Form Of Spherical Harmonics Research Articles

А particle model of interaction between weakly non-spherical bubbles

A mathematical model of interaction of weakly non-spherical gas bubbles in liquid is proposed. It is a system оf the second order ODEs in the radii of bubbles, the position vectors of their centers and the amplitudes of their small deformations in the form of spherical harmonics. Compared to the available analogs, the proposed model equations are more accurate in terms of the ratio of the radii of bubbles to the distance between their centers. This allows one to study bubble-bubble interaction at smaller distances between the bubbles. The model equations are quite compact, which simplifies their theoretical analysis and construction of algorithms of their solution. Five problems (dynamics of a single bubble under harmonic excitation, bubble collapse near a plane rigid wall, interaction of three synchronized and nonsynchronized bubbles located in line, and interaction between sixteen bubbles with the centers located on a plane surface) are considered for validation. It is shown that in all cases the proposed model results well agree with the corresponding available experimental data and numerical solutions. For demonstration of the applicability, the proposed model is used to analyze radial oscillations, spatial movements and surface deformations of the bubbles in rectilinear, planar and spatial clusters under harmonic forcing. Initially 125, 121 and 125 bubbles of the clusters are located at the nodes of 1D, 2D and 3D uniform grids, respectively. It is found that the monotonically decaying radial oscillations of the 1D cluster bubbles during one period of forcing are quite similar to those of a single bubble. Unlike this, the attenuation of the radial oscillations of the central bubbles in the 2D and 3D clusters is essentially non-monotonic. The maximum pressures in the central bubbles of the 2D and 3D clusters are significantly higher than in the single and 1D cluster bubbles. In all clusters the maximum deformations of bubbles are in the form of the second spherical harmonics, except for the central bubble of the 3D cluster, the maximum deformations of which are in the form of the fourth harmonics.

Исследование точности глобальных моделей геопотенциала EGM2008, EIGEN-6C4, GECO, SGG-UGM-1, SGG-UGM-2, XGM2019 на территории Российской Федерации

The improvement of the gravimetric knowledge of the Russian Federation and the creation of new models of the Earth's gravitational field (EGF) in the form of spherical harmonics of the geopotential set the task of choosing the most suitable model based on the characteristics of the gravitational field on the territory of Russia. The satellite gravity missions CHAMP, GRACE and GOCE provide a homoge-neous and almost complete global coverage of information about the Earth's gravitational field. The paper presents the results of a study of the accuracy of modern high-degree global geopotential models for determining various characteristics of the Earth's gravitational field on the territory of the Russian Federation. The models EGM2008, EIGEN-6C4, GECO, SGG-UGM-1, SGG-UGM-2 and XGM2019 were used as the tested geopotential models. The results of the study are presented in the form of statistical parameters and histograms of the distribution of differences g between the gravity values obtained from ground-based gravimetric measurements and calculated from model data. The high-degree model EIGEN-6C4 has a smaller standard deviation ( = 5,24 mGal) compared to other geopotential models. Based on the distribution of differences, we can conclude that the EIGEN 6C4 model is the most suitable for the territory of the Russian Federation.

Calculation of barrier potential height using fraction parameter

Nucleus deformation refers to a change in the nucleus surface shape. The expression of the radius of the nucleus can represent this deformation. Generally, these radii are expansions involving spherical harmonics. In addition, some divide the nucleus shape into three parts. The left and right ends are spherical, while the middle or neck uses the hyperbolic function. The deformation shape does not contain the fraction of the nucleus fragments. In fact, including all possible fraction values will provide all the probability that occurs. These probabilities can be accumulated to determine the value of the potential barrier height. Based on this reason, a technique was developed to include the fragment fraction as part of the input variable. This technique is intended to be able to use experimental data from fission products. Namely, as input to get a potential barrier height. By changing the function expression for the radius of the nucleus, the fragment fraction can insert into the function. For this purpose, the method is to form the three parts of the nuclear radius. The first part is the head in the form of spherical harmonics. The second part is the neck which has a shape like hyperbolic cosines. By applying the continuity condition between the neck and the head, the fraction parameter becomes the expansion coefficient of the spherical harmonics. This technique shows fairly good calculation results. The figure of the potential barrier height shows the results of these calculations. It is through comparisons between this work and several other works that two main conclusions are drawn. First, the insertion of fraction parameters at the nucleus radius was successfully carried out. Second, the fission yield data can be used as input for calculating the fission barrier height.

A least-squares method for estimating the correlated error of GRACE models

SUMMARY A new least-squares method is developed for estimating and removing the correlated errors (stripes) from the Gravity Recovery and Climate Experiment (GRACE) and GRACE Follow-On (GRACE-FO) mission data. This method is based on a joint parametric model of the correlated errors and temporal trends in the spherical harmonic coefficients of GRACE models. Three sets of simulation data are created from the Global Land Data Assimilation System (GLDAS), the Regional Atmospheric Climate Model 2.3 (RACMO2.3) and GRACE models and used to test it. The results show that the new method improves the decorrelation method by Swenson & Wahr significantly. Its application to the release 5 (RL05) and new release 6 (RL06) spherical harmonic solutions from the Center for Space Research (CSR) at The University of Texas at Austin demonstrates its effectiveness and provides a relative assessment of the two releases. A comparison to the Swenson & Wahr and Kusche et al. methods highlights the deficiencies in past destriping methods and shows how the inclusion and decoupling of temporal trends helps to overcome them. A comparison to the CSR mascon and JPL mascon solutions demonstrates that the new method yields global trends that have greater amplitude than those produced by the CSR RL05 mascon solution and are of comparable quality to the JPL RL06 mascon solution. Furthermore, these results are obtained without the need for a priori information, scale factors or complex regularization methods and the solutions remain in the standard form of spherical harmonics rather than discrete mascons. The latter could introduce additional discretization error when converting to the spherical harmonic model, upon which many post-processing methods and applications are built.

Deformation of Cavitation Bubbles during Implementation of Their Supercompression in a Cluster

The deformations of cavitation bubbles in the central region of clusters under their superstrong expansion and compression are studied. The conditions close to those in the well-known experiments on the acoustic cavitation of deuterated acetone are assumed. The simplest model of a cluster composed of seven initially spherical bubbles, the centers of which are located on three mutually perpendicular lines intersecting each other at a single point, is applied. The lateral bubbles are equidistant from the central one, the center of which is the intersection point of the lines. For comparison, the simplest model of a streamer composed of three bubbles on the same line is also considered. It has been established that the maximum deformations of the central bubble in the case of a cluster proves to be two orders of magnitude smaller than for a streamer. At that, in both a cluster and a streamer, the central bubble is deformed only in the form of spherical harmonics with even numbers and, and while the maximum deformations turn out to be in the form of second harmonics in the case of a streamer, the fourth-harmonic deformations are maximal in the case of a cluster.

Deformation of cavitation bubbles at their strong expansion and collapse in a streamer

Deformation of initially spherical cavitation bubbles is considered in the case of their strong single expansion and subsequent collapse in a streamer consisting of three bubbles located in one straight line. The central bubble is equidistant from the others. The influence of the distance between the bubbles is studied in the range, in which the non-sphericity of the central bubble during the collapse does not exceed 20%. The expansion of the bubbles and the most part of their collapse is governed by the ordinary differential equations of the second order in the radii of the bubbles, the positions of their centers on the line of their interaction, and the amplitudes of their axisymmetric non-sphericity in the form of spherical harmonics. In doing so, the interaction between bubbles is taken into account, the assumptions of weak liquid compressibility, homobaricity of bubbles, and their small non-sphericity are used. The final of the collapse of each bubble is governed by the partial differential equations of gas dynamics. In doing so, the interaction of bubbles is ignored, but the axial symmetry of bubbles, liquid compressibility, non-uniformity of vapor, heat conductivity of vapor and liquid are taken into account, wide-range equation of states for vapor and liquid are applied. It is shown that with decreasing the distance between the bubbles, the non-sphericity of the central bubble at the end of its collapse first slowly increases from zero and then quite rapidly grows. Along with that, the central bubble non-sphericity level of 1% is attained at the distance between the bubbles equal to about 20 times their radius at the moment of their maximal expansion. Simulation of the final high-speed stage of the bubble collapse by the model destined to the low-speed stage of expansion and collapse leads to significant overestimation of the non-sphericity of the central bubble at the end of its collapse.

Spherical Harmonics Reveal Standing EEG Waves and Long-Range Neural Synchronization during Non-REM Sleep.

Previous work from our lab has demonstrated how the connectivity of brain circuits constrains the repertoire of activity patterns that those circuits can display. Specifically, we have shown that the principal components of spontaneous neural activity are uniquely determined by the underlying circuit connections, and that although the principal components do not uniquely resolve the circuit structure, they do reveal important features about it. Expanding upon this framework on a larger scale of neural dynamics, we have analyzed EEG data recorded with the standard 10–20 electrode system from 41 neurologically normal children and adolescents during stage 2, non-REM sleep. We show that the principal components of EEG spindles, or sigma waves (10–16 Hz), reveal non-propagating, standing waves in the form of spherical harmonics. We mathematically demonstrate that standing EEG waves exist when the spatial covariance and the Laplacian operator on the head's surface commute. This in turn implies that the covariance between two EEG channels decreases as the inverse of their relative distance; a relationship that we corroborate with empirical data. Using volume conduction theory, we then demonstrate that superficial current sources are more synchronized at larger distances, and determine the characteristic length of large-scale neural synchronization as 1.31 times the head radius, on average. Moreover, consistent with the hypothesis that EEG spindles are driven by thalamo-cortical rather than cortico-cortical loops, we also show that 8 additional patients with hypoplasia or complete agenesis of the corpus callosum, i.e., with deficient or no connectivity between cortical hemispheres, similarly exhibit standing EEG waves in the form of spherical harmonics. We conclude that spherical harmonics are a hallmark of spontaneous, large-scale synchronization of neural activity in the brain, which are associated with unconscious, light sleep. The analogy with spherical harmonics in quantum mechanics suggests that the variances (eigenvalues) of the principal components follow a Boltzmann distribution, or equivalently, that standing waves are in a sort of “thermodynamic” equilibrium during non-REM sleep. By extension, we speculate that consciousness emerges as the brain dynamics deviate from such equilibrium.

Equations of spatial hydrodynamic interaction of weakly nonspherical gas bubbles in liquid in an acoustic field

A mathematical model of spatial hydrodynamic interaction of gas bubbles in liquid in an acoustic field taking into account small deformations of their surfaces is proposed. It is a system of ordinary differential equations of the second order in radii of the bubbles, the position vectors of their centers and the amplitudes of deviation of their shape from the spherical one in the form of spherical harmonics. The equations derived are of the first order of accuracy in A / R and of the fourth order in R / D, where R is the characteristic radius of the bubbles, A is the amplitude of characteristic deviation of their surface from the spherical one in the form of spherical harmonics, D is the characteristic distance between bubbles. The derivation of the equations is carried out by the method of spherical functions with the use of the Bernoulli integral, the kinematic and dynamic boundary conditions on the surface of the bubbles. The effects of viscosity and compressibility of the liquid are considered approximately, the gas in the bubbles is assumed homobaric.

Analytical and Laser Scanning Techniques to Determine Shape Properties of Aggregates

The fundamental shape attributes of aggregates used in pavements (i.e., form, angularity, and surface texture) have not been accurately quantified historically because of their irregular and nonideal shapes. This paper presents selected results from the use of a laser-based scanning technique to determine the form of aggregates employed in construction of pavements in South Africa. A three-dimensional (3-D) laser scanning system was used to examine aggregate materials from various sources, and the data were processed to reconstruct 3-D models of the aggregate particles. The models were further analyzed to determine the properties of the forms. Two analysis approaches based on physical properties of the aggregate and spherical harmonic analysis were employed to determine the form indices, and the results were compared. The indices based on the physical properties included (a) sphericity computed from the surface area and volume, (b) sphericity computed from three orthogonal dimensions, and (c) the flat-and-elongated ratio computed from the longest and shortest dimensions of an aggregate particle. Good correlations were observed between the form indices from the aggregate physical properties and the spherical harmonic form index. One conclusion of the research is that the laser scanning technique can be employed to quantify better the form properties of aggregate materials used in pavements.

A new invariant descriptor for action recognition based on spherical harmonics

The aim of this paper is to introduce a new descriptor for the spatio-temporal volume (STV). Human motion is completely represented by STV (action volume) which is constructed over successive frames by stacking human silhouettes in consecutive frames. Action volume comprehensively contains spatial and temporal information about an action. The main contribution of this paper is to propose a new affine invariant action volume descriptor based on a function of spherical harmonic coefficients. This means, it is invariant under rotation, non-uniform scaling and translation. In the 3D shape analysis literature, there have been a few attempts to use coefficients of spherical harmonics to describe a 3D shape. However, those descriptors are not affine invariant and they are only rotation invariant. In addition, the proposed approach employs a parametric form of spherical harmonics that handles genus zero surfaces regardless of whether they are stellar or not. Another contribution of this paper is the way that action volume is constructed. We applied the proposed descriptor to the KTH, Weizmann, IXMAS and Robust datasets and compared the performance of our algorithm to competing methods available in the literature. The results of our experiments show that our method has a comparable performance to the most successful and recent existing algorithms.

Nonlinear non-spherical oscillations of gas bubble under a periodic variation of ambient liquid pressure

A mathematical model is constructed for the bubble dynamics, in which the interphase surface variation is presented in the form of a series in spherical harmonics, and the equations are written with the accuracy up to the squared amplitude of the distortion of the spherical shape of the bubble. In the oscillation regimes close to periodic sonoluminescence of a single bubble in a standing acoustic wave, the character of air bubble oscillations in water was studied depending on the bubble initial radius and the amplitude of the liquid pressure variation. It was found that non-spherical oscillations of bounded amplitude can take place outside the region of linearly stable spherical oscillations. Both the oscillations with a period equal to one or several periods of the liquid pressure variation and aperiodic oscillations are observed. It is shown that neglecting the distortions in the form of spherical harmonics with large numbers (i > 3) may lead to a change of oscillation regimes. The influence of distortions on the bubble surface shape for the harmonics with i > 8 is insignificant.

GRACE's spatial aliasing error

SUMMARY The GRACE satellite mission provides a near-continuous sequence of approximately 30-d gravity field solutions in the form of spherical harmonics (SH). Because SH functions are global while GRACE measurements are sensitive mainly to variations along the ground-track, undersampling (alias contamination) occurs. Here we investigate how geophysical signals are likely to cause alias error in GRACE gravity fields. We use actual GRACE orbits and systematically sample several types of time-varying signals that might represent either errors in geophysical models such as tide models, or unmodelled geophysical signals. We show how error in semi-diurnal tides like S 2 can alias into long period variations in particular harmonics, particularly as a possible error source in the degree 2, order 0 term (C 20) of GRACE fields. We also show that aliasing associated with non-tidal geophysical model errors is significant at order 15 or multiples of 15, due to the GRACE ground track spacing in longitude. This can be predicted from Kaula’s resonance formula and might be reduced by suppressing amplitudes of affected harmonics.

A non-conforming generalization of Raviart–Thomas elements to the spherical harmonic form of the even-parity neutron transport equation

The Raviart–Thomas finite elements provide an appropriate spatial discretization of the mixed-dual form of the diffusion equation. This discretization can then be coupled to an efficient solution method. The high performances achieved by such an approach triggered research on its possible generalization to the transport equation using a spherical harmonic (or P N) angular approximation. In view of the difficulty of developing a straightforward generalization within the mixed-dual framework, we here consider 2D non-conforming (i.e., allowing interface discontinuities) finite elements coupled to the second-order form of the transport equation. This non-conforming approach keeps the mixed-dual property of the relaxation of the flux interface continuity constraint. We investigate different non-conforming elements and compare them to the well-known Lagrangian conforming elements.

Equatorial thermospheric wind changes during the solar cycle: Measurements at Arequipa, Peru, from 1983 to 1990

Fabry‐Perot interferometer measurements of Doppler shifts in the nightglow 630‐nm emission line have been used to determine near‐equatorial thermospheric wind velocities at Arequipa, Peru, over ∼2/3 of a solar cycle. Monthly‐average nocturnal variations in the meridional and zonal wind components were calculated from the nightly data to remove short term (day‐to‐day) variability, facilitating display of seasonal changes in the wind patterns, as well as any additional changes introduced by the progression of the solar cycle. The measured seasonal variations in the wind patterns are more pronounced than the solar cycle variations and are more readily understandable in terms of the expected, underlying forcing and damping processes. For most of the years, at the winter solstice, there is a weak (≤100 m/s) transequatorial flow from the summer to the winter hemisphere in the early and the late night, with essentially zero velocities in between. At the equinoxes, an early‐night poleward (southward) flow at solar minimum (1986) is replaced by an equatorward (northward) flow at solar maximum (1989–1990). The zonal flows are predominantly eastward throughout the night, except for the solar minimum equinoxes, where brief westward flows appear in the early and the late night. The peak eastward velocities increase toward solar maximum; at the winter solstice, they are ∼100–130 m/s in 1983, 1984 and 1986, reaching ∼200 m/s in 1988, 1989 and 1990. The present equatorial thermospheric wind determinations agree in some respects with the satellite‐data‐based horizontal wind model HWM‐87 and the vector spherical harmonic form of the thermospheric general circulation model. They also are generally consistent with earlier studies at Arecibo, Puerto Rico, and at Kwajalein, Marshall Islands, when compared at similar parts of the solar cycle.

Seasonal variations in the equatorial thermospheric wind measured at Arequipa, Peru

Studies have been carried out at Arequipa, Peru, of the seasonal variations in the thermospheric winds at moderate solar flux levels and low geomagnetic activity. Fabry‐Perot interferometer measurements of the Doppler shifts in the 630.0 nm nightglow emission line from March to August 1983 and from April to October 1988 have yielded monthly‐average meridional winds that are nearly zero (<50 m/s) and possibly fluctuating in direction through much of the night but develop a southward flow at 50–100 m/s in the early and the late night from May onward. The average zonal winds are eastward throughout the night, reaching peak velocities before local midnight and then decreasing. The peak velocities increase to a maximum around the June solstice. The winds are generally stronger in 1988 than in 1983, even though the solar EUV fluxes are comparable for both years. Comparison of the present results with earlier satellite measurements, as embodied in the Horizontal Wind Model of Hedin et al., reveals generally satisfactory agreement at the equinox and June solstice, except for the June 1988 period. Killeen et al.'s vector spherical harmonic form of the NCAR Thermospheric General Circulation Model, calculated for similar solar flux levels, yields meridional and zonal wind variations which exhibit the same temporal behaviors but generally smaller values than the present measurements. The present wind patterns are also compared with those measured during the same years at Arecibo, Puerto Rico; one finds oppositely directed meridional flows and similar, eastward zonal flows at the two locations, as expected from a consideration of solar EUV heating as the principal driving force.

Distortion of high-degree solar p-mode eigenfunctions by latitudinal differential rotation

view Abstract Citations (63) References (5) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Distortion of High-Degree Solar p-Mode Eigenfunctions by Latitudinal Differential Rotation Woodard, Martin F. Abstract The eigenfunctions of nonradial oscillation of the sun depart from their customarily assumed spherical harmonic form as a result of solar rotation. This paper treats the effect on oscillation eigenfunctions of slow, axisymmetric differential rotation which is also mirror symmetric across the solar equatorial plane. A given oscillation eigenfunction, perturbed by rotation, can be expressed as a superposition of eigenfunctions of a (hypothetical) nonrotating state of the sun. The assumption of axialsymmetry implies that each of the superposed functions has the same azimuthal order, m, while mirror symmetry dictates that the values of l, the spherical harmonic degree, involved in the superposition be either all odd or all even. The treatment here is further specialized to the case of high-degree p- and f-mode oscillations, on which the effect of the Coriolis acceleration is small. It is shown that, by ignoring the Coriolis effect and making other sensible approximations, an analytic expression can be obtained for the rotationally distorted mode eigenfunctions, which is expected to be a good approximation for typical solar oscillations of degree l greater than 100. The effect of the calculated mode distortion on the measurement of solar rotation is explored. Publication: The Astrophysical Journal Pub Date: December 1989 DOI: 10.1086/168206 Bibcode: 1989ApJ...347.1176W Keywords: Eigenvectors; Solar Oscillations; Solar Rotation; Axes Of Rotation; Formalism; Spherical Harmonics; Solar Physics; SUN: OSCILLATIONS; SUN: ROTATION full text sources ADS |

Forced asymmetric vibrations of an axisymmetric body in contact with an elastic half-space—A Global-Local Finite Element Approach

Forced asymmetric vibrations of an axisymmetric structure in contact with an elastic half-space are investigated by means of an extended version of the Global-Local Finite Element Method (GLFEM). In GLFEM, the structure and part of the surrounding medium are modeled by conventional finite elements. The behavior beyond this finite element model, i.e., in the remainder of the half-space region, is represented by analytical functions (called global functions) in the form of spherical harmonics. The solution process requires that both displacement and traction continuity at the finite element interface with the outer field be satisfield. Also, to satisfy traction-free surface conditions, the non-vanishing spherical harmonic tractions on the free surface beyond the finite element mesh are constrained in the form of weighted-average integrals that express zero net work due to these stress components. The results for rocking vibration of a rigid circular plate in frictionless contact with the half-space compare well with existing analytical data. Response results that illustrate the effects of various embedment conditions of a rigid circular plate completely bonded to the half-space are presented. Lastly, an example on flexible transducers fully bonded to a half-space is considered. Data describing the transducers' contact displacement distribution near resonance are given.

Axisymmetric soil–structure interaction by global–local finite elements

AbstractA version of the global–local finite element method is presented for studying dynamic steady‐state soil–structure interaction wherein the soil medium extends to infinity. Herein, only axisymmetric behaviour is considered. In this approach, conventional finite elements are used to model the structure and some portion of the surrounding soil medium considered to be homogeneous and isotropic. A complete set of outgoing waves in the form of spherical harmonics for the entire space is used to represent the behaviour in the half‐space beyond the finite element mesh and these are termed the global functions. Full traction and displacement continuity is enforced at the finite element mesh interface with the outer region. On the free surface of the half‐space in the outer field, traction‐free surface conditions are enforced by demanding that a sequence of integrals of the weighted‐average tractions must vanish. Numerical examples are presented for the response of different shaped foundations, resting on the free surface or at various submerged levels, due to a normal seismic plane compressional wave. Plots of differential scattering cross‐sections show the angular distribution of the energy (its directional nature) of the scattered field.

Relationship between Terrestrial and Satellite Doppler Systems

AbstractClassical horizontal geodetic networks are commonly combined with space observations, mostly satellite Doppler, in order to optimize the accuracy of geodetic control points and, thus, satisfy as many types of users as possible. Since satellite Doppler observations refer to a fully defined three-dimensional reference system and terrestrial observations, through the presence of Laplace stations (astronomical longitude and azimuth), contribute also to the pole and longitude orientations, it is imperative to ensure the highest possible degree of compatibility between the astronomical and satellite Doppler systems to maintain optimization of the accuracy of control points. Since gravity and geopotential (in the form of spherical harmonics) data are usually combined to evaluate geoid undulations and deflections of the vertical which are in turn used to reduce terrestrial angular and range observations, it is equally imperative to ensure that the satellite Doppler system and that of the geopotential solution are truly geocentric and thus compatible with the gravity data which should refer to a single equipotential surface. In order to estimate the degree of compatibility in terms of longitude orientation between satellite Doppler and geodetic astronomical systems as realized by current observations, astrogeodetic (based on CIO pole, BIH longitudes, and NWL9D satellite Doppler system) and gravimetric deflections of the vertical were compared at several hundred stations of the Canadian geodetic framework and U.S. transcontinental traverse. It was found that, when using the U.S. data subset only, incompatibility between the zero geodetic meridian plane of the NWL9D system and the zero astronomic meridian plane of the BIH was of the order of 08, which is in good agreement with previous results. However, inter-comparisons between various North American subsets revealed inconsistencies between areas of up to 08 (between Canadian and U.S. geodetic astronomical longitude observations). These results are based on the assumption that gravimetric deflections are bias free. The geocentricity of the NWL9D system with respect to other systems such as the Goddard Earth Models and SAO Standard Earths is also analyzed by comparing satellite Doppler derived geoid undulations with GEM and SAO SE undulations. An incompatibility of 4 m in Z (axis) exists between the origin of the NWL9D system and that of the other systems.

An Experiment in Nonlinear Statistical Weather Forecasting

Abstract We inquire whether an empirical weather forecasting scheme can profitably incorporate a possible nonlinear relationship between observed predictands and predictors. We analyze a set of twice–daily hemispheric 500 mb height fields into truncated series of spherical harmonies. From each act of spherical-harmonic coefficients, we predict the coefficients 24 h in advance by integrating the barotropic vorticity equation in spherical-harmonic form. We then establish linear regression equations for predicting the same coefficients, using as predictors the coefficients which represent the observed height fields, and, in some instances, the numerically predicted height fields. We find that the empirical schemes which incorporate nonlinearity by using the numerically predicted fields perform considerably better than those which do not.