Phase-shift migration with a variable-length spatial transform–an algorithm for moderately varying lateral velocities1

A migration algorithm appropriate for moderately varying lateral velocity changes is developed as an extension of phase-shift migration by using a variable-length spatial transform. This process significantly reduces the number of lateral wave-numbers necessary to downward continue the data, and it replaces the spatial FFT with a simple recursion relationship. For a given frequency and position x, ten lateral wavenumbers are typically sufficient, and the migration algorithm produces accurate images when the velocity structure V (X, z) changes over a few depth intervals of thickness Δz, with lateral velocity gradients up to 1.4 to 1.0.     

Ground motion modelling with a stochastically perturbed excitation

We present a new hybrid method combining deterministic and stochastic features. The aim is to describe the crustal propagation better than deterministic or stochastic methods can do separately. We start from the deterministic hybrid method based on Discrete- Wavenumber and Finite-Difference techniques (DW–FD). First we modify the DW–FD procedure by introducing topographical variations and a spatially varying Q factor. Then, to take into account effects due to small-scale heterogeneities of the crust, we add a stochastic noise (perturbation) to the deterministic signal propagated through the crust. The stochastic noise is constructed using a kind of Markov-like process generator with two physical constraints: to have the Brune spectrum, and to reproduce the spatial decay of coherence reported in literature for real sites. We have chosen a Markov-like technique because it allows us to get stochastic noise, with the given coherence spatial decay, directly in time domain. This new hybrid method is applied in a numerical test, the parameters of which approximate the case of the 12 June, 1995 Rome earthquake. It is found that the coherence decay with distance at the alluvial valley surface is slower than the prescribed coherence decay inside the bedrock.     

A note on magnetic modelling with remanence

The general problem of magnetic modelling involves accounting for the effect of both remanent magnetization and the application of an external magnetic field. However, as far as the disturbing field of a magnetic body in a magnetic environment is concerned, there is an equivalence between the effects of these two causations that allows the remanence to be represented in terms of an equivalent primary magnetic H field. Moreover, due to the linearity of the magnetic field in terms of its causations, the general modelling problem involving an applied magnetic field in the presence of remanence can be simply and more efficiently dealt with in terms of an equivalent primary field acting in the absence of any remanent magnetization.     

Unified global and regional wave model on a multi-resolution grid

Models for ocean surface wave forecasting in weather centres comprise global and regional systems in order to efficiently meet service demands. Regional models cannot run alone and have to use large area or global models to provide boundary wave spectra. The modern two-way nesting technique is to run the two models together with the regional model domain covered by both resolutions. An alternative method is to use a single multi-resolution grid that fits irregular coastlines and provides refined resolutions in selected regions. This paper presents a multi-resolution model based on a spherical multiple-cell (SMC) grid, which is designed to relax the CFL restriction of Eulerian advection time step at high latitudes by merging the conventional latitude-longitude grid cells. The implementation of the SMC grid in WAVEWATCH III is described, and a multi-resolution (6, 12 and 25 km) global SMC configuration is compared with a suite of nested grid ocean surface wave models, including 35-km global, 8-km European and 4-km UK regional models. Verification against buoy and platform wave observations indicates that the unified model is better than the 35-km global and very close in performance to the two regional models.     

Three-dimensional finite-element modelling of magnetotelluric data with a divergence correction

A finite-element method for computing the electric field in a 3-D conductivity model of the Earth for plane wave sources, thus enabling magnetotelluric responses to be calculated, is presented. The method incorporates in the iterative solution of the electric-field system of equations the divergence correction technique introduced for finite-difference solutions by Smith (1996). The correction technique accelerates the development of the discontinuity of the normal component of the approximate electric field across conductivity discontinuities. The convergence rate of the iterative solution is improved significantly, especially for low frequencies. The correction technique involves computing the divergence of the current density for the approximate electric field, computing the static potential whose source is this divergence of the current density, and ‘correcting’ the approximate electric field by subtracting from it the gradient of the potential. This is repeated at regular intervals during the iterative solution of the electric-field system of equations. For the method presented here, the Earth model is discretised using a rectilinear mesh comprising uniform cells. Edge-element basis functions are used to approximate the electric field and nodal basis functions are used to approximate the correction potential. The Galerkin method is used to derive the systems of equations for the approximate electric field and correction potential from the respective differential equations. A bi-conjugate gradient solver was found to be adequate for the system of equations for the correction potential; a generalised minimum residual solver was found to be better for the electric-field system of equations. The method is illustrated using the COMMEMI 3D-1A and 3D-2A models.     

Impacts of spatially and temporally varying snowmelt on subsurface flow in a mountainous watershed: 1. Snowmelt simulation

Abstract

The dominant source of streamflow in many mountainous watersheds is snowmelt recharge through shallow groundwater systems. The hydrological response of these watersheds is controlled by basin structure and spatially distributed snowmelt. The purpose of this series of two papers is to simulate spatially varying snowmelt and groundwater response in a small mountainous watershed. This paper examines the spatially and temporally variable snowmelt to be used as input to the groundwater flow modelling described in the second paper. Snowmelt simulation by the Simultaneous Heat and Water (SHAW) model (a detailed process model of the interrelated heat, water and solute movement through vegetative cover, snow, residue and soil) was validated by applying the model to two years of data at three sites ranging from shallow transient snow cover on a west-facing slope to a deep snow drift on a north-facing slope. The simulated energy balances for several melt periods are presented. Snow depth, density, and the magnitude and timing of snow cover outflow were simulated well for all sites.     

A study for improving precipitation occurrences modelling with a Markov chain

Daily precipitation occurrences and their monthly wet-days' sums of precipitation-measuring stations in Greece are modelled with a Markov chain. The order of the chain is taken to be seasonally varying in accordance with the precipitation station's meteorological conditions and geographical location. The modelling efficiency of the Markov chain is significantly improved when it is conjunctively used with a second-order autoregressive stochastic model fitted on the monthly wet-days' sums.     

Reflections from a randomly grooved interface: ultrasonic modelling and finite-diff erence calculation1

This study utilizes ultrasonic water tank modelling to examine the complexities which can result when seismic energy is diffracted by even a simple 3D interface structure. Ultrasonic laboratory modelling is carried out in our in-house water tank, using an aluminum block to simulate very simple ocean-bottom topography. The topography consists of parallel depressions which simulate valleys running parallel to a mid-ocean spreading ridge. This study shows that the phase and amplitude of primary ocean-bottom reflections can vary strongly with the azimuth of propagation. The trailing energy coda also varies dramatically as a function of propagation direction, showing the importance of developing a seismic model which includes the full 3D azimuthal distribution of interface structure.     

On propagation of love-type waves on a spherical model with rigidity and density both varying exponentially with the radial distance

Summary The propagation of Love type waves on a spherical model' in the mantle of which the rigidity and density vary exponentially with the radial distance while in the core they remain constant — has been investigated. The frequency equation that has been worked out has been examined in detail for the existence of root in the particular cases when they involve Bessel Functions of smallest and largest orders.     

Note on the torsional vibration of a thin beam of varying cross-section

Summary In the present note the problem of torsional vibration of a bar in which the moment of inertia and the torsional rigidity are proportional to 0 + 1s, wheres is the spanwise co-ordinate and ₀ and ₁ are real numbers, has been considered.     

Elastic properties of seafloor sediments from the modelling of amplitudes of multiple water waves recorded on the seafloor off Bear Island,North Atlantic

Wide‐angle seismic data acquired by use of air‐guns and ocean bottom seismometers (OBS) contain strong direct water arrivals and multiples, generally considered as noise and thus not included in the modelling. However, a recent study showed that standard ray‐tracing modelling of the water multiples recorded off the Bear Island, North Atlantic, provided a reliable estimate of the velocity distribution in the water layer. Here, we demonstrate that including the amplitudes in the modelling provide valuable information about the V_P contrast at the seafloor, as well as the V_P/V_S ratio and attenuation (Q_P) of the uppermost sediments. The V_P contrast at the seafloor is estimated at about 250 m/s, within a precision of approximately ±30 m/s. The V_P/V_S ratio in the uppermost sedimentary layer is modelled in the range 2.25–2.50 and the Q_P factor is estimated at 1000 for the water, 30–50 for the uppermost layer and 40–50 for the second sedimentary layer. The values obtained for the sediments suggest a lithology dominated by silty clays, with porosity below average.     

Improving modelling and inversion in refraction seismics with a first-order Eikonal solver

A first-order Eikonal solver is applied to modelling and inversion in refraction seismics. The method calculates the traveltime of the fastest wave at any point of a regular grid, including head waves as used in refraction. The efficiency, robustness and flexibility of the method give a very powerful modelling tool to find both traveltimes and raypaths. Comparisons with finite-difference data show the validity of the results. Any arbitrarily complex model can be studied, including the exact topography of the surface, thus avoiding static corrections. Later arrivals are also obtained by applying high-slowness masks over the high-velocity zones. Such an efficient modelling tool may be used interactively to invert for the model, but a better method is to apply the refractor-imaging principle of Hagedoorn to obtain the refractors from the picked traveltime curves. The application of this principle has already been tried successfully by previous authors, but they used a less well-adapted Eikonal solver. Some of their traveltimes were not correct in the presence of strong velocity variations, and the refractor-imaging principle was restricted to receiver lines along a plane surface. With the first-order Eikonal solver chosen, any topography of the receiving surface can be considered and there is no restriction on the velocity contrast. Based on synthetic examples, the Hagedoorn principle appears to be robust even in the case of first arrivals associated with waves diving under the refractor. The velocities below the refractor can also be easily estimated, parallel to the imaging process. In this way, the model can be built up successively layer by layer, the refractor-imaging and velocity-mapping processes being performed for each identified refractor at a time. The inverted model could then be used in tomographic inversions because the calculated traveltimes are very close to the observed traveltimes and the raypaths are available.     

岩石应力对钻孔附近的弹性波波速的影响

目前,地球物理学界普遍采用岩石的细观力学模型来解释岩石的声弹现象(即岩石应力状态的改变会造成岩石的弹性波波速的改变),认为岩石内部的微裂隙分布和取向在加载应力的作用下会发生改变,使岩体的宏观物理性质发生变化,从而影响到岩石的弹性波波速.     

Using the pseudospectral technique on curved grids for 2D acoustic forward modelling1

The Fourier pseudospectral method has been widely accepted for seismic forward modelling because of its high accuracy compared to other numerical techniques. Conventionally, the modelling is performed on Cartesian grids. This means that curved interfaces are represented in a ‘staircase fashion‘causing spurious diffractions. It is the aim of this work to eliminate these non-physical diffractions by using curved grids that generally follow the interfaces. A further advantage of using curved grids is that the local grid density can be adjusted according to the velocity of the individual layers, i.e. the overall grid density is not restricted by the lowest velocity in the subsurface. This means that considerable savings in computer storage can be obtained and thus larger computational models can be handled. One of the major problems in using the curved grid approach has been the generation of a suitable grid that fits all the interfaces. However, as a new approach, we adopt techniques originally developed for computational fluid dynamics (CFD) applications. This allows us to put the curved grid technique into a general framework, enabling the grid to follow all interfaces. In principle, a separate grid is generated for each geological layer, patching the grid lines across the interfaces to obtain a globally continuous grid (the so-called multiblock strategy). The curved grid is taken to constitute a generalised curvilinear coordinate system, where each grid line corresponds to a constant value of one of the curvilinear coordinates. That means that the forward modelling equations have to be written in curvilinear coordinates, resulting in additional terms in the equations. However, the subsurface geometry is much simpler in the curvilinear space. The advantages of the curved grid technique are demonstrated for the 2D acoustic wave equation. This includes a verification of the method against an analytic reference solution for wedge diffraction and a comparison with the pseudospectral method on Cartesian grids. The results demonstrate that high accuracies are obtained with few grid points and without extra computational costs as compared with Cartesian methods.     

Deformation and stresses in different earth models with a rigid core having varying elastic parameters

Summary In the present paper the small strain theory has been applied to find the stresses and deformation in the interior of the earth models corresponding to different non-uniform density distributions when the elastic parameters are not constant. The case of uniform density distribution in the interior of the earth assumed to be a self-gravitating isotropic sphere has also been considered in the light of the same theory.     

The effect of Newtonian cooling on vertically propagating magneto-acoustic waves in a thermally conducting isothermal atmosphere (II)

In this article, we examine reflection, dissipation and attenuation of vertically propagating waves in an isothermal atmosphere under the combined effect of Newtonian cooling, thermal conduction and viscosity with a weak horizontal magnetic field. We consider the case in which the combined effect of viscosity and magnetic field is dominated by that of the thermal conduction and for small values of the Newtonian parameter. As a result, the atmosphere can be divided into three distinct regions that are connected by two transition regions. The lower and middle regions are connected by a semi-transparent barrier and the middle and upper regions are connected by an absorbing and reflecting barrier. In the connecting barriers the reflection and transmission of the waves takes place. The presence of Newtonian cooling effects on the adiabatic region, produces attenuation in the amplitudes of the waves and reduces the energy absorption in the transition regions. The reflection coefficient is determined in the lower and middle regions and the results are discussed in the context of the heating of the solar atmosphere.     

The effect of nested grid sampling on the parameter estimation of a spatial Gompertz diffusion

This paper evaluates the effects of using data observed on regular nested grids on the parameter estimates of a two-parameter Gompertz diffusion model. This new spatial diffusion process represents a technically more complex stage of Gompertz modeling. Firstly, the diffusion model is introduced through an appropriate transformation of a two-parameter Gaussian diffusion process. Probabilistic characteristics of this model, such as the transition densities and the trend functions, are obtained. Secondly, statistical estimation is considered using data obtained on a regular or irregular grid; the explicit expression of the likelihood equations and the parameter estimators are given for regular grids. Finally, a simulation experiment illustrates the results of this paper.     

提高规则网格最短路径方法反射波走时计算精度的走时校正技术

最短路径射线追踪方法是计算地震波走时的主要方法之一,该方法基于惠更斯原理和费玛原理,具有稳健、适于复杂介质模型的优点.为处理方便,最短路径方法中的介质模型通常以规则网格进行剖分,界面节点(界面与网格的交点)以其邻近的模型单元节点(即边界单元节点)近似表示.界面近似将导致计算误差,对于反射波尤为严重.反射波的走时精度可通过减小网格的尺寸提高,但这样会大大增加计算时间,为高精度和高效率地计算地震反射波走时,我们提出了一种基于规则网格的走时校正技术.地震波传播至或起始于边界单元节点的走时校正为地震波传播至或起始于该边界单元节点所对应的界面节点的走时.数值模型计算结果表明,走时校正方法可使反射波的走时精度提高约1～2个数量级,而其计算时间则和常规算法基本上在相同量级.