Modeling Full Seismogram Envelopes Using Radiative Transfer Theory with Born Scattering Coefficients

The equation of radiative transfer is used to model the transport of seismic energy in 2-D and 3-D acoustic random media. Monte-Carlo solutions of this equation using non-isotropic Born scattering coefficients are compared to three analytical solutions: Markov approximation, radiative transfer theory with isotropic scattering coefficients, and diffusion approximation. Additionally, we compare to finite differences solutions of the full wave equation in 2-D. We find a good correspondence of radiative transfer theory to Markov approximation for the case of multiple forward scattering. The comparison to radiative transfer theory with isotropic scattering coefficients, a model frequently used in data analysis, demonstrates that in the case of forward scattering the isotropic scattering model is not better than a diffusion approach. To compare radiative transfer theory with non-isotropic scattering coefficients to finite differences solutions of the full wave equation, the finite source duration and the bandpass filter process as well as the normalization of absolute amplitudes are explicitely taken into account. We find a good coincidence of both theories for scattering parameters, which are realistic for usual Earth crust. The theory correctly describes the unscattered direct wavefront, the envelope broadening caused by multiple forward scattering, as well as the late coda caused by multiple wide angle scattering. For strong scattering, which can be expected for very heterogeneous media such as strato volcanoes, the solutions of radiative transfer differ from the more complete solutions of the full wave equation.     

A simple numerical model for inlet sedimentation at intermittently open–closed coastal lagoons

This contribution aims to model the dominant processes that control sedimentation within the ocean inlet to intermittently open–closed coastal lagoons; focussing on the role of infilling, backfilling and morphodynamic feedback. The key elements that have been included in the present model are: (1) the delivery of sediment to the mouth of the estuary by littoral processes; (2) sediment transport processes within the inlet due to non-linear tidal flow; and (3) the down-slope diffusion of sediment. The model discussed here includes a simple one-dimensional (1-D), non-linear flow parameterisation that predicts the spatial variability in the magnitude and non-linearity of the tidal flow. The predicted third and fourth velocity moments are used to drive a morphodynamic module. Down-slope diffusion of sediment is dealt with in a separate diffusion term in the bed-evolution equation. Feedback between the evolving morphology and the flow field are integral to the model. Numerical simulations are used to investigate different modes of evolution for this type of inlet system. Inlet closure due to infilling is critically controlled by the balance between sediment addition at the mouth due to littoral processes and the removal of sediment by non-linear tidal flow. Rapid widening or deepening of the inlet at its landward margin leads to the deposition of a flood shoal. Under conditions of high sedimentation (and low diffusion) the flood shoal can become sufficiently well developed to present a physical barrier to sediment entering the lagoon. Under these circumstances backfilling can become significant. The infilling and backfilling processes are ameliorated by efficient down-slope diffusion which is controlled in the present model by a diffusion parameter, D. High diffusion coefficients slow inlet closure and allow sediment to propagate further into the lagoon.     

Determining drag coefficients and their application in modelling of turbulent flow with submerged vegetation

Vegetation is a key aspect of water resources and ecology in natural rivers, floodplains and irrigation channels. The hydraulic resistance of the water flow is greatly changed when submerged vegetation is present. Three kinds of drag coefficients, i.e., the drag coefficient for an isolated cylinder, the bulk drag coefficient of an array of cylinders and the vertically distributed or local drag coefficient, have been commonly used as parameters to represent the vegetation drag force. In this paper, a comprehensive experimental study of submerged stems in an open channel flow is presented. Empirical formulae for the three drag coefficients were obtained based on our experimental results and on data from previous studies. A two-layer model was developed to solve the mean momentum equation, which was used to evaluate the vertical mean velocity profile with each of the drag coefficients. By comparing the velocity distribution model predictions and the measurement results, we found that the model with the drag coefficient for an isolated cylinder and the local drag coefficient was good fit. In addition, the model with the bulk drag coefficient gave much larger velocity values than measurements, but it could be improved by adding the bed friction effect and making choice of the depth-averaged velocity within the canopy layer.     

Prediction of diffusion coefficients in porous media using tortuosity factors based on interfacial areas

Determination of aqueous phase diffusion coefficients of solutes through porous media is essential for understanding and modeling contaminant transport. Prediction of diffusion coefficients in both saturated and unsaturated zones requires knowledge of tortuosity and constrictivity factors. No methods are available for the direct measurement of these factors, which are empirical in their definition. In this paper, a new definition for the tortuosity factor is proposed, as the real to ideal interfacial area ratio. We define the tortuosity factor for saturated porous media (tau5) as the ratio S/S(o) (specific surface of real porous medium to that of an idealized capillary bundle). For unsaturated media, tortuosity factor (tau(a)) is defined as a(aw)/a(aw),o (ratio of the specific air-water interfacial area of real and the corresponding idealized porous medium). This tortuosity factor is suitably measured using sorptive tracers (e.g., nitrogen adsorption method) for saturated media and interfacial tracers for unsaturated media. A model based on this new definition of tortuosity factors, termed the interfacial area ratio (IAR) model, is presented for the prediction of diffusion coefficients as a function of the degree of water saturation. Diffusion coefficients and diffusive resistances measured in a number of saturated and unsaturated granular porous media, for solutes in dilute aqueous solutions, agree well with the predictions of the IAR model. A comparison of permeability of saturated sands estimated based on tau(s) and the same based on the Kozeny-Carman equation confirm the usefulness of the tau(s) parameter as a measure of tortuosity.     

An analysis of the Chow-Kulandaiswamy GHS model

The Chow-Kulandaiswamy general hydrologic system (GHS) model is revisited. Based on a mathematical study by Singh and McCann the GHS model is simplified. Explicit solutions are obtained for special cases which can satisfactorily determine watershed surface runoff response due to given rainfall excess. A rational criterion is developed to determine the number of derivative terms to be retained in the model. In order to determine the coefficients in the GHS model the method of moments is proposed. Criteria are developed to determine complex roots and oscillations for these coefficients. By analysing Chow-Kulandaiswamy's results it is found that in a majority of cases which they studied roots are complex. Moreover, for the cases which have complex roots, a majority of the solutions oscillate. A brief sensitivity analysis of the GHS model is performed with regard to: (a) its leading coefficient, and (b) the order of the differential equation. Finally, the peak characteristics are specified for the second order case and their qualitative properties are shown for the third order case.     

On a simplified radiative-conductive heat transfer equation

Summary A simplified equation purported to represent the joint influence of radiative and turbulent transfers of heat in the atmosphere is derived by dividing the absorption spectrum of terrestrial radiation into strongly and weakly absorbed regions, classified according to the local scale of variation or to the local heating rate, and introducing two mean absorption coefficients for these two groups of regions. Assurning the validity of theK-theory of turbulent diffusion of heat, it is found that the temperature of the atmosphere is governed by a sixth-order partial differential equation in the heightz. This equation can be simplified to the second order if the mean absorption coefficient of the strongly absorbed regions is much larger while that of the weakly absorbed regions is much smaller than the local scale variation, and the influence of the former is equivalent to an added diffusion while that of the latter is a newtonian cooling, and these two influences are present simultaneously. The values of the two coefficients and their dependencies on the concentration of the absorbing material have been obtained. The equation has been applied to the problem of the thermal interaction between the atmosphere and the underlying earth, as related to the diurnal heat wave, and it was found that the temperature changes within the first few hundred meters from the earth can be predicted accurately by the model when the partition of the two groups is adjusted to the heating rate and the eddy transfer coefficient is allowed to increase very rapidly within the lowest 10 m.     

Quantifying the effect of thermospheric parameterization in a global model

Global climate models have become useful tools for studying the important physical processes that affect the Earth's upper atmosphere. However, the results produced by all models contain uncertainty that stems for the manner in which the model is driven, as well as in the treatment of the internal physics and numerics. In order to fully understand the scientific value of the model results then, it is necessary to have a quantitative understanding of the uncertainty in the model. In this study, the global ionosphere–thermosphere model is used to investigate how uncertainty in the use of parameters in a large scale model can affect the model results. Eight parameters are studied that ultimately have an effect on the thermospheric temperature equation. It is found that among these, uncertainty in the thermal conductivity, NO cooling, and NO binary diffusion coefficients most strongly translate to uncertainty in the temperature and density results. In addition, variations in the eddy diffusion coefficient are shown to result in significant uncertainty in the thermospheric composition, and ultimately the electron density.     

Multi-scale micro-structure generation strategy for up-scaling transport in clays

Clays and claystones are used as backfill and barrier materials in the design of waste repositories, because they act as hydraulic barriers and retain contaminants. Transport through such barriers occurs mainly by molecular diffusion. There is thus an interest to relate the diffusion properties of clays to their structural properties. In previous work, we have developed a concept for up-scaling pore-scale molecular diffusion coefficients using a grid-based model for the sample pore structure. Here we present an operational algorithm which can generate such model pore structures of polymineral materials. The obtained pore maps match the rock’s mineralogical components and its macroscopic properties such as porosity, grain and pore size distributions. Representative ensembles of grains in 2D or 3D are created by a lattice Monte Carlo (MC) method, which minimizes the interfacial energy of grains starting from an initial grain distribution. Pores are generated at grain boundaries and/or within grains. The method is general and allows to generate anisotropic structures with grains of approximately predetermined shapes, or with mixtures of different grain types. A specific focus of this study was on the simulation of clay-like materials. The generated clay pore maps were then used to derive upscaled effective diffusion coefficients for non-sorbing tracers using a homogenization technique. The large number of generated maps allowed to check the relations between micro-structural features of clays and their effective transport parameters, as is required to explain and extrapolate experimental diffusion results. As examples, we present a set of 2D and 3D simulations and investigated the effects of nanopores within particles (interlayer pores) and micropores between particles. Archie’s simple power law is followed in systems with only micropores. When nanopores are present, additional parameters are required; the data reveal that effective diffusion coefficients could be described by a sum of two power functions, related to the micro- and nanoporosity. We further used the model to investigate the relationships between particle orientation and effective transport properties of the sample.     

A method for estimating spatially variable seepage and hydraulic conductivity in channels with very mild slopes

Infiltration along ephemeral channels plays an important role in groundwater recharge in arid regions. A model is presented for estimating spatial variability of seepage due to streambed heterogeneity along channels based on measurements of streamflow‐front velocities in initially dry channels. The diffusion‐wave approximation to the Saint‐Venant equations, coupled with Philip's equation for infiltration, is connected to the groundwater model MODFLOW and is calibrated by adjusting the saturated hydraulic conductivity of the channel bed. The model is applied to portions of two large water delivery canals, which serve as proxies for natural ephemeral streams. Estimated seepage rates compare well with previously published values. Possible sources of error stem from uncertainty in Manning's roughness coefficients, soil hydraulic properties and channel geometry. Model performance would be most improved through more frequent longitudinal estimates of channel geometry and thalweg elevation, and with measurements of stream stage over time to constrain wave timing and shape. This model is a potentially valuable tool for estimating spatial variability in longitudinal seepage along intermittent and ephemeral channels over a wide range of bed slopes and the influence of seepage rates on groundwater levels. Copyright © 2012 John Wiley & Sons, Ltd.     

基于复阻尼理论的混合结构Rayleigh阻尼模型

混合结构的阻尼矩阵不满足经典阻尼条件,导致传统的模态叠加法无法适用。复阻尼理论无法适用于时域计算,其自由振动响应中存在发散现象。针对混合结构的阻尼矩阵非比例性和复阻尼理论的时域发散性,基于频域等效原则构建了求解Rayleigh阻尼系数的数学优化模型,进而得到与复阻尼理论等效的Rayleigh阻尼运动方程。算例分析表明:依据位移时程响应和结构等效阻尼比可证明Rayleigh阻尼运动方程的正确性。基于本文研究成果,等效复阻尼理论的混合结构Rayleigh阻尼运动方程可直接采用模态叠加法,结合其确定的结构等效阻尼比,为混合结构的振型分解反应谱法提供理论依据。     

Diffusion in Landscape Development Models: On The Nature of Basic Transport Relations

In constructing large-scale landscape development models, processes must be appropriately represented. The diffusion equation is rewritten with separate coefficients for slow, continuous mass movements and rapid, episodic mass movements. Using available transport data, transport/gradient relations are assessed and estimates of diffusivities are given. Diffusivities estimated in this study are compared with values derived in scarp studies and values adopted in landscape development models. The relation between transport and gradient may be non-linear, which would require modification of the simple diffusion equation normally accepted in modelling. New approaches are required to place these essential relations on a firmer foundation. © 1997 by John Wiley & Sons, Ltd.     

Self-diffusion studies of pore fluids in unconsolidated sediments by PFG NMR

The self-diffusion of water and hexadecane in medium and coarse sands from glacial sand deposits in central Germany were investigated by pulsed field gradient nuclear magnetic resonance (PFG NMR). Due to the restriction of the diffusion path at the pore/grain interface, the measured apparent self-diffusion coefficients (D(Δ)) in the pore space depend on the observation time (Δ) in the PFG NMR experiment. Although the bulk self-diffusion coefficients of water and hexadecane differ by about one order of magnitude, the apparent self-diffusion coefficients in the pore space obey the same characteristic time-behaviour, which depends only on geometrical properties of the pore system. Using the “short-time diffusion” model, surface-to-volume (S/V) ratios and inherent self-diffusion coefficients (D₀) of the pore fluids were extracted from these diffusion measurements. The S/V ratios obtained are independent of the pore fluid used and agree with known geometrical properties of the sand grains. Moreover, the D₀ values are consistent with the corresponding bulk self-diffusion coefficients measured separately. In contrast to these results of PFG NMR, simultaneous investigations of longitudinal (T₁) nuclear magnetic relaxation reveal that the relaxation time of the pore fluid is a less suitable parameter for a quantitative estimation of geometrical properties of the pore/grain interface in these unconsolidated sediments since it depends on chemical properties of the fluid/grain interface.     

A statistical–dynamical method for predicting estuary morphology

A statistical–dynamical model for estuary morphodynamics is presented and demonstrated with a case study on the Humber Estuary, UK. The model presented here is hybrid in nature where simplified process dynamics are combined with a data-driven approach. The modelling methodology uses an inverse technique to construct an unknown source function in the model-governing equation, using historic measurements of estuary bathymetry. Spatial and temporal variability of the source function was established using empirical orthogonal eigenfunction analysis. Predictions of estuary morphology are obtained by extrapolating the temporal coefficients of the EOFs, in order to construct a predicted source function which is then used in the hybrid morphodynamic equation to obtain the predicted estuary morphology. The model was applied to predict the morphological evolution of the Humber Estuary over a period of a decade. Predicted results were compared with measured bathymetry data. The results reveal that the model captured the decadal scale morphodynamic response of the estuary with a good qualitative accuracy, despite the simplified approach used in the model. The accuracy of model predictions can be greatly improved if high-frequency bathymetry data are available.     

Two‐dimensional solute transport for periodic flow in isotropic porous media: an analytical solution

In this article, a mathematical model is presented for the dispersion problem in finite porous media in which the flow is two‐dimensional, the seepage flow velocity is periodic, and dispersion parameter is proportional to the flow velocity. In addition to these, first‐order decay and zero‐order production parameters have also been considered directly proportional to the velocity. Retardation factor is taken into account in the present problem. First‐type boundary condition of periodic nature is considered at the extreme end of the boundary. Mixed‐type boundary condition is assumed at the origin of the domain. A classical mathematical substitution transforms the original advection–dispersion equation into diffusion equation in terms of other dependent and independent variables, with constant coefficients. Laplace transform technique is used to obtain the analytical solution. Copyright © 2011 John Wiley & Sons, Ltd.     

Analytical approximations for flow in compressible,saturated, one-dimensional porous media

A nonlinear model for single-phase fluid flow in slightly compressible porous media is presented and solved approximately. The model assumes state equations for density, porosity, viscosity and permeability that are exponential functions of the fluid (either gas or liquid) pressure. The governing equation is transformed into a nonlinear diffusion equation. It is solved for a semi-infinite domain for either constant pressure or constant flux boundary conditions at the surface. The solutions obtained, although approximate, are extremely accurate as demonstrated by comparisons with numerical results. Predictions for the surface pressure resulting from a constant flux into a porous medium are compared with published experimental data.     

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.     

Transport and Biological Fate of Toluene in Low-Permeability Soils

The effect of simultaneous sorption, diffusion, and biodegradation on the fate and transport of toluene in low-permeability soil formations was examined. A transport model accounting for vapor and liquid sorption, vapor diffusions, and first-order biodegradation was developed to describe the movement of volatile solute in unsaturated soils. Modeling studies were followed with laboratory batch and column studies on fine-grained soil samples obtained from a gasoline-contaminated site. Batch experiments yielded the sorption and diffusion coefficients for generating theoretical solute transport profiles. Column studies were conducted to examine toluene sorption, diffusion, and biodegradation under aerobic and denitrifying conditions. Results from the column studies indicated that vapor sorption onto the soil was minimal due to the high moisture content of the soil. Comparison of model predictions with experimental results indicated that the SASK model, which is based on the resistivity theory, provided a more accurate prediction of the vapor phase tortuosity than the frequently used Millington-Quirk equation. Laboratory results of toluene concentration profiles matched well with the model predictions and yielded degradation rates comparable to those obtained in the field. Column studies, examining toluene biodegradation under aerobic and denitrifying conditions in low-permeability soils, indicated that the presence of excess nitrate in aerobic environments yielded higher solute degradation rates than those observed under exclusively aerobic systems.     

INDUCED POLARIZATION RESPONSE OF A HORIZONTALLY MULTILAYERED EARTH WITH NO RESISTIVITY CONTRAST*

The induced polarization response of a horizontally multilayered earth with no resistivity contrast can rapidly be calculated on a desk calculator or minicomputer for any electrode array. The formulation is a simple series summation of the products of weighting coefficients and the true induced polarization responses for each of the layers. The coefficients are directly derivable from the corresponding resistivity model. This series approach to IP formulation was originally described by Seigel but has not been treated extensively in the present-day geophysical literature. This method can be applied to either time or frequency domain induced polarization measurements. Once the coefficients are known, apparent induced polarization response can readily be obtained by judicious substitution of known, suspected, or assumed values of the true induced polarization of each layer. Basic formulation is presented for the IP potential coefficients (pole-pole or two array) with no resistivity contrast between the layers. From these coefficients, response of any number of layers for any electrode array can be obtained by suitable differentiation. Some examples of Wenner array for a three-layered earth and dipole-dipole array for a four-layered earth are used to illustrate the application. The results of this technique are valid for many natural situations of modest resistivity contrast. However, they definitely cannot be used if there are highly contrasting resistivity layers present. Such an approach is conceptually simple and is useful for survey planning, checking or setting the “depth-of-penetration”of a given array. For field induced polarization data that fits reasonably well to the no-resistivity-contrast model, this simple approach facilitates quantitative interpretation.