For purposes of statistical analyses, oceanographic fields are usually assumed to be Gaussian, stationary, and spatially homogeneous, and their statistical description is limited to the calculation of wavenumber spectra. However, since oceanographic stochastic partial differential equations (see Chapter 2) are nonlinear or bilinear, the statistics of the fields depart from such simple models. The nonlinearity is due mainly to advective terms such as (**u**·∇)**u** where **u** is the velocity vector for water motion. In some cases, specifically for surface gravity waves, the nonlinear nature of the fluid motion is due to nonlinear boundary conditions: water motion is described by a function and is governed by the Laplace equation, while the (kinematic) boundary condition expressing the continuity of the free surface is nonlinear. As a result, closed equations for various statistical moments of the fields cannot be rigorously derived. Pertinent definitions and statistical problems are reviewed in two comprehensive volumes on statistical fluid dynamics by Monin and Yaglom (1971, 1975). A review of statistical geometry and kinematics of turbulent flows is given by Corrsin (1975). Walsh (1986) and Rozovskii (1990) provide introductions to stochastic partial differential equations.

One of the most important and least understood features of oceanographic processes is the intermittent (rare) occurrence of special or catastrophic events. These include (in order of increasing scale) appearance of white caps at the crests of exceedingly steep and breaking surface gravity waves, patches of small-scale turbulence left by breaking internal waves, the shedding of mesoscale rings and eddies by large-scale currents (such as the Gulf Stream or the Agulhas current), and the occurrence of localized anomalies in SST including El Niño events with a time interval on the order of years. Such events play a very important role in the overall dissipation of kinetic energy, and in the transport of heat, salt, and other quantities by ocean currents, as well as in the exchange of energy, momentum, and chemical quantities across the air-sea interface. In terms of the primitive equations describing individual realizations of oceanographic fields, such events may often be viewed as singularities developing in the process of a field’s evolution. Statistical analysis and modeling of such events are highly desirable. The use of quantile estimates might be investigated, especially for information in the tail of the distribution. The statistical geometry of these intermittent events is poorly understood, and improved understanding can be achieved by accounting more fully for the non-Gaussian nature of oceanographic fields.

Considerable progress in statistical modeling of geophysical “turbulent” fields has been achieved using ideas of multifractal processes (e.g., Schmitt et al., 1992). However, most of this work is related to atmospheric phenomena (Lovejoy and Schertzer, 1986; Schertzer and Lovejoy, 1987). A review of various problems arising in remote sensing, geophysical fluid dynamics, solid earth geophysics, and ocean, atmosphere, and climate studies can be found in Schertzer and Lovejoy (1991).

The special case of weak turbulence (when the nonlinear terms are of second order with respect to the linear terms in the governing equations) deserves particular attention, for it is encountered in many oceanographic problems and can be treated by small-perturbation

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Statistics and Physical Oceanography
8
NON-GAUSSIAN RANDOM FIELDS
For purposes of statistical analyses, oceanographic fields are usually assumed to be Gaussian, stationary, and spatially homogeneous, and their statistical description is limited to the calculation of wavenumber spectra. However, since oceanographic stochastic partial differential equations (see Chapter 2) are nonlinear or bilinear, the statistics of the fields depart from such simple models. The nonlinearity is due mainly to advective terms such as (u·∇)u where u is the velocity vector for water motion. In some cases, specifically for surface gravity waves, the nonlinear nature of the fluid motion is due to nonlinear boundary conditions: water motion is described by a function and is governed by the Laplace equation, while the (kinematic) boundary condition expressing the continuity of the free surface is nonlinear. As a result, closed equations for various statistical moments of the fields cannot be rigorously derived. Pertinent definitions and statistical problems are reviewed in two comprehensive volumes on statistical fluid dynamics by Monin and Yaglom (1971, 1975). A review of statistical geometry and kinematics of turbulent flows is given by Corrsin (1975). Walsh (1986) and Rozovskii (1990) provide introductions to stochastic partial differential equations.
One of the most important and least understood features of oceanographic processes is the intermittent (rare) occurrence of special or catastrophic events. These include (in order of increasing scale) appearance of white caps at the crests of exceedingly steep and breaking surface gravity waves, patches of small-scale turbulence left by breaking internal waves, the shedding of mesoscale rings and eddies by large-scale currents (such as the Gulf Stream or the Agulhas current), and the occurrence of localized anomalies in SST including El Niño events with a time interval on the order of years. Such events play a very important role in the overall dissipation of kinetic energy, and in the transport of heat, salt, and other quantities by ocean currents, as well as in the exchange of energy, momentum, and chemical quantities across the air-sea interface. In terms of the primitive equations describing individual realizations of oceanographic fields, such events may often be viewed as singularities developing in the process of a field’s evolution. Statistical analysis and modeling of such events are highly desirable. The use of quantile estimates might be investigated, especially for information in the tail of the distribution. The statistical geometry of these intermittent events is poorly understood, and improved understanding can be achieved by accounting more fully for the non-Gaussian nature of oceanographic fields.
Considerable progress in statistical modeling of geophysical “turbulent” fields has been achieved using ideas of multifractal processes (e.g., Schmitt et al., 1992). However, most of this work is related to atmospheric phenomena (Lovejoy and Schertzer, 1986; Schertzer and Lovejoy, 1987). A review of various problems arising in remote sensing, geophysical fluid dynamics, solid earth geophysics, and ocean, atmosphere, and climate studies can be found in Schertzer and Lovejoy (1991).
The special case of weak turbulence (when the nonlinear terms are of second order with respect to the linear terms in the governing equations) deserves particular attention, for it is encountered in many oceanographic problems and can be treated by small-perturbation

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Statistics and Physical Oceanography
techniques. Examples of weak turbulence include two-dimensional and geostrophic turbulence and surface gravity waves. Weak turbulence theory in its present form (Zakharov et al., 1992) permits derivation of kinetic equations describing energy exchanges (and exchanges of other quantities) among Fourier components, as well as derivation of higher-order spectra (bispectra, etc.) representing Fourier transforms of various statistical moments. Initially, this theory was developed for surface gravity and capillary waves (Hasselmann, 1962; Zakharov, 1984). However, statistical phenomena in waves (e.g., the existence of Kolmogorov-type spectra, the intermittency of breaking waves, and so on) have analogies in other oceanographic fields. The elegant Hamiltonian formulation of nonlinear wave dynamics (Zakharov, 1984; Zakharov et al., 1992) is a powerful tool for studies of fundamental statistical properties of turbulent fields.
To better characterize the scope of statistical issues that the weak turbulence theory or alternative statistical approaches could address, a brief review of some issues related to wind-generated surface gravity waves is in order. Until recently, statistical studies of field geometry were dominated by the work on Gaussian fields. Longuet-Higgins (1957, 1962, 1984) studied a large variety of geometrical properties of such fields with application to sea surface waves. Among other problems, he considered statistics of specular points (the points at which the gradient of the field is either zero or is specified depending on a viewing angle) and of the wave envelope, which play an important role in wave dynamics and analysis of sun glitter and radar backscatter from a wind-disturbed sea surface. A rigorous mathematical analysis of envelope statistics, high-level excursions, field maxima, and other geometrical properties of random two- and multi-dimensional Gaussian fields is presented by Adler (1981). Some of these results have been successfully employed in sea wave studies. Specifically, the theory of level crossings by two- and three-dimensional Gaussian- and Rayleigh-distributed fields was employed to estimate statistics of whitecaps (breaking waves) and of wave trains (Glazman, 1986; Glazman and Weichman, 1989; Glazman, 1991). Observations indicate that whitecaps occur in clusters. Hence, the use of a simple Poisson distribution (Glazman, 1991) for whitecap occurrence, which is known from the theory of high-level excursions by the (Gaussian) wave slope field, may be insufficient. The statistical theory of cluster point processes may be of great help here.
Linear methods are intrinsic for Gaussian stationary processes, and Fourier analysis is a natural tool to use in the resolution of stationary random fields. These yield a global resolution. However, in many situations, a resolution that is better adapted to local behavior would be more appropriate and interesting. This could be local behavior in time or local spatial behavior. One attempt in this direction makes use of wavelet transforms, which are in effect local filters of the field (Farge, 1992). Such a method amounts to a linear analysis of the field, although it could presumably be adapted to types of nonlinearity.
In the last few years, significant research effort in probability and statistics has been directed toward the development of models of non-Gaussian and time-varying random fields. Examples include stable fields; functionals of Gaussian, stable, and other fields represented via multiple integrals; density processes and measure-valued diffusions; and fields described by nonlinear stochastic differential equations. Applications of this research to oceanographic phenomena would be of interest to oceanographers since the fields they study are frequently non-Gaussian and time-varying random fields.

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One of the questions that arises in ocean remote sensing concerns the probability density function (pdf) for the heights of specular points and for the slopes and curvature radii of the surface. These pdfs are essentially non-Gaussian. A particularly interesting problem is statistically characterizing the asymmetry of the sea surface shape about the horizontal plane coincident with the mean sea level. This asymmetry is responsible for the deviation of the mean height of the specular points from the mean (zero-valued) height of the surface itself. As a result, an error bias (known as the sea-state bias) appears in altimeter measurements of the sea level. Mathematical analysis of such non-Gaussian surface properties is based on approximate joint pdfs for surface height and slopes. Following the work by Longuet-Higgins (1963) in which a truncated Gram-Charlier series expansion for the joint pdf was derived, the sea-state bias has been related to various spectral moments (Jackson, 1979; Srokosz, 1986) and ultimately expressed in terms of wind-wave generation conditions. While a simplified case of a one-dimensional surface has been studied, a two-dimensional case needs additional effort. The estimation of joint pdfs for dependent random sequences is reviewed, e.g., by Rosenblatt (1991). Further statistical effort in this direction could greatly facilitate analysis of biological and other oceanographic multidimensional processes.
The arrival of supercomputers opens new avenues for numerical modeling of complex processes. Now, for instance, numerical simulation of electromagnetic scattering by individual realizations of the random sea surface has become feasible. In this regard, simulated non-Gaussian random fields that satisfy basic conservation laws of fluid dynamics represent a great interest. A possible way of constructing individual realizations of a random field might be via the use of Wiener-Hermite polynomials (i.e., the Wiener-Ito expansion (Major, 1980)) in which the functional coefficients are determined on the requirement that the field yields the correct cumulants up to a certain order. Although bispectra (in the frequency domain) for surface gravity waves have been known since the work by Hasselmann et al. (1963), cumulants above second order for the surface’s spatial variation have not been studied. In the literature on large-scale ocean dynamics (two-dimensional and geostrophic turbulence), the Wiener-Ito expansion has never been used, although it appears to be most relevant. Estimation of the cumulant spectra is discussed in the pioneering work of Brillinger and Rosenblatt (1967). See also Rosenblatt (1985) and more recent material in Lii and Rosenblatt (1990).
STATISTICAL RESEARCH OPPORTUNITIES
There are many statistical research opportunities in the realm of non-Gaussian physical oceanographic random fields on which progress would be desirable. Some specific topics worthy of investigation are the following (also see related issues in Chapter 2):
Models of non-Gaussian and time-varying random fields: (a) probabilistic analysis of different models of non-Gaussian or nonstationary or time-varying random processes and fields (e.g., stable fields, measure-valued diffusions, density

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processes, non-Gaussian generalized fields, and so on), (b) structure of random fields with long-range dependence, and (c) non-Gaussian time series;
Theoretical models and techniques of simulation of non-Gaussian random fields with prescribed statistical properties, for example, (a) known moments up to some order, (b) known tail behavior of multivariate probability density functions, and (c) known statistics of extremes;
Extrema, sample path behavior, and geometry for non-Gaussian random processes and fields;
Inference and analysis of point processes with applications to oceanographic data;
Analysis of the Navier-Stokes system driven by Gaussian and non-Gaussian white noise;
Analysis of random fields that appear as solutions of stochastic partial differential equations (of special interest are equations driven by non-Gaussian noise or noises over a product of time-space and location-space);
Wavelet analysis of random fields with application to oceanographic problems; and
Statistical problems for non-Gaussian data (see models of particular interest in 2. above): (a) modeling (model identification, parameter estimation, and so on), (b) data analysis of irregularly sampled points on a field, (c) quantile estimation from dependent stationary processes and fields, (d) estimation problems for random fields given the types of sampling or observational layouts that are typical in oceanography, and (e) estimation problems for samples from non-Gaussian random fields.