Isotropic Increments Research Articles

Complexity of Gaussian Random Fields with Isotropic Increments

Complexity of Gaussian Random Fields with Isotropic Increments

A Comparison of the Impacts of Radiosonde and AMSU Radiance Observations in GSI Based 3DEnsVar and 3DVar Data Assimilation Systems for NCEP GFS

The impact of observations can be dependent on many factors in a data assimilation (DA) system including data quality control, preprocessing, skill of the model, and the DA algorithm. The present study focuses on comparing the impacts of observations assimilated by two different DA algorithms. A three-dimensional ensemble-variational (3DEnsVar) hybrid data assimilation system was recently developed based on the Gridpoint Statistical Interpolation (GSI) data assimilation system and was implemented operationally for the National Center for Environmental Prediction (NCEP) Global Forecast System (GFS). One question to address is, how the impacts of observations on GFS forecasts differ when assimilated by the traditional GSI-three dimensional variational (3DVar) and the new 3DEnsVar. Experiments were conducted over a 6-week period during Northern Hemisphere winter season at a reduced resolution. For both the control and data denial experiments, the forecasts produced by 3DEnsVar were more accurate than GSI3DVar experiments. The results suggested that the observations were better and more effectively exploited to increment the background forecast in 3DEnsVar. On the other hand, in GSI3DVar, where the observation will be making mostly local, isotropic increments without proper flow dependent extrapolation is more sensitive to the number and types observations assimilated.

High-dimensional Gaussian fields with isotropic increments seen through spin glasses

We study the free energy of a particle in (arbitrary) high-dimensional Gaussian random potentials with isotropic increments. We prove a computable saddle point variational representation in terms of a Parisi-type functional for the free energy in the infinite-dimensional limit. The proofs are based on the techniques developed in the course of the rigorous analysis of the Sherrington-Kirkpatrick model with vector spins.

Covariance and spectral properties of the wavelet transform and discrete wavelet coefficients of second-order random fields

Abstract The second-order properties of the wavelet transform and of the discrete wavelet coefficients, in the orthonormal series representation, of second-order random fields on Rn are determined. Both weakly homogeneous random fields as well as random fields with weakly homogeneous increments are considered. Weakly isotropic fields and fields with weakly isotropic increments are also considered. Applications to fractional Brownian fields on Rn are given.

Covariance and spectral properties of the wavelet transform and discrete wavelet coefficients of second-order random fields

The second-order properties of the wavelet transform and of the discrete wavelet coefficients, in the orthonormal series representation, of second-order random fields on R n are determined. Both weakly homogeneous random fields as well as random fields with weakly homogeneous increments are considered. Weakly isotropic fields and fields with weakly isotropic increments are also considered. Applications to fractional Brownian fields on R n are given.

Scaling of random fields by means of truncated power variograms and associated spectra

An interpretation is offered for the observation that the log hydraulic conductivity of geologic media often appears to be statistically homogeneous but with variance and integral scale which grow with domain size. We first demonstrate that the power (semi)variogram and associated spectra of random fields, having homogeneous isotropic increments, can be constructed as weighted integrals from zero to infinity (an infinite hierarchy) of exponential or Gaussian variograms and spectra of mutually uncorrelated homogeneous isotropic fields (modes). We then analyze the effect of filtering out (truncating) high‐ and low‐frequency modes from this infinite hierarchy in the real and spectral domains. A low‐frequency cutoff renders the truncated hierarchy homogeneous with an autocovariance function that varies monotonically with separation distance in a manner not too dissimilar than that of its constituent modes. The integral scales of the lowest‐ and highest‐frequency modes (cutoffs) are related, respectively, to the length scales of the sampling window (domain) and data support (sample volume). Taking each relationship to be one of proportionality renders our expressions for the integral scale and variance of a truncated field dependent on window and support scales in a manner consistent with observations. The traditional approach of truncating power spectral densities yields autocovariance functions that oscillate about zero with finite (in one and two dimensions) or vanishing (in one dimension) integral scales. Our hierarchical theory allows bridging across scales at a specific locale, by calibrating a truncated variogram model to data observed on a given support in one domain and predicting the autocovariance structure of the corresponding multiscale field in domains that are either smaller or larger. One may also venture (we suspect with less predictive power) to bridge across both domain scales and locales by adopting generalized variogram parameters derived on the basis of juxtaposed hydraulic and tracer data from many sites.

Hausdorff dimension of the graph of a Gaussian random field

We shall find the Hausdorff dimensions of the image and the graph of a Gaussian random field with isotropic increments.

Second order random fields over l p with homogeneous and isotropic increments

A random field over l p is a stochastic process X(t), where t is an element of l p .It is said to have homogeneous and isotropic increments if E(X(t) − X(s)) 2 is a function of ∥t-s∥. The subject of this work is the spectral theory of such processes. The main results are: a representation of the field as a series of filtered, orthogonal processes with a real time parameter; a representation as a white noise integral over l p ;limit theorems for the average of X over a sphere; and, finally, filtering of the orthogonal components. In particular, we mention: (1) The averages over spheres of increasing dimension converge in quadratic mean for p=2 but not for 0<p<2. (2) The limiting distribution of a fixed coordinate of a point uniformly distributed over the l p -unit sphere in n-space, n → ∞, has the density [2γ(1/p+1)p 1/p ]−1exp(-¦x¦ p /P).