ω-k FILTER DESIGN*

Two-dimensional band-pass filters can be constructed by a simple extension of the theory of one-dimensional band-pass filters. Similarly to the one-dimensional analogue the shape of the two-dimensional filter is important in determining its effectiveness. The band-pass filter formulation can be further refined so that the filter will concentrate its rejection energies in certain areas of the ω, k plane. Such band-pass, band-reject filters are found by solving a set of simultaneous equations.     

ω-k FILTER DESIGN*

Two-dimensional band-pass filters can be constructed by a simple extension of the theory of one-dimensional band-pass filters. Similarly to the one-dimensional analogue the shape of the two-dimensional filter is important in determining its effectiveness. The band-pass filter formulation can be further refined so that the filter will concentrate its rejection energies in certain areas of the ω, k plane. Such band-pass, band-reject filters are found by solving a set of simultaneous equations.     

On the hyperbolic nature of the equations of alluvial river hydraulics and the equivalence of stable and energy dissipating shocks

Abstract

The depth-averaged hydraulic equations augmented with a suitable bed-load sediment transport function form a closed system which governs the one-dimensional flow in an alluvial river or channel. In this paper, it is shown that this system is hyperbolic and yields three families of shock-wave solutions. These are determined to be temporally stable in restricted regions of the (H, F ₀)-plane, via the Lax shock inequalities. Further, it is demonstrated that this criterion is equivalent to the energy dissipation criterion developed by Needham and Hey (1991).     

On the Mathematical Analysis of an Elastic-gravitational Layered Earth Model for Magmatic Intrusion: The Stationary Case

In the early eighties Rundle (1980, 1981a,b, 1982) developed the techniques needed for calculations of displacements and gravity changes due to internal sources of strain in layered linear elastic-gravitational media. The approximation of the solution for the half space was obtained by using the propagator matrix technique. The Earth model considered is elastic-gravitational, composed of several homogeneous layers overlying a bottom half space. Two dislocation sources can be considered, representing magma intrusions and faults. In recent decades theoretical and computational extensions of that model have been developed by Rundle and co-workers (e.g., Fernández and Rundle, 1994a,b; Fernández et al., 1997, 2005a; Tiampo et al., 2004; Charco et al., 2006, 2007a,b). The source can be located at any depth in the media. In this work we prove that the perturbed equations representing the elastic-gravitational deformation problem, with the natural boundary and transmission conditions, leads to a well-posed problem even for varied domains and general data. We present constructive proof of the existence and we show the uniqueness and the continuous dependence with respect to the data of weak solutions of the coupled elastic-gravitational field equations.     

Die Bestimmung der Frontgeschwindigkeit in Einem Einachsig Inhomogenen Körper aus Seismischen Reflexionsmessungen

Summary In this paper a new direct method for calculating the wave velocityc(z) from the recorded travel-time-curves is developed. Further a comparison of the well-known average ofc(z) is made with the exact average. Finally a concrete example is given.     

Exact Solutions for Spine Reconnective Magnetic Annihilation

Solutions for spine reconnective annihilation are presented which satisfy exactly the three-dimensional equations of steady-state resistive incompressible magnetohydrodynamics (MHD). The magnetic flux function ( A ) and stream function have the form $$A = A_{0}(R) \sin \phi + A_{1}(R)z, \qquad \Psi = \Psi _{0}(R) \sin \phi + \Psi _{1}(R)z,$$ in terms of cylindrical polar coordinates ( R , { , z ). First of all, two non-linear fourth-order equations for A 1 and 1 are solved by the method of matched asymptotic expansions when the magnetic Reynolds number is much larger than unity. The solution, for which a composite asymptotic expansion is given in closed form, possesses a weak boundary layer near the spine ( R = 0). These solutions are used to solve the remaining two equations for A 0 and 0 . Physically, the magnetic field is advected across the fan separatrix surface and diffuses across the spine curve. Different members of a family of solutions are determined by values of a free parameter n and the components ( B Re , B ze ) and ( v Re , v ze ) of the magnetic field and plasma velocity at a fixed external point ( R , { , z ) = (1, ~ /2,0), say.     

Smart base isolated buildings with variable friction systems: H∞ controller and SAIVF device

A new control algorithm is developed for reducing the response of smart base isolated buildings with variable friction semiactive control systems in near‐fault earthquakes. The central idea of the control algorithm is to design a H_∞ controller for the structural system and use this controller to determine the optimum control force in the semiactive device. The H_∞ controller is designed using appropriate input and output weighting filters that have been developed for optimal performance in reducing near‐fault earthquake responses. A novel semiactive variable friction device is also developed and with the H_∞ controller shown to be effective in achieving response reductions in smart base isolated buildings in near‐fault earthquakes. The new variable friction device developed consists of four friction elements and four restoring spring elements arranged in a rhombus configuration with each arm consisting of a friction–stiffness pair. The level of friction force can be adjusted by varying the angle of the arms of the device leading to smooth variation of friction force in the device. Experimental results are presented to verify the proposed analytical model of the device. The H_∞ algorithm is implemented analytically on a five storey smart base isolated building with linear elastomeric isolation bearings and variable friction system located at the isolation level. The H_∞ controller along with the weighting filters leads to the smooth variation of friction force, thus eliminating the disadvantages associated with rapid switching. Several recent near‐fault earthquakes are considered in this study. The robustness of the H_∞ controller is shown by considering a stiffness uncertainty of ±10%. Copyright © 2006 John Wiley & Sons, Ltd.     

Separation of base flow from streamflow using groundwater levels—illustrated for the Pang catchment (UK)

A new filter to separate base flow from streamflow has developed that uses observed groundwater levels. To relate the base flow to the observed groundwater levels, a non‐linear relation was used. This relation is suitable for unconfined aquifers with deep groundwater levels that do not respond to individual rainfall event. Because the filter was calibrated using total streamflow, an estimate of the direct runoff was also needed. The direct runoff was estimated from precipitation and potential evapotranspiration using a water balance model. The parameters for the base flow and direct runoff were estimated simultaneously using a Monte Carlo approach. Instead of one best solution, a range of satisfactory solutions was accepted. The filter was applied to data from two nested gauging stations in the Pang catchment (UK). Streamflow at the upstream station (Frilsham) is strongly dominated by base flow from the main aquifer, whereas at the downstream station (Pangbourne) a significant component of direct runoff also occurs. The filter appeared to provide satisfactory estimates at both stations. For Pangbourne, the rise of the base flow was strongly delayed compared with the rise of the streamflow. However, base flow exceeded streamflow on several occasions, especially during summer and autumn, which might be explained by evapotranspiration from riparian vegetation. To evaluate the results, the base flow was also estimated using three existing base‐flow separation filters: an arithmetic filter (BFI), a digital filter (Boughton) and another filter based on groundwater levels (Kliner and Knĕz̆ek). Both the BFI and Boughton filters showed a much smaller difference in base flow between the two stations. The Kliner and Knĕz̆ek filter gave consistently lower estimates of the base flow. Differences and lack of clarity in the definition of base flow complicated the comparison between the filters. An advantage of the method introduced in this paper is the clear interpretation of the separated components. A disadvantage is the high data requirement. Copyright © 2004 John Wiley & Sons, Ltd.     

An LMI approach to H∞-control of time-delay systems for the benchmark problem

In this paper, three H_∞-control design methods are developed and applied to a three-storey building with an active mass damper as a control mechanism. The system of equations of the structural system, including the actuator and sensors, has been developed directly from experimentally derived data which forms the basis of the benchmark study discussed in this paper. The building plus the damper are modelled as a nominally linear system with input as well as state delays. Feedback control synthesis are first performed by using either of the two forms; the first is a pure state feedback and the other is a static output feedback. The analytical results are cast into a Linear Matrix Inequality (LMI) framework which can be solved numerically by efficient interior-point methods. The developed system is subjected to two historical earthquake excitation inputs (ElCentro and Hachinohe) and to the Kanai–Tajimi filter. The response is given in the form of indices in order to compare with other solutions of the benchmark problem. In addition, simulation results pertinent to the developed techniques are presented. © 1998 John Wiley & Sons, Ltd.     

A FREQUENCY DOMAIN APPROACH TO TWO-DIMENSIONAL MIGRATION *

The problem of the propagation of acoustic waves in a two-dimensional layered medium can be easily solved in the frequency domain if the Dix approximation is used, i.e. when only the primary reflections are considered. The migrated data at a depth z are obtained by convolving the time section with a proper two-dimensional operator dependent on z. The same result can be obtained by multiplying their two-dimensional spectra and summing for all the values of the temporal frequency. The aspect of the operator in the time-space domain has the classic hyperbolic structure together with the prescribed temporal and spatial decay. The main advantages of the frequency domain approach consist in the noticeable computer time savings and in the better approximation. On the other hand lateral velocity variations are very difficult to be taken into account. This can be done if a space variant filter is used in the time-space domain. To reduce computer time, this filter has to be recursive; the problem has been solved by Claerbout by transforming the hyperbolic partial differential equation into a parabolic one, and using the latter to generate the recursion operator. In the presentation a method is given for the generation of recursive filters with a better phase characteristics that have a pulse response with the requested hyperbolic shape instead of the parabocli one. This allows a better migration of steeper dips.     

A model for the simulation of turbulent and eddy diffusion processes at heights of 0–65 km

A generalized turbulent diffusion model has been developed which evaluates the time rate of growth of a simulated cloud of particles released into a turbulent (i.e. diffusive) atmosphere. The general model, in the form of second-order differential equations, computes the three-dimensional size of the cloud as a function of time. Parameters which influence the cloud growth, and which are accounted for in the model equations, are: (1) length scales and velocity magnitudes of the diffusive field, (2) rate of viscous dissipation , (3) vertical stability as characterized by the relative adiabatic lapse rate (1/T)(g/C _p +T/z), and (4) vertical shear in the mean horizontal winds , and , for a given height and of spatial extent equal to that of the diffusing cloud. Sample results for near ground level and for upper stratospheric heights are given. For the atmospheric boundary layer case, the diffusive field is microscale turbulence. In the upper stratospheric case it is considered to be a field of highly interactive and dispersive gravity waves.     

Filter formulation and wavefield separation of cross-well seismic data

Multichannel filtering to obtain wavefield separation has been used in seismic processing for decades and has become an essential component in VSP and cross-well reflection imaging. The need for good multichannel wavefield separation filters is acute in borehole seismic imaging techniques such as VSP and cross-well reflection imaging, where strong interfering arrivals such as tube waves, shear conversions, multiples, direct arrivals and guided waves can overlap temporally with desired arrivals. We investigate the effects of preprocessing (alignment and equalization) on the quality of cross-well reflection imaging wavefield separation and we show that the choice of the multichannel filter and filter parameters is critical to the wavefield separation of cross-well data (median filters, f–k pie-slice filters, eigenvector filters). We show that spatial aliasing creates situations where the application of purely spatial filters (median filters) will create notches in the frequency spectrum of the desired reflection arrival. Eigenvector filters allow us to work past the limits of aliasing, but these kinds of filter are strongly dependent on the ratio of undesired to desired signal amplitude. On the basis of these observations, we developed a new type of multichannel filter that combined the best characteristics of spatial filters and eigenvector filters. We call this filter a ‘constrained eigenvector filter’. We use two real data sets of cross-well seismic experiments with small and large well spacing to evaluate the effects of these factors on the quality of cross-well wavefield separation. We apply median filters, f–k pie-slice filters and constrained eigenvector filters in multiple domains available for these data sets (common-source, common-receiver, common-offset and common-midpoint gathers). We show that the results of applying the constrained eigenvector filter to the entire cross-well data set are superior to both the spatial and standard eigenvector filter results.     

Book review of ENCYCLOPEDIA OF NONLINEAR SCIENCE,edited by Alwyn Scott,Routledge, Taylor and Francis Group, 2005, VII +1053 pp., 135, $225 hardback (ISBN 1-57958-385-7).

We present three-dimensional (3D) solutions of the magnetohydrostatic equations in the co-rotating frame of reference outside a magnetized rigidly rotating cylinder. We make no symmetry assumption for the magnetic field, but to be able to make analytical progress we neglect outflows and specify a particular form for the current density. The magnetohydrostatic equations can then be reduced to a single linear partial differential equation for a pseudo-potential U, from which the magnetic field can be calculated by differentiation. The equation for U can be solved by standard methods. The solutions can also be used to determine the plasma pressure, density and temperature as functions of all three spatial coordinates. Despite the obvious limitations of this approach, it can, for example, be used as a simple tool to create 3D models for the closed field line regions of rotating magnetospheres without rotational symmetry.     

Planetary structures in general relativity

Summary For a static single-zone planet assuming that 1) the pressure and density are connected by the equationP =K _Q ^1+1/n–s, 2) a complete spherical symmetry is preserved and the system is in hydrostatic equilibrium; in section II, the expressions for the field equations have been obtained in suitable dimensionless forms. In section III the solution of the field equations forn=0 (which represents a homogeneous liquid) has been given in explicit form; for other prescribed value ofn=2/5 it has been pointed out that solutions must be performed by numerical integrations. Expressions for the mass-radius relation, the ratio of central to average density, the total energy, the proper energy and gravitational potential energy, which give some informations about the internal structure of the planet, have also been given in this section. Section IV discusses the velocity of sound at the centre of the planet. A few concluding remarks regarding the structure of the planet have been given in section V.     

Refinement of predictive reaeration equations for a typical Indian river

Dissolved oxygen mass balance has been computed for different reaches of River Kali in western Uttar Pradesh (India) to obtain the reaeration coefficient (K₂). A total of 270 field data sets have been collected during the period from March 1999 to February 2000. Eleven most popular predictive equations, used for reaeration prediction and utilizing mean stream velocity, bed slope, flow depth, friction velocity and Froude number, have been tested for their applicability in the River Kali using data generated during field survey. The K₂ values computed from these predictive equations have been compared with the K₂ values observed from dissolved oxygen balance measurements in the field. The performance of predictive equations have been evaluated using error estimation, namely standard error (SE), normal mean error (NME), mean multiplicative error (MME) and correlation statistics. The equations developed by Smoot and by Cadwallader and McDonnell showed comparatively better results. Moreover, a refined predictive equation has been developed using a least‐squares algorithm for the River Kali that minimizes error estimates and improves correlation between observed and computed reaeration coefficients. Copyright © 2001 John Wiley & Sons, Ltd.     

Recent progress in estimating uncertainty in geomagnetic field modeling

Abstract

Recent work pertaining to estimating error and accuracies in geomagnetic field modeling is reviewed from a unified viewpoint and illustrated with examples. The formulation of a finite dimensional approximation to the underlying infinite dimensional problem is developed. Central to the formulation is an inner product and norm in the solution space through which a priori information can be brought to bear on the problem. Such information is crucial to estimation of the effects of higher degree fields at the Core-Mantle boundary (CMB) because the behavior of higher degree fields is masked in our measurements by the presence of the field from the Earth's crust. Contributions to the errors in predicting geophysical quantities based on the approximate model are separated into three categories: (1) the usual error from the measurement noise; (2) the error from unmodeled fields, i.e. from sources in the crust, ionosphere, etc.; and (3) the error from truncating to a finite dimensioned solution and prediction space. The combination of the first two is termed low degree error while the third is referred to as truncation error.

The error analysis problem consists of “characterizing” the difference δz = z—z, where z is some quantity depending on the magnetic field and z is the estimate of z resulting from our model. Two approaches are discussed. The method of Confidence Set Inference (CSI) seeks to find an upper bound for |z—?|. Statistical methods, i.e. Bayesian or Stochastic Estimation, seek to estimate E(δz² ), where E is the expectation value. Estimation of both the truncation error and low degree error is discussed for both approaches. Expressions are found for an upper bound for |δz| and for E(δz² ). Of particular interest is the computation of the radial field, B., at the CMB for which error estimates are made as examples of the methods. Estimated accuracies of the Gauss coefficients are given for the various methods. In general, the lowest error estimates result when the greatest amount of a priori information is available and, indeed, the estimates for truncation error are completely dependent upon the nature of the a priori information assumed. For the most conservative approach, the error in computing point values of B_r at the CMB is unbounded and one must be content with, e.g., averages over some large area. The various assumptions about a priori information are reviewed. Work is needed to extend and develop this information. In particular, information regarding the truncated fields is needed to determine if the pessimistic bounds presently available are realistic or if there is a real physical basis for lower error estimates. Characterization of crustal fields for degree greater than 50 is needed as is more rigorous characterization of the external fields.     

Magnetic Rossby waves in a stably stratified layer near the surface of the Earth's outer core

Abstract

The stratification profile of the Earth's magnetofluid outer core is unknown, but there have been suggestions that its upper part may be stably stratified. Braginsky (1984) suggested that the magnetic analog of Rossby (planetary) waves in this stable layer (the ‘H’ layer) may be responsible for a portion of the short-period secular variation. In this study, we adopt a thin shell model to examine the dynamics of the H layer. The stable stratification justifies the thin-layer approximations, which greatly simplify the analysis. The governing equations are then the Laplace's tidal equations modified by the Lorentz force terms, and the magnetic induction equation. We linearize the Lorentz force in the Laplace's tidal equations and the advection term in the magnetic induction equation, assuming a zeroth order dipole field as representative of the magnetic field near the insulating core-mantle boundary. An analytical β-plane solution shows that a magnetic field can release the equatorial trapping that non-magnetic Rossby waves exhibit. A numerical solution to the full spherical equations confirms that a sufficiently strong magnetic field can break the equatorial waveguide. Both solutions are highly dissipative, which is a consequence of our necessary neglect of the induction term in comparison with the advection and diffusion terms in the magnetic induction equation in the thin-layer limit. However, were one to relax the thin-layer approximations and allow a radial dependence of the solutions, one would find magnetic Rossby waves less damped (through the inclusion of the induction term). For the magnetic field strength appropriate for the H layer, the real parts of the eigenfrequencies do not change appreciably from their non-magnetic values. We estimate a phase velocity of the lowest modes that is rather rapid compared with the core fluid speed typically presumed from the secular variation.     

The vertical structure of tidal currents

Abstract

The accuracy to which the vertical structure of tidal currents can be predicted is examined. Theoretical models for current structure are developed employing (a) a constant eddy viscosity E = ε and (b) an eddy viscosity varying linearly with height above the sea bed z; E(z)=βz. By requiring these models to satisfy the commonly accepted quadratic friction law, the condition ε>½k is deduced where k is the bed friction coefficient, W a representative velocity and D the depth.

The sense of rotation of a current ellipse is shown to be related to the configuration of co-tidal charts. The vertical structure of the current ellipse is illustrated from the theoretical models and the sensitivity of this structure is examined for the following variables: (a) eddy viscosity ε or βz, (b) the bed friction parameter kW, (c) rotation of the prescribed pressure gradients and (d) tidal period. While reasonable agreement between observed and calculated current profiles may often be reported, precise agreement is shown to depend upon accurate specification of both eddy viscosity and the bed stress condition.     

On Rossby wave propagation in a meridionally stratified channel

Rossby wave propagation in the presence of a nonseparable Brunt-Väisälä frequency,N(y,z), and the associated geostrophic zonal flow,U(y,z), is examined in this paper. The usual quasi-geostrophic potential vorticity equation only includes vertical variations in Brunt-Väisälä frequency (i.e.N(z)). We derive a linearised quasi-geostrophic potential vorticity equation which explicitly includesN(y, z), where variations inN may occur on the internal Rossby radius length scale. A mixed layer distribution that monotonically deepens in the poleward direction leads to a nonseparableN(y,z). The resulting meridional pressure gradient is balanced by an eastward zonal geostrophic flow.By assuming mixed layer depth changes occur slowly, relative to a typical horizontal wavelength of a Rossby wave, a local analysis is presented. The Rossby wave is found to have a strongly modulated meridional wavenumber,l, with amplitude proportional to |l|^–1/2. To elucidate whether the modulations of the Rossby wave are caused by the horizontal variations inN orU we also consider the cases where eitherN orU vary horizontally. Mixed layer depth changes lead to largestl where the mixed layer is deepest, whereasl is reduced in magnitude whereU is nonzero. When bothU(y,z) andN(y,z) are present, the two effects compete with one another, the outcome determined by the size of |c|/U _max, wherec is the Rossby wave phase speed. Finally, the slowly varying assumption required for the analytical approach is removed by employing a numerical model. The numerical model is suitable for studying Rossby wave propagation in a rectangular zonal channel with generalN(y, z) andU(y, z).