Statistics and Physical Oceanography (1993)

Oceanographic fields and processes possess certain features that are not commonly encountered in some other areas of science and engineering. One of these is a wide range of scales (wavenumbers and frequencies) in which observed fields exhibit spatial and temporal variation. In other words, a “typical” time (space) scale is absent, and there exists a broad band of frequencies (wavenumbers) of roughly equal importance. This is the reason for the term “multiple-scale variability.” Oceanographic processes include coupling across a large range of scales (i.e., nonlocal interactions) and linkage between a number of factors of different nature. In Figure 2.1 (from Dickey, 1990, 1991), typical spatial and temporal scales of some Oceanographic processes are sketched.

FIGURE 2.1 A schematic diagram illustrating the relevant time and space scales of several physical and biological processes important to the physics and ecosystem of the upper ocean. Reprinted from Dickey (1990, 1991) with permission.

From the statistical standpoint, a random field is a stochastic process with multidimensional parameters (e.g., time and position) or a more complicated parameter such as a function. The fields of primary interest have four parameters: one dimension of time and three dimensions of space. Examples of such time-varying fields include fluid velocity, pressure, water density, temperature, and salinity. Fields with only two spatial dimensions include sea surface height (sea level), wind velocity and wind stress at the surface, sea surface temperature (SST), ocean color, and sea ice. Wavenumber spectra of these fields are usually very broad, covering several decades of wavenumbers (e.g., Fu, 1983; Freilich and Chelton, 1986), and the spectral density function can be approximated by a power law. Characteristic values of exponents in the power laws indicate a fractal regime in the geometry of the fields. For instance, the sea surface elevation field, for scales related to wind-generated surface gravity waves (from a decimeter to several hundred meters), is characterized by a two-dimensional wavenumber spectrum that falls off roughly as k^-7/2. This corresponds to a cascade pattern in surface topography (a hierarchy of randomly superimposed waves with decreasing amplitude and wavelength). A characteristic property of this field is its statistical self-affinity (Glazman and Weichman, 1989). The corresponding Hausdorff dimension, for an assumed Gaussian distribution, is 2.25.

The fluid velocity field, whose kinetic energy spectrum is characterized by k^-5/3, exhibits a Hausdorff dimension of 2.666. A typical geometrical feature of such fields is a hierarchy of eddies. Such cascade patterns in a field’s geometry are related to the cascade nature of the energy transfer along the spectrum through nonlinear interactions among different scales of fluid motion. Other physical quantities, e.g., momentum, enstrophy (i.e., half the square of vorticity), and wave action, may also be transferred either up or down the spectrum. The spectral cascades of these quantities are not necessarily conservative: interactions between different oceanographic fields (occurring within certain limited ranges of scales—the “generation and dissipation subranges”—and resulting in energy and momentum exchange) provide energy sources or sinks in various spectral bands. For instance, at meter scales wind provides the energy input into surface gravity waves that in turn exchange momentum and energy with larger-scale motions (e.g., mesoscale eddies, Langmuir circulations, internal waves). Mesoscale oceanic eddies are caused by the barotropic instability of basin-scale currents. Seasonal heating and cooling of the ocean surface causes convection and vertical mixing, while differential (across the oceanic basins) heating, evaporation, precipitation, and ice melting cause density-driven currents. Ocean circulation on basin scales is caused by large-scale curl of the wind stress. This multiplicity of the energy sources and sinks and the interactions between different scales and individual components of ocean dynamics are responsible for the extreme complexity of patterns of ocean circulation, sea surface temperature, sea level, and so on as observed both in satellite images and in highly complicated trajectories of free-drifting floats. Apparently, the interaction of motions with different scales implies statistical dependence between corresponding Fourier components or between corresponding eigenvectors in the empirical orthogonal functions (EOF) series (Karhunen-Loeve expansion; see, e.g., Lorenz, 1956; Davis, 1976; Preisendorfer, 1988). Identifying and accounting for such correlations in statistical models are important problems of oceanographic data analysis.

The difficulties mentioned above need not defeat efforts to understand ocean dynamics. In contrast to economics, demography, biology, and many other fields, physical oceanography is based on the comparatively reliable and universal quantitative physical models summarized in Chapter 1.

Initial and boundary conditions complete the formulation of specific oceanographic problems. Since the boundary conditions (e.g., the distribution of wind stress over the sea surface) and the coefficients in the equations (e.g., ocean current velocities in the heat-transfer equations) are intrinsically random, oceanographic problems are actually those for stochastic partial differential equations (SPDEs). Many of the issues related to SPDEs are also encountered in analysis of oceanographic observations. These include, for instance, the impact of subscale (microscopic) motions on the (macroscopic) behavior of the mean fields (analogous to the dependence of measured quantities on the spatial, temporal, or spatiotemporal resolution of a measuring technique). On a more fundamental level, the justification of the “macroscopic” equations remains a difficult problem.

These problems that present opportunities for statisticians are also central to eventually understanding the structure of turbulent flow. Turbulent fields of fluid velocity, pressure, and temperature are highly inhomogeneous and include compact regions where these fields or their spatial derivatives attain extreme values. Regions with large fluid velocity gradients are particularly important, because most dissipation of the mechanical energy into heat occurs in these localized regions. Due to an irregular spatial and temporal distribution of such regions, the occurrences of extreme events are often referred to as intermittency. Intermittency becomes pronounced at high Reynolds numbers associated with the onset of turbulence. The Reynolds number is a measure of the relative importance of inertial forces in the fluid as compared to viscous forces (viz., it is the ratio of the inertia of fluid particles to the fluid’s viscous friction). At high Reynolds numbers, when the inertia of fluid particles is no longer balanced by friction forces, particle trajectories become tremendously complicated. This results from the frictionless fluid particles having an unrestrained ability to continue their motion in whatever may be the direction they were launched (by some initial disturbance) or deflected (by interactions with neighboring particles). No matter how small the differences in initial directions and velocities between individual particles, their trajectories quickly diverge. An observer sees a highly chaotic pattern of flow, including intermittent events with particularly large velocity gradients. What is the probability structure of the dissipation field and related field gradients in a turbulent flow? No rigorous deductions based on the governing N-S equations have been reported, although a number of heuristic models have been proposed (e.g., Novikov and Stewart, 1964; Novikov, 1966; Yaglom, 1966; Mandelbrot, 1974).

Satellite instruments measure at different incidence angles the electromagnetic characteristics of the emitted radiation (passive instruments working in visible, infrared, and microwave ranges of the electromagnetic spectrum) and backscattered radar pulses (active instruments working in the microwave range) that come from the ocean surface. These

characteristics (e.g., the intensity of visible and infrared radiation at various wavelengths, radio brightness temperature, radar cross section, round-trip travel time of a reflected pulse, the shape of the pulse distorted by a random sea surface, and so on) are interpreted in terms of oceanographic parameters (pigment and chlorophyll-A concentrations, sea surface temperature, wind speed and direction at the surface, sea level height, and others). The interpretations are typically obtained from empirical algorithms based on incomplete or approximate physical models. For instance, empirical relationships based on a limited set of coincidental radar and buoy observations are routinely employed to derive wind speed from altimeter and scatterometer radar cross sections. Such relationships are called geophysical model functions (GMFs). The available GMFs are based on rather simple linear or nonlinear regression models, and considerable improvement might be possible in this area with the use of more advanced statistical methods.

Instrument footprint sizes, swath widths, and other characteristics of typical satellite instruments are summarized in Table 2.1. The footprint is a spot on the surface from which reflected or emitted radiation is collected by satellite antenna to produce the observed radar cross section, brightness temperature, and so on. Spatial coverage (which depends on swath width, footprint size, sampling rate, and satellite orbit geometry) varies from one instrument to another. The spatial sampling rate, i.e., the distance between individual satellite footprints, may cause aliasing of the data. Other factors leading to aliasing are the spatial separation of satellite orbits and the specific time interval between repeat tracks (see Figure 6.1 in Chapter 6). All these factors raise issues regarding correct interpretation of satellite measurements and their use in numerical models of ocean circulation. Spatial inhomogeneity of surface properties on scales within and beyond the footprint size, and these properties varying nonlinearly along any direction within a footprint, produce an appreciable dependence of satellite measurements upon the instrument employed. The case of wind speed measurements is most instructive. Wind speed maps for the same period of time but based on measurements by different satellite techniques exhibit appreciable differences—regardless of the fact that the root-mean-square measurement errors characterizing individual instruments are very similar. Pandey (1987) compared wind fields based on satellite scatterometer, altimeter, and microwave radiometer data and found that the discrepancy locally may exceed 2 m/s. Statistical distributions of wind velocities derived from different instruments can also differ.

Statistical models of oceanographic fields with prescribed statistical properties might prove useful for analysis of satellite and other measurements (e.g., Ropelewski, 1992). In Chapter 6, additional problems arising in connection with the spatial inhomogeneity, statistical anisotropy and intermittency observed in oceanographic fields are reviewed. Those include transferring (binning) the satellite-produced data onto geographic grids, filling gaps in the data, and interpolating, extrapolating, smoothing, and filtering the data.

There are important open questions associated with sampling at different rates: how does sampling at different rates relate to aliasing, and to interaction of processes occurring

at different scales? What can and cannot be inferred about the continuous process within which sampling is done? These concerns also involve different types of estimates such as second- and higher-order spectral estimates, probability density estimates, and regression estimates. Such questions should be considered under the assumptions of both stationary and nonstationary processes. These problems are connected with those involving non-Gaussian observations (see Chapter 8). Suitably selected and designed multiscale wavelets may be helpful in this situation.

There are statistical research opportunities in modeling a random field given:

1. observational data representing averages over regions (pixels) of a given size (as determined, e.g., by a satellite footprint), and

2. observational data obtained by irregular sampling (spatial and temporal data gaps, etc.) of a random field.

An analysis of extrema of non-Gaussian fields is needed. It will depend partly on what one can say in the stationary case about the tails of the instantaneous distributions. Such an analysis will have both a probabilistic and a statistical aspect; i.e., given a nice probabilistic characterization, can some aspect of it be effectively estimated from data? Progress on these questions may also carry over to notions of intermittency. Specific issues for focus include:

1. analysis of asymptotics of extrema of a non-Gaussian field,

2. analysis of behavior of outlying observations in a case of non-Gaussian data, and

3. modeling of a random field with given statistics of extrema.

Additional issues and problems concerning non-Gaussian random fields and processes are listed at the end of Chapter 8.