The gravitational field of topographic-isostatic masses and the hypothesis of mass condensation II-the topographic-isostatic geoid

In Part I we focussed on a convergent representation of the gravitational potential generated bytopographic masses on top of the equipotential surface atMean Sea Level, thegeoid, and by those masses which compensate topography. Topographic masses have also been condensated, namely represented by a single layer. Part II extends the computation of the gravitational field of topographic-isostatic masses by a detailed analysis of itsforce field in terms ofvector-spherical harmonic functions. In addition, the discontinuous mass-condensated topographic gravitational force vector (head force) is given. Once we identify theMoho discontinuity asone interface of isostatically compensated topographical masses, we have computed the topographic potential and the gravitational potential which is generated by isostatically compensated masses atMean Sea Level, the geoid, and illustrated by various figures of geoidal undulations. In comparison to a data oriented global geoid computation ofJ. Engels (1991) the conclusion can be made that the assumption of aconstant crustal mass density, the basic condition for isostatic modeling, does not apply. Insteaddensity variations in the crust, e.g. betweenoceanic and continental crust densities, have to be introduced in order to match the global real geoid and its topographic-isostatic model. The performed analysis documents that thestandard isostatic models based upon aconstant crustal density areunreal.     

Geoid and Sea Surface Topography Derived from ERS-1 Altimeter Data by the Adjoint Method

A method for splitting sea surface height measurements from satellite altimetry into geoid undulations and sea surface topography is presented. The method is based on a combination of the information from altimeter data and a dynamic sea surface height model. The model consists of geoid undulations and a quasi-geostrophic model for expressing the sea surface topography. The goal is the estimation of those values of the parameters of the sea surface height model that provide a least-squares fit of the model to the data. The solution is accomplished by the adjoint method which makes use of the adjoint model for computing the gradient of the cost function of the least-squares adjustment and an optimization algorithm for obtaining improved parameters. The estimation is applied to the North Atlantic. ERS-1 altimeter data of the year 1993 are used. The resulting geoid agrees well with the geoid of the EGM96 gravity model.     

Crustal density structure in the Spanish Central System derived from gravity data analysis (Central Spain)

Shallow and deep sources generate a gravity low in the central Iberian Peninsula. Long-wavelength shallow sources are two continental sedimentary basins, the Duero and the Tajo Basins, separated by a narrow mountainous chain called the Spanish Central System. To investigate the crustal density structure, a multitaper spectral analysis of gravity data was applied. To minimise biases due to misleading shallow and deep anomaly sources of similar wavelength, first an estimation of gravity anomaly due to Cenozoic sedimentary infill was made. Power spectral analysis indicates two crustal discontinuities at mean depths of 31.1 ± 3.6 and 11.6 ± 0.2 km, respectively. Comparisons with seismic data reveal that the shallow density discontinuity is related to the upper crust lower limit and the deeper source corresponds to the Moho discontinuity. A 3D-depth model for the Moho was obtained by inverse modelling of regional gravity anomalies in the Fourier domain. The Moho depth varies between a mean depth of 31 km and 34 km. Maximum depth is located in a NW–SE trough. Gravity modelling points to lateral density variations in the upper crust. The Central System structure is described as a crustal block uplifted by NE–SW reverse faults. The formation of the system involves displacement along an intracrustal detachment in the middle crust. This detachment would split into several high-angle reverse faults verging both NW and SE. The direction of transport is northwards, the detachment probably being rooted at the Moho.     

大地水准面异常在湖南地区的地球物理解释

以湖南地区为例,利用超高阶地球重力位模型EGM2008计算了研究区的重力大地水准面,并采用棱柱体公式和球体公式相结合的方法分别进行了完全地形改正和Airy-Heiskanen局部均衡改正,得到布格大地水准面和均衡大地水准面.对三种大地水准面进行不同波长分量的分离处理,得到包含不同深度异常信息的剩余大地水准面,并结合其他地球物理资料对研究区进行了详细的地球物理解释.结果表明,剩余重力大地水准面可以有效地反映出研究区内的深部构造特征,如深大断裂带分布、构造块体位置、上地幔密度横向分布等,但对地壳内异常结构反映不明显;研究区岩石圈密度变化相对平缓,厚度由东向西增加;根据剩余均衡大地水准面及研究区Airy局部均衡莫霍面,可以大致推测出研究区的莫霍面起伏形态以及均衡状态,可作为一种有用的参考信息.     

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

INTRODUCTIONThegeoidistheiargeopotentials~econfidingmostlywiththemeanseasurfaceandisdenotedastheheightrelativetotheidealelliPSes~eoftheearth.Thegeoidundulationsinglobalaceareupto100m.TheunevenstructureOftheearthgivesrisetotheunevenfeatureofthecitysot...     

Water wave scattering by bottom undulations in an ice-covered two-layer fluid

The scattering of plane surface waves by bottom undulations in an ice-covered ocean modelled as a two-layer fluid consisting of a layer of fresh water of lesser density above a deep layer of salt water, is investigated here by using a simplified perturbation analysis. In such a two-layer fluid there exist waves of two different modes, one with higher mode propagates along the interface and the other with lower mode propagates along the ice-cover. An incident wave of a particular mode gets reflected and transmitted by the bottom undulations into waves of both the modes so that transfer of wave energy from one mode to another takes place. The first-order reflection and transmission coefficients of two different modes are obtained due to incident waves of again two different modes by employing Fourier transform technique in the mathematical analysis. For sinusoidal bottom topography these coefficients are depicted graphically against the wavenumber. These figures show how the transfer of energy from one mode to another takes place.     

A new hybrid geoid model for Japan, GSIGEO2000

 The latest gravimetric geoid model for Japan, JGEOID2000, was successfully combined with the nationwide net of GPS at benchmarks, yielding a new hybrid geoid model for Japan, GSIGEO2000. The least-squares collocation (LSC) method was applied as an interpolation for fitting JGEOID2000 to the GPS/leveling geoid undulations. The GPS/leveling geoid undulation data were reanalyzed in advance, in terms of three-dimensional positions from GPS and orthometric heights from leveling. The new hybrid geoid model is, therefore, compatible with the new Japanese geodetic reference frame. GSIGEO2000 was evaluated internally and independently and the precision was estimated at 4 cm throughout nearly the whole region. Received: 15 October 2001 / Accepted: 27 March 2002 Acknowledgments. Messrs. Toshio Kunimi and Tadashi Saito at the Third Geodetic Division of the Geographical Survey Institute (GSI) mainly carried out the computations of most of the updated leveled heights. With regard to the reanalysis of GPS data, the discussions with Messrs. Yuki Hatanaka and Shoichi Matsumura of GSI were of great help in building the analysis strategy. Messrs. Kazuyuki Tanaka and Hiromi Shigematsu collaborated in the preparatory stages of GPS data computation. The authors' thanks are extended to these colleagues. Some plots were made by GMT software (Wessel and Smith 1991). Correspondence to: Y. Kuroishi     

Topography of the Moho undulations in France from gravity data: their age and origin

The complete gravity data set from France and part of the neighboring countries has been analyzed to compute the topography of the Moho undulations. This work is based on an improved filtering technique and an appropriate assumed density contrast between the crust and the upper mantle. Comparison with deep seismic refraction data reveals that this relief map expresses the continuity and geometry of the Moho undulations better than the sparsely distributed seismic refraction data in France. This gravity Moho map, though may not give absolute depths at places, provides a far better correlation with surface geology than the result from other geophysical techniques. Four domains have been recognized: (a) the Alpine domain where all the Moho undulations are concentric with the Alps; (b) the Armorican domain in which all the undulations are north-west/south-east oriented; (c) the Pyrenean domain, in which the undulations are parallel with the Mountain chain; and (d) the Massif Central Domain which does not show clear structural orientation because of the influence of the strong heat flow located at the lower crust/upper mantle interface. Study of the topography and of the superficial structures associated with these undulations reveals that the undulations delineated in the Alpine Domain result from the Tertiary compression which shaped the Alps. The Armorican Domain was first created during the Lower to Middle Cretaceous opening of the Bay of Biscay. It is now slightly affected by the Tertiary to Quaternary closure of this Bay. The Pyrenean Domain was successively shaped by the Lower Cretaceous oblique opening of the Bay of Biscay and by the Upper Cretaceous to Eocene northward displacement of Spain. Comparison between the Moho undulations map and the stress map of France reveals that most of the undulations are perpendicular to the actual shortening directions. This observation suggests that the Mesozoic, Cenozoic and Quaternary stress directions were roughly the same. Massif Central is characterized by the convergence of these three sets of undulations. Its Post-Oligocene uplift was probably the result of the converging stresses recognized in the three surrounding domains. When the Moho undulations and the topography are compared, two types of periodic crustal instabilities can be recognized. One corresponds to the buckling of the crust developed under compression, the other to boudinage which was associated with extension. Both phenomena show a typical wavelength of 200–250 km which is in agreement with the results of the actual physical and numerical modeling currently available.     

利用卫星测高数据推求中国近海及邻域大地水准面起伏和重力异常研究水

首先对中国近海及邻域（0°-40°N,105°─135°E）内4年的T／P卫星测高数据（1992－10-03-1996－10－09）和1年多的ERS－1卫星测高数据（1992－10－23-1994－01－20）进行了预处理以剔除数据中的粗差影响．然后,对卫星的重复轨迹采用"共线"处理方法得到该海域的平均海平面．在扣除海面地形的影响后得到该海域30'×X30'大地水准面起伏。再分别采用Stokes公式逆运算加FFT技术和最小二乘配置法恢复出该海域30'×30'海洋重力异常．所得到的海洋大地水准面起伏精度为8．5cm,恢复出的重力异常的精度为35×10－5m／s2．     

An estimation of the height system bias parameter <Emphasis Type="Italic">N</Emphasis><Subscript>0</Subscript> using least squares collocation from observed gravity and GPS-levelling data

This paper deals with the analysis of gravity anomaly and precise levelling in conjunction with GPS-Levelling data for the computation of a gravimetric geoid and an estimate of the height system bias parameter N_o for the vertical datum in Pakistan by means of least squares collocation technique. The long term objective is to obtain a regional geoid (or quasi-geoid) modeling using a combination of local data with a high degree and order Earth gravity model (EGM) and to determine a bias (if there is one) with respect to a global mean sea surface. An application of collocation with the optimal covariance parameters has facilitated to achieve gravimetric height anomalies in a global geocentric datum. Residual terrain modeling (RTM) technique has been used in combination with the EGM96 for the reduction and smoothing of the gravity data. A value for the bias parameter N_o has been estimated with reference to the local GPS-Levelling datum that appears to be 0.705 m with 0.07 m mean square error. The gravimetric height anomalies were compared with height anomalies obtained from GPS-Levelling stations using least square collocation with and without bias adjustment. The bias adjustment minimizes the difference between the gravimetric height anomalies with respect to residual GPS-Levelling data and the standard deviation of the differences drops from 35 cm to 2.6 cm. The results of this study suggest that N_o adjustment may be a good alternative for the fitting of the final gravimetric geoid as is generally done when using FFT methods.