On the gravimetric undulations of the geoid and the deflections of the vertical

Summary The author mentions the aims of the World-wide gravity project he established in the Ohio State University in Columbus, in 1950. He outlines the practical procedure of the gravimetric computations of the undulationsN and the vertical deflection components and and emphasizes that only by the global international cooperation and additional gravity observations at sea carried out during the last decade it has been possible to gather to Columbus the needed gravity material. Since there exist still large gravimetrically unsurveyed areas it is of vital significance to study what gravity anomalies are best to be used for these regions. The given figures concerning the accuracy of theN, and , estimated theoretically and obtained in practice, indicate that in the gravimetrically well surveyed parts of the world like in Europe and the United States we can get gravimetrically on basis of existing gravity material theN-values with accuracy of about 5–10 meters, and and with the accuracy of about 1. The geoid undulationsN are already computed in Columbus for more than 6000 points of the northern hemisphere. The sample maps show the interesting geoid of Europe and vicinity between the latitudes 60° and 30° and longitude 5° W and 30° E, drafted on basis of more than 1000N-values computed at the corners of 1°×1° squares. It is interesting to realize that the geoid undulations in all this area are positive, the extreme values being between 40 and 50 meters. The geophysical significance of the geoid maps of this kind is pointed out.     

Lunar deflections of the vertical and their elementary interpretation

Summary The selenopotential was determined at the Apollo 12 landing site (A12) using the selenocentric constant, Stokes' constants of the Moon up to n=13, the angular velocity of rotation of the Moon and the value of gravity directly observed at A12. Using and the constants mentioned above, the radius-vector of the equiselenopotential surface passing through A12 was derived. The fundamental selenocentric parameters, based on this surface, were computed, as well as the deflections of the vertical especially in some strongly anomalous regions of the Moon. For some mascons an elementary interpretation has been carried out.Dedicated to Academician Alois Zátopek on His 65th BirthdayPresented at the XVth Plenary Meeting of COSPAR, Madrid, May 10–24, 1972, under the title:Selenocentric Reference Parameters and Deflections of the Vertical Related to the Equipotential Surface Passing through the Apollo 12 Landing Site.     

Upward continuation of the deflections of the vertical by spherical integral formulas

Zusammenfassung In dieser Arbeit werden sph?rische Integralformeln für die Fortsetzung auf der Kugel gegebener Lotabweichungskomponenten ξ, η in den Au?enraum gegeben.

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Address: Hardenbergstrasse 34, 1 Berlin 12.     

Ellipsoidal vertical deflections and ellipsoidal gravity disturbance: Case studies

First, we present three different definitions of the vertical which relate to (i) astronomical longitude and astronomical latitude as spherical coordinates in gravity space, (ii) Gauss surface normal coordinates (also called geodetic coordinates) of type ellipsoidal longitude and ellipsoidal latitude and (iii) Jacobi ellipsoidal coordinates of type spheroidal longitude and spheroidal latitude in geometry space. Up to terms of second order those vertical deflections agree to each other. Vertical deflections and gravity disturbances relate to a reference gravity potential. In order to refer the horizontal and vertical components of the disturbing gravity field to a reference gravity field, which is physically meaningful, we have chosen the Somigliana-Pizzetti gravity potential as well as its gradient. Second, we give a new closed-form representation of Somigliana-Pizzetti gravity, accurate to the sub Nano Gal level. Third, we represent the gravitational disturbing potential in terms of Jacobi ellipsoidal harmonics. As soon as we take reference to a normal potential of Somigliana-Pizzetti type, the ellipsoidal harmonics of degree/order (0,0), (1,0), (1, − 1), (1,1) and (2,0) are eliminated from the gravitational disturbing potential. Fourth, we compute in all detail the gradient of the gravitational disturbing potential, in particular in orthonormal ellipsoidal vector harmonics. Proper weighting functions for orthonormality on the International Reference Ellipsoid are constructed and tabulated. In this way, we finally arrive at an ellipsoidal harmonic representation of vertical deflections and gravity disturbances. Fifth, for an ellipsoidal harmonic Gravity Earth Model (SEGEN: http://www.uni-stuttgart.de/gi/research/paper/coefficients/coefficients.zip) up to degree/order 360/360 we compute the global maps of ellipsoidal vertical deflections and ellipsoidal gravity disturbances which transfer a great amount of geophysical information in a properly chosen equiareal ellipsoidal map projection.     

Accuracy of deflections of vertical determined from discrete solution of the geodetic boundary value problem

Summary The general problem of determining the figure of the earth leads to the solution of the geodetic boundary value problem. By its discrete approximation we obtain the discrete disturbing potential that maintains all properties of the original problem. Thus, the discrete approximation of the disturbing potential can be used in studying the behaviour of the earth's gravity field outside the disturbing masses. The deflections of the vertical are one of the quantities describing the behaviour of the earth's gravity field. A method for their computation from the discrete solution of the geodetic boundary value problem is put forth and estimates for its accuracy are given.     

Determination of the parameters of a selenocentric reference system and the deflections of the vertical at the lunar surface

Summary Stokes' constants and, the selenocentric constant, and the angular velocity of the rotation of the Moon define the shape of the external equiselenopotential surfaces, generalized in dependence on the degree N of the harmonics preserved. The scale factor for lengths was computed on the basis of absolute gravity measurement made by the first lunarlanding mission Apollo11 at the landing site[1] under the assumption of a sufficient accuracy of the Stokes' constants used[2, 15]. Anyway, the numerical solution here is only to be considered as an example of the application of the outlined theoretical method, inclusive of the parameters of the lunar reference system, which will be made considerably more accurate when gravity measurements at more points of the lunar surface are available.Presented at the XVth IUGG General Assembly, Moscow, July 30 – August 14, 1971.     

Calculation of the second vertical derivative of gravity field

Summary Two formulas, using Taylor's series expansion method, have been developed by means of which the second vertical derivative of gravity field may be computed at any point in the horizontal plane of observation. A comparative study of the resolving powers of different approaches by performing numerical and filter response tests suggests that the formulas provide a definite improvement over many existing methods.NGRI Contribution No. 69-130.     

Slowness domain offset and traveltime approximations in layered vertical transversely isotropic media

Considering horizontally layered transversely isotropic media with vertical symmetry axis and all types of pure‐mode and converted waves we present a new wide‐angle series approximation for the kinematical characteristics of reflected waves: horizontal offset, intercept time, and total reflection traveltime as functions of horizontal slowness. The method is based on combining (gluing) both zero‐offset and (large) finite‐offset series coefficients. The horizontal slowness is bounded by the critical value, characterised by nearly horizontal propagation within the layer with the highest horizontal velocity. The suggested approximation uses five parameters to approximate the offset, six parameters to approximate the intercept time or the traveltime, and seven parameters to approximate any two or all three kinematical characteristics. Overall, the method is very accurate for pure‐mode compressional waves and shear waves polarised in the horizontal plane and for converted waves. The application of the method to pure‐mode shear waves polarised in the vertical plane is limited due to cusps and triplications. To demonstrate the high accuracy of the method, we consider a synthetic, multi‐layer model, and we plot the normalised errors with respect to numerical ray tracing.     

关于垂直分量磁变仪的灵敏度问题

目前国内地磁台垂直分量磁变以的灵敏度都普遍偏低，不能适应一些研究工作的需要，必须予以适当的提高，本文在基本不改变仪器固有灵敏度的前提下，采取记录光线二次反射法，以提高仪器的记录灵敏度，利用这种方法既使仪器的灵敏度提高一倍同时又保证了仪器原有的稳定性。     

On the solution of the Molodenskii boundary value problem using successive approximations

Gravimetric methods of determining the figure of the Earth can be divided into two groups. The first employs an auxiliary surface—the geoid. The second employs the so-called quasigeoid as an auxiliary surface. Its main advantage is in that it only uses surface measurements and that it does not require knowledge of the structure of the Earth's crust. This method was treated in [1]. The results display an accuracy of the order of the Earth's flattening. The purpose of this paper is to show that this method can be applied theoretically to reach an arbitrary accuracy.     

On the hemispheric correlations of vertical and meridional wind components

Summary Using the values of vertical velocities computed byJensen (1960) and the values of the meridional component of the wind for the months of January and April 1958, the covariances of both quantities were evaluated at different levels at various latitudes. These covariances are tabulated and the terms associated with transient eddies are compared to the standing eddy terms, the relative magnitudes of both contributions being finally investigated.

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Résumé En employant les valeurs du mouvement vertical calculées parJensen (1960) et les valeurs des composantes méridionales des vents pour les mois de Janvier et Avril 1958, les covariances des deux quantités ont été evaluées aux différents niveaux dans différentes latitudes. Ces covariances sont tabulées et les valeurs associées aux perturbations transientes et stationnaires sont comparées afin de donner un aperçu sur l'importance relative des deux procès.

     

On the vertical ozone and wind profiles near the tropopause

Pure and Applied Geophysics -     

Location of secondary faults from cross-correlation of the second vertical derivative of gravity anomalies1

A method is described to locate secondary faults, which can be difficult to identify on the Bouguer gravity map. The method is based on cross-correlation between the theoretical anomaly due to a vertical step and the second vertical derivative of the Bouguer anomaly. Faults are located from the closed maxima and minima on the cross-correlation contour map calculated for two perpendicular directions. One-dimensional model computations show that the magnitude of the extremum of the cross-correlation is related to the depth to the top of the hanging wall and the throw of the fault. Application of the method to the Bouguer gravity map of the former mouth of the Yellow River in the Shengli Oilfield area near the Bo Hai Sea shows the effectiveness of the method.