Gravity,Mean Sea Surface,and Bathymetry Models Comparison in the Canary Islands

In geodetic and oceanographic studies generally, some reference surfaces are needed. These surfaces must represent as much as possible the gravity field of the Earth and the height/bathymetry systems. In the last years, several gravimetric, bathymetric, and mean sea surface models have appeared. Analyzing them it is possible to see that there are significant discrepancies between the models provided by different authors or organizations; there are also differences between the models and data obtained by independent measurements. We present the analysis of such differences and determine the most representative choice of models, in our opinion, for the Canary Islands region.     

大地水准面异常地震层析和地慢热动力模型

当人们使用长波长的地震层析数据利用地震波速和密度的实验关系外推而直接计算地球大地水准面异常时惊奇地发现，计算大地水准面异常和观测大地水准面异常无论在异常型态或者异常幅度上均相差甚远。因而不能将地幔看做“刚性”地幔，而真实的地幔应当是一“动力”的地幔，它的内部，甚至被看做固体地幔内部也存在物质流动，否则无法解释上述两种水准面的差异。地幔，特别是下地幔远比人们设想的要活跃得多。地幔内部存在一个复杂的热动力系统。全球规模的大尺度对流、上地幔范围的二级对流、层状对流、核－幔边界上部Ｄ″层内很小尺度对流以及地幔热柱形式的对流可能同时存在于地幔之中，它们相互独立又相互影响，而形成多层次、多形态的运动格局。为了进一步完善、丰富人类对其赖以生存的地球的认识，除了需要建立更为合理的非线性的地幔热动力学模型之外，更加精细的、更加精确的大地测量、地球物理和地球动力学观测资料是必需的。     

A rapid method to map the crustal and lithospheric thickness using elevation, geoid anomaly and thermal analysis. Application to the Gibraltar Arc System, Atlas Mountains and adjacent zones

We present a method based on the combination of elevation and geoid anomaly data together with thermal field to map crustal and lithospheric thickness. The main assumptions are local isostasy and a four-layered model composed of crust, lithospheric mantle, sea water and the asthenosphere. We consider a linear density gradient for the crust and a temperature dependent density for the lithospheric mantle. We perform sensitivity tests to evaluate the effect of the variation of the model parameters and the influence of RMS error of elevation and geoid anomaly databases. The application of this method to the Gibraltar Arc System, Atlas Mountains and adjacent zones reveals the presence of a lithospheric thinning zone, SW–NE oriented. This zone affects the High and Middle Atlas and extends from the Canary Islands to the eastern Alboran Basin and is probably linked with a similarly trending zone of thick lithosphere constituting the western Betics, eastern Rif, Rharb Basin, and Gulf of Cadiz. A number of different, even mutually opposite, geodynamic models have been proposed to explain the origin and evolution of the study area. Our results suggest that a plausible slab-retreating model should incorporate tear and asymmetric roll-back of the subducting slab to fit the present-day observed lithosphere geometry. In this context, the lithospheric thinning would be caused by lateral asthenospheric flow. An alternative mechanism responsible for lithospheric thinning is the presence of a hot magmatic reservoir derived from a deep ancient plume centred in the Canary Island, and extending as far as Central Europe.     

Sea-level changes, geoid and gravity anomalies due to Pleistocene deglaciation by means of multilayered, analytical Earth models

A new class of analytical, multilayered, viscoelastic Earth models based on PREM, with an incompressible, linear, viscoelastic Maxwell rheology, is applied to the modeling of global sea-level changes due to Pleistocene deglaciation. Until now, analytical schemes based on normal mode theory, have dealt with at most five layers, an elastic lithosphere, a three layered mantle including a transition zone, and a core (Spada et al., 1992. Geophys. J. Int. 109, 683–700). The novelty of our approach, used for the first time in sea-level studies, stands on an analytical scheme that can reproduce continuous elastic and rheological stratification when a sufficient number of layers is taken into account. We specifically assess the importance of our results for the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) satellite mission. GOCE will resolve the gravity field with a spatial resolution (half-wavelength) of 75 km and amplitude of 1.5 mgal, with a uniform coverage over the Earth, including presently unsurveyed, remote areas. Our models lead to post-glacial rebound induced free air gravity anomalies of a few mgals peak-to-peak in the harmonic degree range l=80–200, which will be discernible by GOCE. This finding demonstrates that post-glacial rebound has a high frequency component in the gravity field that can in principle be resolved by high resolution gravity satellite missions. We show that post-glacial rebound can contribute a substantial fraction to present-day sea-level variations and point out that for the Mediterranean Sea they are of the same order of magnitude as those induced by tectonic processes.     

Updating and computing the geoid using two dimensional fast Hartley transform and fast T transform

In the analyses of 2D real arrays, fast Hartley (FHT), fast T (FTT) and real-valued fast Fourier transforms are generally preferred in lieu of a complex fast Fourier transform due to the advantages of the former with respect to disk storage and computation time. Although the FHT and the FTT in one dimension are identical, they are different in two or more dimensions. Therefore, first, definitions and some properties of both transforms and the related 2D FHT and FTT algorithms are stated. After reviewing the 2D FHT and FTT solutions of Stokes' formula in planar approximation, 2D FHT and FTT methods are developed for geoid updating to incorporate additional gravity anomalies. The methods are applied for a test area which includes a 64×64 grid of 3^′×3^′ point gravity anomalies and geoid heights calculated from point masses. The geoids computed by 2D FHT and FTT are found to be identical. However, the RMS value of the differences between the computed and test geoid is ±15 mm. The numerical simulations indicate that the new methods of geoid updating are practical and accurate with considerable savings on storage requirements. Received: 15 February 1996; Accepted: 22 January 1997     

A general model for modifying Stokes’ formula and its least-squares solution

Today the combination of Stokes formula and an Earth gravity model (EGM) for geoid determination is a standard procedure. However, the method of modifying Stokes formula varies from author to author, and numerous methods of modification exist. Most methods modify Stokes kernel, but the most widely applied method, the remove compute restore technique, removes the EGM from the gravity anomaly to attain a residual gravity anomaly under Stokes integral, and at least one known method modifies both Stokes kernel and the gravity anomaly. A general model for modifying Stokes formula is presented; it includes most of the well-known techniques of modification as special cases. By assuming that the error spectra of the gravity anomalies and the EGM are known, the optimum model of modification is derived based on the least-squares principle. This solution minimizes the expected mean square error (MSE) of all possible solutions of the general geoid model. A practical formula for estimating the MSE is also presented. The power of the optimum method is demonstrated in two special cases. AcknowledgementsThis paper was partly written whilst the author was a visiting scientist at The University of New South Wales, Sydney, Australia. He is indebted to Professor W. Kearsley and his colleagues, and their hospitality is acknowledged.     

Transforming ellipsoidal heights and geoid undulations between different geodetic reference frames

Transforming height information that refers to an ellipsoidal Earth reference model, such as the geometric heights determined from GPS measurements or the geoid undulations obtained by a gravimetric geoid solution, from one geodetic reference frame (GRF) to another is an important task whose proper implementation is crucial for many geodetic, surveying and mapping applications. This paper presents the required methodology to deal with the above problem when we are given the Helmert transformation parameters that link the underlying Cartesian coordinate systems to which an Earth reference ellipsoid is attached. The main emphasis is on the effect of GRF spatial scale differences in coordinate transformations involving reference ellipsoids, for the particular case of heights. Since every three-dimensional Cartesian coordinate system ‘gauges’ an attached ellipsoid according to its own accessible scale, there will exist a supplementary contribution from the scale variation between the involved GRFs on the relative size of their attached reference ellipsoids. Neglecting such a scale-induced indirect effect corrupts the values for the curvilinear geodetic coordinates obtained from a similarity transformation model, and meter-level apparent offsets can be introduced in the transformed heights. The paper explains the above issues in detail and presents the necessary mathematical framework for their treatment. An erratum to this article can be found at     

海域航空重力快速构建区域大地水准面

我国在海域开展了大规模的航空重力勘探,这些资料对构建高精度大地水准面具有重要价值.基于此,本文提出一种利用海域航空重力测量数据快速构建大地水准面的方法.该方法基于移去-恢复法思想,利用位场最小曲率方法对航空重力数据进行高精度向下延拓并获取相应的扰动位,实现航空重力测量快速构建海域大地水准面.与斯托克斯积分计算相比,采用了处理效率更高的频率域位场转换,解决了向下延拓及垂向积分时航空重力异常数据空白及扩边问题,具有较高的位场转换精度.本文应用EGM2008模拟航空重力数据进行模型验证,计算结果与其给出的水准面的精度相当;同时,也选取GRAV-D计划的航空重力数据进行实际验证,计算结果与xGEOID18B水准面模型精度基本一致.模型验证和实际应用验证了本方法的实用性.

     

The CARIB97 high-resolution geoid height model for the Caribbean Sea

A 2×2 arc-minute resolution geoid model, CARIB97, has been computed covering the Caribbean Sea. The geoid undulations refer to the GRS-80 ellipsoid, centered at the ITRF94 (1996.0) origin. The geoid level is defined by adopting the gravity potential on the geoid as W ₀=62 636 856.88 m²/s² and a gravity-mass constant of GM=3.986 004 418×10¹⁴ m³/s². The geoid model was computed by applying high-frequency corrections to the Earth Gravity Model 1996 global geopotential model in a remove-compute-restore procedure. The permanent tide system of CARIB97 is non-tidal. Comparison of CARIB97 geoid heights to 31 GPS/tidal (ITRF94/local) benchmarks shows an average offset (h–H–N) of 51 cm, with an Root Mean Square (RMS) of 62 cm about the average. This represents an improvement over the use of a global geoid model for the region. However, because the measured orthometric heights (H) refer to many differing tidal datums, these comparisons are biased by localized permanent ocean dynamic topography (PODT). Therefore, we interpret the 51 cm as partially an estimate of the average PODT in the vicinity of the 31 island benchmarks. On an island-by-island basis, CARIB97 now offers the ability to analyze local datum problems which were previously unrecognized due to a lack of high-resolution geoid information in the area. Received: 2 January 1998 / Accepted: 18 August 1998