Efi Foufoula-Georgiou

Wavelet analysis for geophysical applications

Wavelet transforms originated in geophysics in the early 1980s for the analysis of seismic signals. Since then, significant mathematical advances in wavelet theory have enabled a suite of applications in diverse fields. In geophysics the power of wavelets for analysis of nonstationary processes that contain multiscale features, detection of singularities, analysis of transient phenom- ena, fractal and multifractal processes, and signal com- pression is now being exploited for the study of several processes including space-time precipitation, remotely sensed hydrologic fluxes, atmospheric turbulence, can- opy cover, land surface topography, seafloor bathymetry, and ocean wind waves. It is anticipated that in the near future, significant further advances in understanding and modeling geophysical processes will result from the use of wavelet analysis. In this paper we review the basic properties of wavelets that make them such an attractive and powerful tool for geophysical applications. We dis- cuss continuous, discrete, orthogonal wavelets and wavelet packets and present applications to geophysical processes.

Channel network source representation using digital elevation models

Methods for identifying the size, or scale, of hillslopes and the extent of channel networks from digital elevation models (DEMs) are examined critically. We show that a constant critical support area, the method most commonly used at present for channel network extraction from DEMs, is more appropriate for depicting the hillslope/valley transition than for identifying channel heads. Analysis of high-resolution DEMs confirms that a constant contributing area per unit contour length defines the extent of divergent topography, or the hillslope scale, although there is considerable variance about the average value. In even moderately steep topography, however, a DEM resolution finer than the typical 30 m by 30 m grid size is required to accurately resolve the hillslope/valley transition. For many soil-mantled landscapes, a slope-dependent critical support area is both theoretically and empirically more appropriate for defining the extent of channel networks. Implementing this method for overland flow erosion requires knowledge of an appropriate proportionality constant for the drainage area-slope threshold controlling channel initiation. Several methods for estimating this constant from DEM data are examined, but acquisition of even limited field data is recommended. Finally, the hypothesis is proposed that an inflection in the drainage area-slope relation for mountain drainage basins reflects a transition from steep debris flow-dominated channels to lower-gradient alluvial channels.

Mountaintop Mining Consequences

Damage to ecosystems and threats to human health and the lack of effective mitigation require new approaches to mining regulation. There has been a global, 30-year increase in surface mining (1), which is now the dominant driver of land-use change in the central Appalachian ecoregion of the United States (2). One major form of such mining, mountaintop mining with valley fills (MTM/VF) (3), is widespread throughout eastern Kentucky, West Virginia (WV), and southwestern Virginia. Upper elevation forests are cleared and stripped of topsoil, and explosives are used to break up rocks to access buried coal (fig. S1). Excess rock (mine “spoil”) is pushed into adjacent valleys, where it buries existing streams.

A multicomponent decomposition of spatial rainfall fields: 1. Segregation of large‐ and small‐scale features using wavelet transforms

Issues of scaling characteristics in spatial rainfall have attracted increasing attention over the last decade. Several methods based on simple and multiscaling and multifractal ideas have been proposed and parameter estimation techniques developed for the hypothesized models. Simulations based on these models have realistic resemblance to “generic rainfall fields.” In this research we analyze rainfall data for scaling characteristics without an a priori assumed model. We look at the behavior of rainfall fluctuations obtained at several scales, via orthogonal wavelet transform of the data, to infer the precise nature of scaling exhibited by spatial rainfall. The essential idea behind the analysis is to segregate large-scale (long wavelength) features from small-scale features and study each of them independently. The hypothesis is set forward that rainfall might exhibit scaling in small-scale fluctuations, if at all, and at large scale this behavior will break down to accommodate the effects of external factors affecting the particular rain-producing mechanism. The validity of this hypothesis is examined. In the first of these papers we develop the methodology for the segregation of large- and small-scale features and apply it to a severe spring time midlatitude squall line storm. The second paper (Kumar and Foufoula-Georgiou, this issue) develops a framework for testing the presence and studying the nature of self-similarity in the fluctuations.

Model for multiscale disaggregation of spatial rainfall based on coupling meteorological and scaling descriptions

The precipitation output of a mesoscale atmospheric numerical model is usually interpreted as the average rainfall intensity over the grid cell of the model (typically 30 × 30 km to 60×60 km). However, rainfall exhibits considerable heterogeneity over subgrid scales (i.e., scales smaller than the grid cell), so it is necessary for hydrologic applications to recreate or simulate the small-scale rainfall variability given its large-scale average. Rainfall disaggregation is usually done statistically. In this paper, a new subgrid scale rainfall disaggregation model is developed. It has the ability to statistically reproduce the rainfall variability at scales unresolved by mesoscale models while being conditioned on large-scale rainfall averages and physical properties of the prestorm environment. The model is based on two extensively tested hypotheses for midlatitude mesoscale convective systems [Perica and Foufoula-Georgiou, 1996]: (1) standardized rainfall fluctuations (defined via a wavelet transform) exhibit simple scaling over the mesoscale, and (2) statistical scaling parameters of rainfall fluctuations relate to the convective available potential energy (CAPE), a measure of the convective instability of the prestorm environment. Preliminary evaluation of the model showed that the model is capable of reconstructing the small-scale statistical variability of rainfall as well as the fraction of area covered with rain at all analyzed subgrid scales. The performance evaluation was based on comparison of summary statistics and spatial pattern measures of simulated fields with those of known fields observed during the Oklahoma-Kansas Preliminary Regional Experiment for Storm-Central (PRE-STORM).

Wavelet Analysis in Geophysics: An Introduction

Abstract Wavelet analysis is a rapidly developing area of mathematical and application-oriented research in many disciplines of science and engineering. The wavelet transform is a localized transform in both space (time) and frequency, and this property can be advantageously used to extract information from a signal that is not possible to unravel with a Fourier or even windowed Fourier transform. Wavelet transforms originated in geophysics in early 1980s for the analysis of seismic signal. After a decade of significant mathematical formalism they are now also being exploited for the analysis of several other geophysical processes such as atmospheric turbulence, space-time rainfall, ocean wind waves, seafloor bathymetry, geologic layered structures, climate change, among others. Due to their unique properties, well suited for the analysis of natural phenomena, it is anticipated that there will be an explosion of wavelet applications in geophysics in the next several years. This chapter provides a basic introduction to wavelet transforms and their most important properties. The theory and applications of wavelets is developing very rapidly and we see this chapter only as a limited basic introduction to wavelets which we hope to be of help to the unfamiliar reader and provide motivation and references for further study.

Profiling risk and sustainability in coastal deltas of the world

Deltas are growing centers of risk Population growth, urbanization, and rising sea levels are placing populations living in delta regions under increased risk. The future resiliency and potential for adaptation by these populations depend on a number of socioeconomic and geophysical factors. Tessler et al. examined 48 deltas from around the globe to assess changes in regional vulnerability (see the Perspective by Temmerman). Some deltas in countries with a high gross domestic product will be initially more resilient to these changes, because they can perform expensive maintenance on infrastructure. However, short-term policies will become unsustainable if unaccompanied by long-term investments in all delta regions. Science, this issue p. 638; see also p. 588 Present-day strategies for risk management may impede long-term sustainability of river deltas. [Also see Perspective by Temmerman and Kirwan] Deltas are highly sensitive to increasing risks arising from local human activities, land subsidence, regional water management, global sea-level rise, and climate extremes. We quantified changing flood risk due to extreme events using an integrated set of global environmental, geophysical, and social indicators. Although risks are distributed across all levels of economic development, wealthy countries effectively limit their present-day threat by gross domestic product–enabled infrastructure and coastal defense investments. In an energy-constrained future, such protections will probably prove to be unsustainable, raising relative risks by four to eight times in the Mississippi and Rhine deltas and by one-and-a-half to four times in the Chao Phraya and Yangtze deltas. The current emphasis on short-term solutions for the world’s deltas will greatly constrain options for designing sustainable solutions in the long term.

Multiscale Statistical Properties of a High-Resolution Precipitation Forecast

Abstract Small-scale (less than ∼15 km) precipitation variability significantly affects the hydrologic response of a basin and the accurate estimation of water and energy fluxes through coupled land–atmosphere modeling schemes. It also affects the radiative transfer through precipitating clouds and thus rainfall estimation from microwave sensors. Because both land–atmosphere and cloud–radiation interactions are nonlinear and occur over a broad range of scales (from a few centimeters to several kilometers), it is important that, over these scales, cloud-resolving numerical models realistically reproduce the observed precipitation variability. This issue is examined herein by using a suite of multiscale statistical methods to compare the scale dependence of precipitation variability of a numerically simulated convective storm with that observed by radar. In particular, Fourier spectrum, structure function, and moment-scale analyses are used to show that, although the variability of modeled precipitation agr...

Scale issues in verification of precipitation forecasts

Precipitation forecasts from numerical weather prediction models are often compared to rain gauge observations to make inferences as to model performance and the “best” resolution needed to accurately capture the structure of observed precipitation. A common approach to quantitative precipitation forecast (QPF) verification is to interpolate the model-predicted areal averages (typically assigned to the center point of the model grid boxes) to the observation sites and compare observed and predicted point values using statistical scores such as bias and RMSE. In such an approach, the fact that the interpolated values and their uncertainty depend on the scale (model resolution) of the values from which the interpolation was done is typically ignored. This interpolation error, which comes from scale effects, is referred to here as the “representativeness error.” It is a nonzero scale-dependent error even for the case of a perfect model and thus can be seen as independent of model performance. The scale dependency of the representativeness error can have a significant effect on model verification, especially when model performance is judged as a function of grid resolution. An alternative method is to upscale the gauge observations to areal averages and compare at the scale of the model output. Issues of scale arise here too, with a different scale dependency in the representativeness error. This paper examines the merits and limitations of both verification methods (area-to-point and point-to-area) in view of the pronounced spatial variability of precipitation fields and the inherent scale dependency of the representativeness error in each of the verification procedures. A composite method combining the two procedures is introduced and shown to diminish the scale dependency of the representativeness error.

Normal and anomalous diffusion of gravel tracer particles in rivers

Received 12 December 2008; revised 16 November 2009; accepted 10 December 2009; published 4 May 2010. [1] One way to study the mechanism of gravel bed load transport is to seed the bed with marked gravel tracer particles within a chosen patch and to follow the pattern of migration and dispersal of particles from this patch. In this study, we invoke the probabilistic Exner equation for sediment conservation of bed gravel, formulated in terms of the difference between the rate of entrainment of gravel into motion and the rate of deposition from motion. Assuming an active layer formulation, stochasticity in particle motion is introduced by considering the step length (distance traveled by a particle once entrained until it is deposited) as a random variable. For step lengths with a relatively thin (e.g., exponential) tail, the above formulation leads to the standard advection‐diffusion equation for tracer dispersal. However, the complexity of rivers, characterized by a broad distribution of particle sizes and extreme flood events, can give rise to a heavy‐tailed distribution of step lengths. This consideration leads to an anomalous advection‐diffusion equation involving fractional derivatives. By identifying the probabilistic Exner equation as a forward Kolmogorov equation for the location of a randomly selected tracer particle, a stochastic model describing the temporal evolution of the relative concentrations is developed. The normal and anomalous advection‐diffusion equations are revealed as its long‐time asymptotic solution. Sample numerical results illustrate the large differences that can arise in predicted tracer concentrations under the normal and anomalous diffusion models. They highlight the need for intensive data collection efforts to aid the selection of the appropriate model in real rivers.