agree at the level of 2 cm in terms of standard deviation in the case of the geoid height prediction. The gravity anomaly prediction results by the same methods vary between 2.18 and 2.54 mGal in terms of standard deviation. In all cases, the spectral techniques have a much higher computational efficiency than the collocation procedure. In order to investigate the importance of satellite altimetry for gravity field modeling, a pure gravimetric geoid solution, carried out in a previous study for our lest area by the fast collocation approach (FCOL), is used in comparison with the combined geoid models. The combined solutions give more accurate results, at the level of about 15 cm in terms of standard deviation, than the gravimetric geoid solution, when the geoid heights derived by each method are compared with TOPEX altimeter sea surface heights (SSHs). Moreover, nonisotropic power spectral density functions (PSDs) can be easily used by IOST, while LSC requires isotropic covariance functions. The results show that higher prediction accuracies are always obtained when using a priori nonisotropic information instead of isotropic information.     

6′x 10′ Free air gravity anomalies of Europe including marine areas

A set of 103 997 free air gravity anomalies in 6′x 10′ blocks has been compiled covering Europe including the Mediterranean Sea, North Sea, Norwegian Sea, Baltic Sea and parts of the North Atlantic Ocean. Concerning sea areas, this data set is based on a collection of point free air gravity anomalies. Anomalies for land areas have been compiled resp. computed from free air gravity anomaly maps or Bouguer anomaly maps and partly from supplied mean values of convenient small block size. Remaining gaps have been interpolated by means of least squares prediction filtering. The precision of the compiled mean free air gravity anomalies is estimated to ±7 mgal, verified by a comparison of independent gravity anomaly sets.     

Calculation of geoid undulations and gravity anomalies in the Northwest Pacific by using the Topex/Poseidon and Geosat altimeter data

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Collocation resolution of the oceanic geoid based on satellite altimetry

The approach presented is directed toward a specific adaptation of the least‐squares collocation with noise, yielding smooth predictions of geophysical quantities. The smoothing corresponds here to a truncated gravity field equivalent to an (n’, n') spherical‐harmonic expansion. This is reflected in the truncation, at the degree n‘, of the pertinent covariance and cross‐covariance functions in most (but not all) instances. The smooth predictions of geophysical quantities, made in an equilateral grid corresponding to the truncation degree n‘, serve in constructing contour maps after having been densified for the needs of a contour routine. Such a densification is carried out efficiently via errorless collocation with the degree truncation n‘ throughout. Consistent with this procedure, “residuals” at observation points (i.e., discrepancies between the contour map and the data) are computed using the same algorithm. The complete collocation approach is utilized for a 2° resolution of the earth's gravity field with emphasis on the oceanic geoid, based on the residuals from a global spherical‐harmonic adjustment of SEASAT altimetry. The presented results include contour maps of geoid undulations and gravity anomalies. They are compared to the results of a point‐mass adjustment, another technique based on the spherical‐harmonic adjustment. The agreement between these two techniques is found to be excellent.     

Residual depth and residual geoid anomalies of the South Atlantic ocean

Abstract

We have obtained the residual depth and residual geoid anomalies of the South Atlantic Ocean and interpreted them in terms of upper mantle processes. Starting from the 5’ X 5’ SYNBAPS data, we computed the 1° X 1° mean water depth. We digitized a recent sediment thickness map of the South Atlantic and corrected for the isostatic sediment loading effects. A plot of the corrected basement depth against crustal ages shows a good match to the depth‐age curve of the plate model. We then subtracted the predicted plate model depths from the corrected basement depths and obtained the 1° X 1° residual depth. The residual depth anomalies have positive values over the topographic highs and relative negative values over the ocean basins. A prominent asymmetry is observed between the residual depth over the Argentine Basin and that over the Cape Basin.

We have obtained the 1° X 1° residual geoid of the South Atlantic by subtracting both the long wavelength features and the geoid variations due to the plate cooling from the 1° X 1° Seasat altimeter derived geoid. The long wavelength features are represented by the degree and order 10 geoid of GEM1OC, and the geoid variations due to the plate cooling effects are predicted by the plate model geoid‐age relationship. The residual geoid anomalies also show an asymmetry although weaker than the case of the residual depth, between the Argentine and Cape basins.

By taking the 5° X 5° averages, we removed possible plate bending effects on the depth and geoid anomalies. We then compared those two data sets with respect to the reported hot spot tracks in the South Atlantic. The residual depth generally shows positive values over the hot spot tracks, whereas the residual geoid does not show any sign of them. A prominent asymmetric feature of depth and geoid anomalies is observed over the Argentine and Cape basins. This asymmetry is probably caused by hotter and less dense materials beneath the Cape Basin. Hot spots or other mantle upwellings may be the heat sources.     

Oceanic gravity by analytical inversion of Hotine's formula

An analytical inversion of the Hotine formula is developed using fast Fourier transform techniques. Detailed mathematical derivations are used to explain the concepts behind the inverse transformation. Three modifications of the analytical inversion of the Hotine formula are compared and tested using both synthetic data from the OSU91A geopotential model and real GEOSAT altimetry data from the Exact Repeat Mission. The stability of this inverse Hotine approach is investigated using simulated data, and numerical tests are done to quantify the stability of this approach. The approach seems to be numerically stable without employing any stabilization technique. Estimated gravity information from GEOSTAT altimetry data is compared to marine gravity data from shipboard measurements in the Orphan Knoll area. The standard deviations and mean values of the differences between satellite and marine gravity disturbances are 8.2 and 2.9 mGal for the planar approximation, 9.2 and 3.7 mGal for the spherical approximation, and 9.5 and 1.9 mGal for the Molodenskii‐like approximation, respectively, indicating that latitude‐dependent errors affect the latter two approximations. Such errors could be eliminated by performing the calculations by the rigorous one‐dimensional fast Fourier transform (FFT) technique, and any data noise could be filtered out by utilizing covariance knowledge about the input geoid undulations and their errors. Simulation studies also showed that the accuracy of the techniques (for all approximations) can reach a root‐mean‐square (RMS) level of only a few mGal when proper treatment of FFT edge effects is employed and a rather wide area of results is disregarded around the edges.     

Comparison of Satellite Altimetric Gravity and Global Geopotential Models with Shipborne Gravity in the Red Sea

The determination of high-resolution geoid for marine regions requires the integration of gravity data provided by different sources, e.g. global geopotential models, satellite altimetry, and shipborne gravimetric observations. Shipborne gravity data, acquired over a long time, comprises the short-wavelengths gravitation signal. This paper aims to produce a consistent gravity field over the Red Sea region to be used for geoid modelling. Both, the leave-one-out cross-validation and Kriging prediction techniques were chosen to ensure that the observed shipborne gravity data are consistent as well as free of gross-errors. A confidence level equivalent to 95.4% was decided to filter the observed shipborne data, while the cross-validation algorithm was repeatedly applied until the standard deviation of the residuals between the observed and estimated values are less than 1.5 mGal, which led to the elimination of about 17.7% of the shipborne gravity dataset. A comparison between the shipborne gravity data with DTU13 and SSv23.1 satellite altimetry-derived gravity models is done and reported. The corresponding results revealed that altimetry models almost have identical data content when compared one another, where the DTU13 gave better results with a mean and standard deviation of ?2.40 and 8.71 mGal, respectively. A statistical comparison has been made between different global geopotential models (GGMs) and shipborne gravity data. The Spectral Enhancement Method was applied to overcome the existing spectral gap between the GGMs and shipborne gravity data. EGM2008 manifested the best results with differences characterised with a mean of 1.35 mGal and a standard deviation of 11.11 mGal. Finally, the least-squares collocation (LSC) was implemented to combine the shipborne gravity data with DTU13 in order to create a unique and consistent gravity field over the Red Sea with no data voids. The combined data were independently tested using a total number of 95 randomly chosen shipborne gravity stations. The comparison between the extracted shipborne gravity data and DTU13 altimetry anomalies before and after applying the LSC revealed that a significant improvement is procurable from the combined dataset, in which the mean and standard deviation of the differences dropped from ?3.60 and 9.31 mGal to ?0.39 and 2.04 mGal, respectively.     

Relation between Gravity Field Feature and Tectonics and Earthquakes in Taiwan and Its Adjacent Seas

Short wave gravity anomaly is correlated to sea floor topography in the gravity field of Taiwan and its adjacent seas. Gravity values of 200 × 10-5ms-2 at Yushang and -160 × 10-5ms-2 at Liuqiu sea trench are respectively the maximum and minimum gravity values in this area.Bouguer gravity anomaly reflects not only Moho interface undulation, but also fault distribution.The inflexion of gradient belt of Bouguer gravity anomaly is a spot liable to earthquakes. Middlelong wave geoid is the best data to invert crustal thickness. We calculate crustal thickness by using geoid data, and the maximum value is 38km; the minimum value is 12km in Taiwan and its adjacent seas.     

On the Impact of Mispointing Error and Hamming Filtering on Altimeter Waveform Retracking and Skewness Retrieval

An adequate conceptual definition of the geoid is essential for the unambiguous combination of satellite tracking data, satellite al‐timetry, and surface gravity measurements to obtain sea surface topography. The factors influencing the selection of a particular level surface of the earth's gravity field include the purpose(s) for which the geoid is to be used at the 5‐cm level, and the types of data to be used in achieving these objectives. The principal reasons for high precision determinations of the shape of the geoid are: the determination of sea surface topography for applications in oceanography; and the unification of leveling datums with a resolution equivalent to that of first order geodetic leveling. A conceptual definition of the geoid acceptable to oceanographers would be: The geoid for a selected epoch of measurement is that level surface of the earth's gravity field in relation to which the average non‐tidal (or quasi‐stationary) sea surface topography is zero as sampled globally in ocean regions. In the geodetic context, it would be convenient, though not essential, to modify this definition in such a way that the global sea surface topography had zero mean as sampled for evaluations of the geodetic boundary value problem. In either case, a basis exists for unifying all leveling datums serving areas in excess of 10⁶ km², using either gravity anomaly data for the regions or precise determinations of position at first order bench marks. Unfavorable signal‐to‐noise ratios can pose problems when dealing with datums serving smaller areas. Elevation and gravity data banks must be correctly referenced to leveling datums prior to use in sea surface topography determinations. A recent attempt to upgrade the Australian gravity anomaly data bank indicates that all current data banks of this type are inadequate for the task. It is unlikely that time variations in the radial position of the geoid as conceptually defined above, will exceed ±5 cm per century, provided the rate of earth expansion was less than 1 part in 10¹⁰ yr^‐l and there is no dramatic change in the present rate of secular change in Mean Sea Level.     

Improvement in the Determination of the Marine Geoid by Estimating the Bathymetry from Altimetry and Depth Soundings

Abstract

The contribution of bathymetry to the estimation of gravity field related quantities is investigated in an extended test area in the Mediterranean Sea. The region is located southwest of the island of Crete, Greece, bounded between 33^? ≤ ? ≤ 35^? and 15^? ≤ λ ≤ 25^?. Gravity anomalies from the KMS99 gravity field and shipborne depth soundings are used with a priori statistical characteristics of depths in a least-squares collocation procedure to estimate a new bathymetry model. Two different global bathymetry models, namely JGP95E and Sandwell and Smith V8, are used to derive the depth a priori statistical information, while the estimated model is compared against both the global ones and the shipborne depth soundings to assess whether there is an improvement. Various marine geoid models are estimated using ERS1 and GEOSAT Geodetic Mission altimetry and shipborne gravity data. In that process, the effect of the bathymetry is computed using both the estimated and the original depths through a residual terrain modeling reduction. The TOPEX/Poseidon Sea Surface Heights, known for their high accuracy and precision, and the GEOMED solution for the geoid in the Mediterranean are used as control for the validation of the new geoid models and to assess the improvement that the estimated depths offer to geoid modeling. The results show that the newly estimated bathymetry agrees better (by about 30 to 300 m) with the shipborne depth soundings and provides smoother residual geoid heights and gravity anomalies (by about 8–20%) than those from global models. Finally, the achieved accuracy in geoid modeling ranges between 6 and 10 cm (1σ).     

Heterogeneous Gravity Data Fusion and Gravimetric Quasigeoid Computation in the Coastal Area of China

ABSTRACT

Different types of gravity observations are available over coastal areas. The main challenge for coastal geoid determination is the proper fusion of heterogeneous gravity data including land, shipborne, airborne, and altimetry-derived gravity data. This paper describes the gravity data fusion and the computation of the gravimetric quasigeoid in the coastal area of mainland China. An iterative procedure of the weighted least-squares prediction based on rectangular harmonic functions is used for merging the land, altimetric, shipborne, and airborne gravity data. Applying the analytical continuation method in Molodensky's theoretic frame, the merged gravity data are then used to determine the gravimetric quasigeoid model by using the generalized Stokes' integral in a remove-compute-restore fashion. The gravimetric quasigeoid model is compared with the height anomalies determined at 662 GPS leveling points over the coastal region of mainland China, where both the geodetic height and the normal height are known. The standard deviations of the differences in the coastal provinces range from 1.8 to 4.4 cm. For the entire computation area, the mean and standard deviation of the differences are 27.9 and 3.9 cm, respectively.     

Improvement in the Determination of the Marine Geoid by Estimating the Bathymetry from Altimetry and Depth Soundings

The contribution of bathymetry to the estimation of gravity field related quantities is investigated in an extended test area in the Mediterranean Sea. The region is located southwest of the island of Crete, Greece, bounded between 33^ˆ ≤ ϕ ≤ 35^ˆ and 15^ˆ ≤ λ ≤ 25^ˆ. Gravity anomalies from the KMS99 gravity field and shipborne depth soundings are used with a priori statistical characteristics of depths in a least-squares collocation procedure to estimate a new bathymetry model. Two different global bathymetry models, namely JGP95E and Sandwell and Smith V8, are used to derive the depth a priori statistical information, while the estimated model is compared against both the global ones and the shipborne depth soundings to assess whether there is an improvement. Various marine geoid models are estimated using ERS1 and GEOSAT Geodetic Mission altimetry and shipborne gravity data. In that process, the effect of the bathymetry is computed using both the estimated and the original depths through a residual terrain modeling reduction. The TOPEX/Poseidon Sea Surface Heights, known for their high accuracy and precision, and the GEOMED solution for the geoid in the Mediterranean are used as control for the validation of the new geoid models and to assess the improvement that the estimated depths offer to geoid modeling. The results show that the newly estimated bathymetry agrees better (by about 30 to 300 m) with the shipborne depth soundings and provides smoother residual geoid heights and gravity anomalies (by about 8-20%) than those from global models. Finally, the achieved accuracy in geoid modeling ranges between 6 and 10 cm (1σ).