Vertical Correlation Length Research Articles

Variational Assimilation of Precipitation Data Using Moist Convective Parameterization Schemes: A 1D-Var Study

Some basic aspects related to the problem of incorporating moist convective processes in a variational data assimilation framework are considered. The methodology is based on inverse problem theory and is formulated in its simplest context where the adjustment of temperature and humidity fields take place only in the vertical. In contrast to previous studies on the subject, the impact of error statistics from prior information and data sources of information is clarified. The accuracy of linearization of convection operators and the resulting impact in a minimization procedure are examined. The former was investigated using Monte Carlo methods. Versions of two schemes are examined: the Kuo–Anthes scheme and the relaxed Arakawa–Schubert scheme (RAS). It is found, in general, that for nonpathological convective points (i.e., points where convection always remains active during the minimization process), a significant adjustment of convection (and precipitation rate) is realizable within the range of realistic background temperature and specific humidity errors and precipitation rate observation errors. Typically, three to five iterations are sufficient for convergence of the variational algorithm for both convective schemes. The degree of nonlinearity of both schemes appears comparable. The vertical correlation length for the background error temperature field is shown to produce a strong interaction with the RAS scheme in the minimization process where significantly different vertical structures of analysis increments for the temperature field are generated in the vicinity of a critical value of the correlation length.

Quantitative in situ x-ray diffraction analysis of magnetic multilayers during annealing

In situ high-angle x-ray diffraction (XRD) measurements during an annealing process and extensive low- and high-angle XRD analysis before and after annealing have been performed to study the influence of annealing on the microstructure of a Ni81Fe19/Ag multilayer film. We concentrate on a detailed description and quantification of induced changes in microstructural properties and on changes in the crystalline structure of sublayers. The observed effects obtained from the high-angle XRD, where superlattice structural refinement and standard powder diffraction methods were used, are utilized to model the changes from microscopic point of view. The low-angle XRD reflectivity measurements were analyzed using a new distorted-wave Born approximation approach which enables to get information about interlayer structure parameters such as interface roughnesses, degree of vertical correlation and lateral correlation length. The presented low-angle and high-angle analysis demonstrates general applicability of x-ray diffraction for complex and quantitative in-situ investigation of structural changes in metallic multilayers during annealing. An effect of lateral scaling for different types of roughness is discussed and a general approach is shown for the case of metallic multilayers.

Transport properties of n-channel Si/SiGe modulation-doped systems with varied channel thickness: effect of the interface roughness

The effect of the interface roughness on the transport properties of modulation-doped strained Si on relaxed is studied. The mobility decreases between 25 K and 300 K with decreasing channel thickness from 200 to 13 Å. When the channel is thicker than 53 Å, back-gated measurements reveal that the mobility is proportional to the carrier density, reflecting the two-dimensionality of the system. When the channel width is about 40 Å, however, interface-roughness scattering is found to strictly limit the mobility below 20 K from the temperature dependence as well as the thickness dependence of the mobility. The vertical and lateral correlation lengths of the roughness are estimated to be 6.5 - 8.0 Å and 120 - 130 Å respectively. In the samples with a channel thickness less than 27 Å, which is comparable to the roughness, it is found that a two-dimensional variable-range hopping process dominates the conductivity below 40 K.

Interface roughness of InAs/AlSb superlattices investigated by x-ray scattering

InAs/AlSb short period superlattices grown either with AlAs-like or with InSb-like interfaces are investigated by grazing incidence x-ray scattering and high resolution diffractometry. The superlattices are grown on a relaxed AlSb buffer layer. It is shown that the two possible stackings of layers in the superlattices resulting in a different degree of lattice relaxation lead also to a different height of interface roughness. The lateral and vertical correlation lengths of the roughness decrease with increasing relaxation of the superlattice. The vertical correlation length corresponds to an almost complete correlation of different interfaces in the case of the nearly perfect superlattice with InSb-like interfaces.

Determination of seabed statistical properties and their effects on acoustic wave propagation in shallow water

Genetic algorithms are used to invert statistical properties of sea bottom/sub-bottom from backscattering data. Backscattering from the seabed is calculated using composite roughness model of Jackson etal. [J. Acoust. Soc. Am. 79(5) (1986)], incorporated with the volume scattering model of Stockhausen [NRE Report 63/9 (1963)]. A parametric study indicates that, for given average sound speed and density of the seabed, bottom roughness parameters such as spectral strength and spectral exponent and sub-bottom parameters such as sound-speed variation, vertical correlation length, and horizontal correlation length can be inverted successfully. Complementary sub-bottom measurement techniques such as core sampling, seismic, and tomographic measurements are also discussed. Acoustic calculations using measured seabed inhomogeneities predict significant coupling to the higher-order modes. This coupling manifests itself as an important loss mechanism.

Determination of vertical correlation lengths in a channel using SWELLEX‐1 thermistor data

In order to properly determine the volume scattering in a channel, it is necessary to know the characteristic vertical and horizontal correlation lengths associated with the random index‐of‐refraction fluctuations. The results we have obtained for characteristic vertical correlation lengths using SWELLEX‐1 vertical‐array thermistor data are discussed. The data have been analyzed for day‐time and night‐time observations. In addition, the results are dependent upon averaging times and this effect is discussed. Graphs are given for the standard deviation and normalized cross correlations of the fluctuations, as a function of depth. [The vertical temperature data analyzed in this presentation were produced by NRL Code 7120 and NRAD Code 541 under ONR sponsorship. The data were supplied to us by B. Pasewark of NRL Code 7120. Our analysis was supported by ONR.]

Fluctuation analysis of tension-controlled undulation forces between giant vesicles and solid substrates.

Using reflection interference contrast microscopy, we studied the thermal fluctuations of giant vesicles that weakly adhere to flat solid substrates. The absolute membrane-substrate separation distance was imaged and the average contact contour, including the contact area, the contact rounding, and the asymptotic contact angle, was determined. The static fluctuations in the flat, adhering part of the vesicle were analyzed. The spectrum of mean square amplitudes yielded the lateral membrane tension and the second derivative of the interaction potential. The vertical roughness and lateral correlation length were measured from the spatial autocorrelation of the undulations. The roughness was shown to obey the behavior predicted by functional renormalization in the observed tension regime of ${10}^{\mathrm{\ensuremath{-}}6}$ to ${10}^{\mathrm{\ensuremath{-}}4}$ J/${\mathrm{m}}^{2}$. Moreover, the measured separation distances can be explained within the framework of undulation and van der Waals forces and confirmed the model of tension-induced adhesion. However, the adhesion energies as well as the measured separation distances exhibit a weaker dependence on the membrane tension than predicted.

Anisotropic roughness in Ge/Si superlattices

We discuss measurements of buried interfaces utilizing x-ray specular reflectivity profiles, and high-resolution x-ray reflectivity measurement of diffuse scattering. Interface structure is compared to atomic force microscopy characterization of the morphology of the Si cap layer overlying the buried structures. The results show that the morphology of roughness on the growth surface is greatly influenced by substrate miscut, and tends to form one-dimensional undulations. Roughness along the miscut direction is highly replicated from interface to interface, but roughness perpendicular to the miscut evolves more rapidly during the growth process. This effect is characterized by a reduced vertical correlation length in x-ray diffuse scattering in one azimuthal orientation.

The crust as a heterogeneous “optical” medium, or “crocodiles in the mist”

Based on petrophysical data, geologic maps, and a well log, we present statistical descriptions of likely upper-, middle-, and lower-crustal rocks to characterize the fine-scale heterogeneity observed in crustal exposures and inferred from deep-crustal seismic data. The statistical models, developed for granitic and metamorphic upper crust, and for an extended metamorphic lower crust, are used to construct whole-crustal models of seismic velocity heterogenity. We present finite-difference synthetic CMP data from several models which compare favorably with field data. The statistical models also permit classification of the seismic reflection experiment and the crustal heterogeneity according to scattering regime. The “optical”, or scattering properties of importance for classification are the velocity fluctuation intensity, the horizontal and vertical correlation lengths of the medium, the correlation function of the medium, and the velocity population function. For the crustal properties we measured, the bandwidth of a typical deep crustal experiment overlaps from the weak to the strong scattering regime, with implications for crustal seismic data processing and imaging. Notably, deep-crustal signals are likely to have experienced multiple scattering, making common seismic imaging techniques of questionable value. Moreover, the details of the unmigrated CMP stacked section bears little resemblance to the underlying medium.

Comparisons Between Seismic Earth Structures and Mantle Flow Models Based on Radial Correlation Functions

Three-dimensional numerical simulations were conducted of mantle convection in which flow through the transition zone is impeded by either a strong chemical change or an endothermic phase change. The temperature fields obtained from these models display a well-defined minimum in the vertical correlation length at or near the radius where the barrier is imposed, even when the fields were filtered to low angular and radial resolutions. However, evidence for such a feature is lacking in the shear-velocity models derived by seismic tomography. This comparison suggests that any stratification induced by phase or chemical changes across the mid-mantle transition zone has a relatively small effect on the large-scale circulation of mantle material.

Internal waves and matched-field processing

The performance of matched-field processing is degraded due to randomness of the propagation medium. With a vertical array, this degradation takes the form of fragmentation and wandering of the peaks (mainlobe and sidelobes) in the range-depth ambiguity surface. The resulting errors in localization are characterized in terms of the rms processor output. Simulations and theory specialized to the cw Bartlett processor are used. First, the case of an ocean waveguide with a quadratic average sound-speed profile and vertically stationary sound-speed fluctuation statistics is examined. Next, approximations are introduced so that a relatively simple analytic model can be abstracted from the theory. This model is checked against Monte Carlo PE computations that avoid some of the simplifying approximations. The simple model contains scaling rules for processor performance as a function of frequency, array length, and medium vertical correlation length. In general, the effects of internal waves become more important as frequency increases, array length increases, medium correlation length decreases, and range increases. The processor is predicted to be most sensitive to internal-wave mismatch for sources that are in convergence zones. Finally, the analytical model is compared to Monte Carlo PE computations using oceanic realizations obtained from a realistic dynamic internal wave model.

Measurement of the correlation spectrum of electrostatic potential fluctuations in an ECRH Helimak plasma

Measurements of the length and time scales of turbulent potential fluctuations were carried out on the MIT Versatile Toroidal Facility (VTF). Plasmas were produced by ECRF heating in a novel toroidal configuration which we have termed the Helimak. The configuration consists of a toroidal fieldB φ of approximately 800 gauss and a vertical fieldB z of typically 10 gauss, produced by a Helmholtz coil.T e is approximately 10 eV. The density exhibited a peaked profile havingn max ≈ 2×1010 cm−3 and a density gradient scale length of ∼10 cm. The fluctuation experiments were conducted using a mobile vertical array of eight Langmuir probes. At major radii outside the density peak, the vertical correlation lengths λc⊥ of fluctuations were found to be on the order of 5–10 cm for fluctuation frequencies below 3–8 kHz, and on the order of 1–2 cm at higher frequencies. At major radii on the inner slope of the density peak, a new feature appears in the spatial coherence function consisting of a second peak at a probe separation which scales linearly with vertical field. This observation indicates that these fluctuations have a correlation length on the order of 2πR 0∼600 cm in the direction parallel to the helical magnetic field lines.

Production of statistically nonstationary stochastic structure realizations for infrared background scene simulations

We describe the development of a 3-D statistically nonstationary earthlimb stochastic structure representation and its application to earthlimb infrared (IR) background structure simulations. An earthlimb viewing geometry is defined by the location of both observer and source in space. The line-of-sight tangent altitude is the minimum altitude of the trajectory, defined at the earth-centered perpendicular to the line of sight. The stochastic structure overlay is constructed from a 2-D diagonal cut through a 3-D matrix of correlated Gaussian deviates; each plane of the matrix represents a statistically constant representation of the 2-D correlation lengths at a given altitude above the earth. Each matrix plane is generated using successive 2-D fast Fourier transform routines that have empirical values of vertical and horizontal correlation lengths as input. The total deviate variance versus altitude is then scaled from empirical measurements of fluctuations in atmospheric density, temperature, and/or emissivity. These structure generators are used both as perturbations on input atmospheric data to IR radiance codes and as high-resolution overlays to earthlimb lR mean-radiance 2-D scenes. An IR structured scene realization and scene validation analysis is presented.

Tomographic measurements of sediment inhomogeneities and determination of statistical parameters critical to sediment volume backscattering

A tomographic method was used for measuring sound-speed fluctuations in the top 5 m of sediments in Biscayne Bay, Florida. Sub-bottom porosity and density fluctuations were estimated from measured sound-speed fluctuations by using Gassmann’s [Geophysics 16, 673–685 (1951)] equations. Statistical parameters for volume backscattering calculations such as vertical correlation length, horizontal correlation length, and variance of acoustic impedance were obtained from tomographic measurements. Volume backscattering strengths for the experiment site were calculated using the theoretical results given by Crowther [AcousticsandtheSeabed, edited by N. G. Pace (Bath University, Bath, 1983), pp. 147–155]. Measurements of acoustic backscattering strengths were also conducted at the same site. Comparisons between measured backscattering strengths and calculated volume backscattering strengths indicate that, for this particular location, scattering is mostly due to bottom roughness. [Work supported by ONR and UM-Research Program.]

Measured wave-front fluctuations in 1000-km pulse propagation in the Pacific Ocean

A 1000-km acoustical transmission experiment has been carried out in the North Pacific, with pulses broadcast between a moored broadband source (250-Hz center frequency) and a moored sparse vertical line of receivers. Two data records are reported: a period of 9 days at a pulse rate of one per hour, and a 21-h period on the seventh day at six per hour. Many wave-front segments were observed at each hydrophone depth, and arrival times were tracked and studied as functions of time and depth. Arrivals within the final section of the pulse are not trackable in time or space at the chosen sampling rates, however. Broadband fluctuations, which are uncorrelated over 10-min sampling and 60-m vertical spacing, are observed with about 40 (ms)2 variance. The variance of all other fluctuations (denoted as low-frequency) is comparable or smaller than the broadband value; this low-frequency variance can be separated into two parts: a wave-front segment displacement (with vertical correlation length greater than 1 km) that varies substantially between rays with different ray identifiers, and a distortion (with vertical correlation length between 60 m and 1 km) of about 2 (ms)2 variance. The low-frequency variance may be explained as the effect of internal waves, including internal tides. The variance of the broadband fluctuations is reduced somewhat but not eliminated if only high-intensity peaks are selected; this selection does not affect the statistics of the low-frequency fluctuations.

Statistics of small‐scale activity in the upper ocean

Observations are presented on the statistical distribution of temperature fluctuations on horizontal scales from 10 cm to 1 km in the seasonal thermocline. This range is especially interesting because it includes internal wave processes on the long end and it approaches microscale processes on the short end. A measure of the activity level is computed as the variance of horizontal gradients of temperature or conductivity normalized by the square of the mean vertical gradient, essentially a pseudo Cox number, in eight wavelength bands across this range of scales. The activity levels are found to be distributed approximately lognormally in these bands, with the mean level increasing with decreasing wavelength in conformance with previously published temperature and displacement spectra. Root‐mean‐square slopes inferred from the mean activity level are found to be slightly less than 1% for wavelengths longer than 300 m and increase only slightly on shorter scales until a wavelength of 2 to 3 m, where they increase sharply. The variance also increases significantly near 3 m and is significantly greater near the peak density gradient under the base of the mixed layer than deeper in the seasonal thermocline. Both horizontal and vertical correlation lengths of the log of activity level are calculated, providing estimates of “patch” sizes which are about 100 m long and about l m high for the shortest wavelengths. These correlation lengths are used with local estimates of the variance of activity level to conclude that there is significant inhomogeneity and/or nonstationarity in the data set.

Propagation in atmospheric convective‐boundary‐layer turbulence

Turbulence in the atmospheric convective boundary layer is inhomogeneous and anisotropic. The vertical correlation length of the sound speed field is typically proportional to height from the ground, whereas the horizontal correlation length is proportional to boundary‐layer thickness. Solutions for the turbulence strength and diffraction parameters, valid for this particular case, are presented. The solutions differ significantly from those for homogeneous, isotropic turbulence. An intercomparison is made between the theoretical solutions, numerical propagation simulations, and experimental data. For the numerical simulations, rays were traced through wind and temperature fields generated by a large‐eddy simulation.

Superintegrable chiral Potts model: Thermodynamic properties, an ?inverse? model, and a simple associated Hamiltonian

The partition function of theN-state Superintegrable chiral Potts model is obtained exactly and explicitly (if not completely rigorously) for a finite lattice with particular boundary conditions. This yields the bulk and surface free energies, and horizontal and vertical correlation lengths and interfacial tensions. The critical exponents are α=1−2/N,μhor=υhor=2/N, and μvert=υvert=1, and the finite-size corrections are obtained at criticality. The eigenvalue spectrum of the column-to-column transfer matrix is that of a direct product ofN byN matrices. Inverting this matrix gives a related solvable model which is a generalization of the free-fermion model. The associated Hamiltobian has a very simple form, suggesting there may be a more direct algebraic method (perhaps a generalized Clifford algebra) for obtaining its eigenvalues.

Theoretical Models For Polarimetric Radar Clutter

The Mueller matrix and polarization covariance matrix are described for polarimetric radar systems. The clutter is modeled by a layer of random permittivity, described by a three-dimensional correlation function, with variance, and horizontal and vertical correlation lengths. This model is applied, using the wave theory with Born approximations carried to the second order, to find the backscattering elements of the polarimetric matrices. It is found that 8 out of 16 elements of the Mueller matrix are identically zero, corresponding to a covariance matrix with four zero elements. Theoretical predictions are matched with experimental data for vegetation fields.

A natural gradient experiment on solute transport in a sand aquifer: Spatial variability of hydraulic conductivity and its role in the dispersion process

The spatial variability of hydraulic conductivity at the site of a long‐term tracer test performed in the Borden aquifer was examined in great detail by conducting permeability measurements on a series of cores taken along two cross sections, one along and the other transverse to the mean flow direction. Along the two cross sections, a regular‐spaced grid of hydraulic conductivity data with 0.05 m vertical and 1.0 m horizontal spatial discretization revealed that the aquifer is comprised of numerous thin, discontinuous lenses of contrasting hydraulic conductivity. Estimation of the three‐dimensional covariance structure of the aquifer from the log‐transformed data indicates that an exponential covariance model with a variance equal to 0.29, an isotropic horizontal correlation length equal to about 2.8 m, and a vertical correlation length equal to 0.12 m is representative. A value for the longitudinal macrodispersivity calculated from these statistical parameters using three‐dimensional stochastic transport theory developed by L. W. Gelhar and C. L. Axness (1983) is about 0.6 m. For the vertically averaged case, the two‐dimensional theory developed by G. Dagan (1982, 1984) yields a longitudinal djspersivity equal to 0.45 m. Use of the estimated statistical parameters describing the ln (K) variability in Dagan's transient equations closely predicted the observed longitudinal and horizontal transverse spread of the tracer with time. Weak vertical and horizontal dispersion that is controlled essentially by local‐scale dispersion was obtained from the analysis. Because the dispersion predicted independently from the statistical description of the Borden aquifer is consistent with the spread of the injected tracer, it is felt that the theory holds promise for providing meaningful estimates of effective transport parameters in other complex‐structured aquifers.