Comparison of satellite and terrestrial gravity data

Summary The integral mean values of gravity on the surface W=W ₀ , obtained from satellite observations with the use of harmonic coefficients[3, 7] and from terrestrial gravity measurements[12], are compared. The squares and products of the harmonic coefficients were neglected, with the exception of [J ₂ ⁽⁰⁾ ] ² , which was taken into account. The Potsdam correction and the geocentric constant are being discussed. The paper ties up with[13–15] and the symbols used are the same. The given problem was treated, e.g., in[2, 4, 6, 8–10]; in the present paper the values of gravity are compared directly.     

On the triaxiality of the earth on the basis of satellite data

Summary The evolution of the opinions as to the problem of the triaxiality of the Earth in the period prior to satellite geodesy can be seen, e.g., in[1–18]. Recently the opinion has been voiced that triaxiality is a result of the mathematical treatment of data rather than reality[19–21], especially since this is a comparatively small parameter. This opinion is not in contradiction with the results of satellite observations[22–28], but the non-zero values of the harmonic coefficients of the second degree and second order are a reality, they yield a value of the equatorial flattening of about1/90 000, and the representation of the equatorial section by an ellipse is justified even if the harmonics n=3, k=1 and n=3, k=3 have amplitudes only about half as small, and some other parameters might occur with just as much justification besides triaxiality.     

Oblique surface wave propagation over a small undulation on the bottom of an ocean

The problem of oblique water wave diffraction by small undulation of the bottom of a laterally unbounded ocean is considered using linear water wave theory. A perturbation analysis is employed to obtain the velocity potential, the reflection and the transmission coefficients up to the first order in terms of integrals involving the shape functions c(x) representing the bottom undulation. Finite cosine transform is used to find the first order potential, and this potential is utilised in obtaining the first order reflection and transmission coefficients. Some particular forms of the shape function representing an exponentially damped undulation, a single hump and a patch of sinusoidal ripples are considered and the integrals for the reflection and transmission coefficients are evaluated. For the exponentially damped undulation, it is observed that the reflection ceases much before transmission while for the single hump, reflection and transmission go hand in hand up to a certain value of the wavenumber, after which they vanish. For the patch of sinusoidal ripples having the same wavenumber, the reflection coefficient up to the first order is found to be an oscillatory function in the quotient of twice the component of the wavenumber along x-axis and the ripple wavenumber. When this quotient becomes one, the theory predicts a resonant interaction between the bed and free surface, and the reflection coefficient becomes a multiple of the number of ripples. High reflection of the incident wave energy occurs if this number is large. Also, when a patch of ripples having different wavenumbers is considered the same result follows. Known results for the normal incidence are recovered as special cases for the patch of sinusoidal ripples. The theoretical observations are shown computationally.     

THE INSENSITIVITY OF REFLECTED SH WAVES TO ANISOTROPY IN AN UNDERLYING LAYERED MEDIUM1

Propagation in the plane of mirror symmetry of a monoclinic medium, with displacement normal to the plane, is the most general circumstance in anisotropic media for which pure shear-wave propagation can occur at all angles. Because the pure shear mode is uncoupled from the other two modes, its slowness surface in the plane is an ellipse. When the mirror symmetry plane is vertical the pure shear waves in this plane are SH waves and the elliptical SH sheet of the slowness surface is, in general, tilted with respect to the vertical axis. Consider a half-space of such a monoclinic medium, called medium M, overlain by a half-space of isotropic medium I with plane SH waves incident on medium M propagating in the vertical symmetry plane of M. Contrary to the appearance of a lack of symmetry about the vertical axis due to the tilt of the SH-wave slowness ellipse, the reflection and transmission coefficients are symmetrical functions of the angle of incidence, and further, there exists an isotropic medium E with uniquely determined density and shear speed which gives exactly the same reflection and transmission coefficients underlying medium J as does monoclinic medium M. This means that the underlying monoclinic medium M can be replaced by isotropic medium E without changing the reflection and transmission coefficients for all values of the angle of incidence. Thus no set of SH seismic experiments performed in the isotropic medium in the symmetry plane of the underlying half-space can reveal anything about the monoclinic anisotropy of that underlying half-space. Moreover, even when the underlying monoclinic half-space is stratified, there exists a stratified isotropic half-space that gives the identical reflection coefficient as the stratified monoclinic half-space for all angles of incidence and all frequencies.     

Adjustment of a spatial network according to conditional observations when various kinds of corrections are introduced

Summary In the present paper the adjustment of the Hradilek's spatial network[3] was carried out in space using the least-squares method according to conditional observations. Triangle, side and base equations of condition are given (see also[1]). In the first method of adjustment (Alternative A) the corrections are assigned to oblique (position) angles and the lengths of sides. In the second method of adjustment (Alternative B) the corrections are assigned to horizontal directions, zenith distances and lengths of sides. The refraction coefficients in both alternatives are introduced as unknown parameters. Neither method of adjustment depends on the directions of the verticals. Theoretically, Alternative B is more correct. However, for practical purposes the results yielded by Alternative A are little better than those yielded by Alternative B. As regards the economical aspect Alternative A is considerably more convenient. Both methods seem suitable for computing the rectangular spatial co-ordinates, less so for determining the refraction coefficients.     

Calculation of reflection and transmission coefficients for qP waves incident on a planar interface between isotropic and triclinic media

In the paper by Chattopadhyay and Rajneesh (2006, “Reflection and refraction of waves at the interface of an isotropic medium over a highly anisotropic medium’, Acta Geophysica, vol. 54, no. 3, pp. 239–249), the authors proposed a process to calculate R/T (reflection and transmission) coefficients at the interface between isotropic and triclinic half-spaces, with incident qP waves in triclinic media. Unfortunately, besides several misprints, the authors made a fatal assumption that there is no transmitted SH wave generated in isotropic media, which led the successive analytical derivations and numerical calculations thoroughly wrong. In this paper, the errors are analyzed at length and corrections are given. Then an alternative approach to solve the problem is proposed and numerical results are shown and discussed.     

Effects of anelasticity on reflection and transmission coefficients

It is known that the reflection and transmission coefficients used in the zeroth order approximation of asymptotic ray theory (ART) are identical to those obtained for the plane wave impinging on a plane interface separating two perfectly elastic half-spaces. We have used ART to compute reflection and transmission coefficients for two viscoelastic media separated by a plane interface. Our method is different from the plane-wave approach because the ART approach requires only a local application of the boundary conditions both for the eikonal and the ray amplitudes. Several types of viscoelastic media were studied. For a given model, the elastic case was emulated by setting all the quality factors Q equal to each other. Several anelastic cases were computed by keeping the same velocities and densities while changing the Qs. The quality factor is a relatively difficult parameter to measure exactly. Hence elastic coefficients are used in most synthetic seismogram computations, and the quality factors are chosen from experimental measurements or simply estimated. From these computations, amplitude and phase differences between elastic coefficients and coefficients for dissipative media are observed in some cases. These differences show the importance of knowing the exact values of Q. Incorrect Q values can lead to unrealistic moduli and to noticeable phase differences of these viscoelastic coefficients.     

Partial derivatives of dispersion curves for higher modes of love waves in a single-layered medium

Summary The partial derivatives of the dispersion curves have been calculated by means of a method founded on the theorem of implicit functions, which was described in[1]. The partial derivatives of the dispersion curves of Love waves are studied in detail for a model of a single-layered medium. Analytical formulae for the partial derivatives are used for deriving the limits of these partial derivatives when the period approaches zero or the critical period.     

Generation of continuous surface soil moisture dataset using combined optical and thermal infrared images

Surface soil moisture (SSM) is a critical variable for understanding water and energy flux between the atmosphere and the Earth's surface. An easy to apply algorithm for deriving SSM time series that primarily uses temporal parameters derived from simulated and in situ datasets has recently been reported. This algorithm must be assessed for different biophysical and atmospheric conditions by using actual geostationary satellite images. In this study, two currently available coarse‐scale SSM datasets (microwave and reanalysis product) and aggregated in situ SSM measurements were implemented to calibrate the time‐invariable coefficients of the SSM retrieval algorithm for conditions in which conventional observations are rare. These coefficients were subsequently used to obtain SSM time series directly from Meteosat Second Generation (MSG) images over the study area of a well‐organized soil moisture network named REMEDHUS in Spain. The results show a high degree of consistency between the estimated and actual SSM time series values when using the three SSM dataset‐calibrated time‐invariable coefficients to retrieve SSM, with coefficients of determination (R²) varying from 0.304 to 0.534 and root mean square errors ranging from 0.020 m³/m³ to 0.029 m³/m³. Further evaluation with different land use types results in acceptable debiased root mean square errors between 0.021 m³/m³ and 0.048 m³/m³ when comparing the estimated MSG pixel‐scale SSM with in situ measurements. These results indicate that the investigated method is practical for deriving time‐invariable coefficients when using publicly accessed coarse‐scale SSM datasets, which is beneficial for generating continuous SSM dataset at the MSG pixel scale.     

Parameters of the normal gravity field deduced from satellite observations

Summary The parameters of the normal gravity field were deduced from the harmonic coefficients[3, 4] upto n=6 and compared with the parameters used hitherto. The symbols used are the same as in papers[5, 6, 8] with which this paper connects up.     

On the problem of determining the vertical gradients of gravity anomalies

Summary Green's theorem on harmonic functions makes it possible to determine the integral relationship between the harmonic function and its derivative with respect to the normal on a closed Lyapunov surface. The conditions of solvability are given by Fredholm's theory of integral equations. The solution for a sphere was presented by Molodenskii[3] and the general solution with the help of Molodenskii's parameter k by Ostach[4]. The present paper indicates a possibility of solving this problem with the help of a system of linear algebraic equations, a simplified modification of the Ostach-Molodenskii solution and, finally, a method, based on Eremeev's solution of the fundamental integral equation[5].     

Scattering of internal waves in a two-layer fluid flowing through a channel with small undulations

Using two-dimensional linear water wave theory, we consider the problem of normal water wave (internal wave) propagation over small undulations in a channel flow consisting of a two-layer fluid in which the upper layer is bounded by a fixed wall, an approximation to the free surface, and the lower one is bounded by a bottom surface that has small undulations. The effects of surface tension at the surface of separation is neglected. Assuming irrotational motion, a perturbation analysis is employed to calculate the first-order corrections to the velocity potentials in the two-layer fluid by using Green’s integral theorem in a suitable manner and the reflection and transmission coefficients in terms of integrals involving the shape function c(x) representing the bottom undulation. Two special forms of the shape function are considered for which explicit expressions for reflection and transmission coefficients are evaluated. For the specific case of a patch of sinusoidal ripples having the same wave number throughout, the reflection coefficient up to the first order is an oscillatory function in the quotient of twice the interface wave number and the ripple wave number. When this quotient approaches one, the theory predicts a resonant interaction between the bed and the interface, and the reflection coefficient becomes a multiple of the number of ripples. High reflection of the incident wave energy occurs if this number is large. Again, when a patch of sinusoidal ripples having two different wave numbers for two consecutive stretches is considered, the interaction between the bed and the interface near resonance attains in the neighborhood of two (singular) points along the x-axis (when the ripple wave number of the bottom undulation become approximately twice as large as the interface wave number). The theoretical observations are presented in graphical form.     

Weak Contrast PP Wave Displacement R/T Coefficients in Weakly Anisotropic Elastic Media

—Approximate PP plane wave displacement coefficients of reflection and transmission for weak contrast interfaces separating weakly but arbitrarily anisotropic elastic media are presented. The PP reflection coefficient for such an interface has been derived recently by Vavry?uk and P?en?ík (1997). The PP transmission coefficient presented in this paper was derived by the same approach. The coefficients are given as a sum of the coefficient for the weak contrast interface separating two nearby isotropic media and a term depending linearly on contrasts of the so-called weak anisotropy (WA) parameters (parameters specifying deviation of properties of the medium from isotropy), across the interface. While the reflection coefficient depends only on 8 of the complete set of the WA parameters describing P-wave phase velocity in weakly anisotropic media, the transmission coefficient depends on their complete set. The PP reflection coefficient depends on "shear-wave splitting parameter" γ. Tests of accuracy of the approximate formulae are presented on several models.     

Compositional variation and the origin of deep crustal reflections

Identifying the sources of crustal reflections is essential for deriving more geological information from deep crustal reflection profiles. Theoretical and model studies help place constraints on the role of compositional variation in producing deep crustal reflections. Analysis of laboratory-derived velocities and densities of rock types typical for the continental crust indicates that reflection coefficients are generally small, but significantly, 17% of the possible reflection coefficients have magnitudes between 0.1 and 0.2. Comparison between reflection coefficients derived from well logs and reflections observed in associated seismic profiles indicates that constructive interference associated with geological layering is at least as important as the magnitude of the reflection coefficients for producing detectable reflections. Constructive interference can increase reflection amplitude by two to three times but is limited to a relatively narrow range of layer thickness. For a typical 10–40 Hz seismic wavelet and typical crustal velocities of about 6 km/s, constructive interference occurs for layer thickness ranging between about 35 and 80 m. Layers thinner than 35 m interfere destructively. If reflections result from compositional variation, seismic models of hypothetical and observed geologic relations provide analogs for interpreting complex reflection patterns observed in deep crustal reflection profiles. Such models show reflection patterns similar to those observed in the reflection profiles. The models indicate that the reflections could originate in the complexly deformed and intruded terranes that are common in the crystalline crust and it may not be necessary to appeal to unobserved phenomena such as special lamellae or fluid-filled fractures to explain the reflections.     

Radioactivity and heat production data from several boreholes in the Bohemian Massif

Summary The radiogenic heat production of rock samples from boreholes in the Bohemian Massif has been calculated from gamma-radiometric determinations of Th, U and K contents. The results, in general, fit the heat flow distribution on the territory of Czechoslovakia[1]. The values of heat production are in the range from 1.1µW m^–3 in the eastern part to 4.4µW m^–3 in the north-western part of the Bohemian Massif.     

A note on the statistics of ordered peaks in stationary stochastic processes

Numerical simulation experiments have been carried out to evaluate the relative performance of three approximate formulations owing to Amini and Trifunac,[1] Gupta and Trifunac[2] and Basu et al.[3] for probability distributions of ordered peaks in stationary stochastic processes. The first two formulations are based on the assumption that the unordered peaks are statistically independent; whereas the formulation of Basu et al.[3] considers the dependence via Markov transition probability. In the formulation of Gupta and Trifunac,[2] the probability distribution of nth order peak is inherently conditioned by the fact that at most (n−1) peaks can occur with higher amplitudes and, thus, the ordered peaks are necessarily dependent. In this paper, ensembles of stationary time-histories are generated for three PSDFs representing a narrow-band, a broad-band and a band-limited white noise type of processes. Comparison of the results for the expected value and standard deviation for various orders of peaks, obtained by averaging over these ensembles, with the corresponding results obtained from the above mentioned three approximations, defined in terms of the moments of the PSDFs, has shown that the formulation of Gupta and Trifunac[2] describes the data well.     

Numerical simulation of reservoir sediment and effects on hydro‐dynamic response of arch dams

Based on the dynamic theory for saturated porous media by Biot (Journal of the Acoustical Society of America 1956; 28 : 168–178), a numerical model is presented to analyse the reflection behaviours of reservoir sediment and compared with those from the visco‐elastic model. It is concluded that the two models give very similar results of reflection coefficient α within the frequency range of interest. Then, using the two models, the change of the reflection coefficients α with various sedimentation parameters and excitation frequencies are studied in detail. The results are further used in the analysis of response functions of hydro‐dynamic pressures on, and structural displacements of the Xiang Hong Dian arch dam, for which some results from a field vibration test are available. It appears that effects of water compressibility with sediment reflection on hydro‐dynamic pressures and structural response are not significant for this specific case. Copyright © 2001 John Wiley & Sons, Ltd.