Donald B. Percival

Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques

Glossary of symbols 1. Introduction to spectral analysis 2. Stationary stochastic processes 3. Deterministic spectral analysis 4. Foundations for stochastic spectral theory 5. Linear time-invariant filters 6. Non-parametric spectral estimation 7. Multiple taper spectral estimation 8. Calculation of discrete prolate spheroidal sequences 9. Parametric spectral estimation 10. Harmonic analysis References Appendix: data and code via e-mail Index.

Wavelet analysis of covariance with application to atmospheric time series

Multiscale analysis of univariate time series has appeared in the literature at an ever increasing rate. Here we introduce the multiscale analysis of covariance between two time series using the discrete wavelet transform. The wavelet covariance and wavelet correlation are defined and applied to this problem as an alternative to traditional cross-spectrum analysis. The wavelet covariance is shown to decompose the covariance between two stationary processes on a scale by scale basis. Asymptotic normality is established for estimators of the wavelet covariance and correlation. Both quantities are generalized into the wavelet cross covariance and cross correlation in order to investigate possible lead/lag relationships. A thorough analysis of interannual variability for the Madden-Julian oscillation is performed using a 35+ year record of daily station pressure series. The time localization of the discrete wavelet transform allows the subseries, which are associated with specific physical time scales, to be partitioned into both seasonal periods (such as summer and winter) and also according to El Nino-Southern Oscillation (ENSO) activity. Differences in variance and correlation between these periods may then be firmly established through statistical hypothesis testing. The daily station pressure series used here show clear evidence of increased variance and correlation in winter across Fourier periods of 16-128 days. During warm episodes of ENSO activity, a reduced variance is observed across Fourier periods of 8-512 days for the station pressure series from Truk Island and little or no correlation between station pressure series for the same periods.

Evaluating scaled windowed variance methods for estimating the Hurst coefficient of time series

Three-scaled windowed variance methods (standard, linear regression detrended, and brdge detrended) for estimating the Hurst coefficient (H) are evaluated. The Hurst coefficient, with 0 < H < 1, characterizes self-similar decay in the time-series autocorrelation function. The scaled windowed variance methods estimate H for fractional Brownian motion (fBm) signals which are cumulative sums of fractional Gaussian noise (fGn) signals. For all three methods both the bias and standard deviation of estimates are less than 0.05 for series having N ≥ 2(9) points. Estimates for short series (N < 2(8)) are unreliable. To have a 0.95 probability of distinguishing between two signals with true H differing by 0.1, more than 2(15) points are needed. All three methods proved more reliable (based on bias and variance of estimates) than Hursts rescaled range analysis, periodogram analysis, and autocorrelation analysis, and as reliable as dispersional analysis. The latter methods can only be applied to fGn or differences of fBm, while the scaled windowed variance methods must be applied to fBm or cumulative sums of fGn.

Analysis of Subtidal Coastal Sea Level Fluctuations Using Wavelets

Abstract Subtidal coastal sea level fluctuations affect coastal ecosystems and the consequences of destructive events such as tsunamis. We analyze a time series of subtidal fluctuations at Crescent City, California, during 1980–1991 using the maximal overlap discrete wavelet transform (MODWT). Our analysis shows that the variability in these fluctuations depends on the season for scales of 32 days and less. We show how the MODWT characterizes nonstationary behavior succinctly and how this characterization can be used to improve forecasts of inundation during tsunamis and storm surges. We provide pseudocode and enough details so that data analysts in other disciplines can readily apply MODWT analysis to other nonstationary time series.

Maximum likelihood estimation of K distribution parameters for SAR data

The K distribution has proven to be a promising and useful model for backscattering statistics in synthetic aperture radar (SAR) imagery. However, most studies to date have relied on a method of moments technique involving second and fourth moments to estimate the parameters of the K distribution. The variance of these parameter estimates is large in cases where the sample size is small and/or the true distribution of backscattered amplitude is highly non-Rayleigh. The present authors apply a maximum likelihood estimation method directly to the K distribution. They consider the situation for single-look SAR data as well as a simplified model for multilook data. They investigate the accuracy and uncertainties in maximum likelihood parameter estimates as functions of sample size and the parameters themselves. They also compare their results with those from a new method given by Raghavan (1991) and from a nonstandard method of moments technique; maximum likelihood parameter estimates prove to be at least as accurate as those from the other estimators in all cases tested, and are more accurate in most cases. Finally, they compare the simplified multilook model with nominally four-look SAR data acquired by the Jet Propulsion Laboratory AIRSAR over sea ice in the Beaufort Sea during March 1988. They find that the model fits data from both first-year and multiyear ice well and that backscattering statistics from each ice type are moderately non-Rayleigh. They note that the distributions for the data set differ too little between ice types to allow discrimination based on differing distribution parameters. >

Analyzing exact fractal time series: evaluating dispersional analysis and rescaled range methods

Precise reference signals are required to evaluate methods for characterizing a fractal time series. Here we use fGp (fractional Gaussian process) to generate exact fractional Gaussian noise (fGn) reference signals for one-dimensional time series. The average autocorrelation of multiple realizations of fGn converges to the theoretically expected autocorrelation. Two methods commonly used to generate fractal time series, an approximate spectral synthesis (SSM) method and the successive random addition (SRA) method, do not give the correct correlation structures and should be abandoned. Time series from fGp were used to test how well several versions of rescaled range analysis (R/S) and dispersional analysis (Disp) estimate the Hurst coefficient (0 < H < 1.0). Disp is unbiased for H < 0.9 and series length N ≥ 1024, but underestimates H when H > 0.9. R/S-detrended overestimates H for time series with H < 0.7 and underestimates H for H > 0.7. Estimates of H(Ĥ) from all versions of Disp usually have lower bias and variance than those from R/S. All versions of dispersional analysis, Disp, now tested on fGp, are better than we previously thought and are recommended for evaluating time series as long-memory processes.

The discrete wavelet transform and the scale analysis of the surface properties of sea ice

The formalism of the one-dimensional discrete wavelet transform (DWT) based on Daubechies wavelet filters is outlined in terms of finite vectors and matrices. Both the scale-dependent wavelet variance and wavelet covariance are considered and confidence intervals for each are determined. The variance estimates are more accurately determined with a maximal-overlap version of the wavelet transform. The properties of several Daubechies wavelet filters and the associated basis vectors are discussed. Both the Mallat orthogonal-pyramid algorithm for determining the DWT and a pyramid algorithm for determining the maximal-overlap version of the transform are presented in terms of finite vectors. As an example, the authors investigate the scales of variability of the surface temperature and albedo of spring pack ice in the Beaufort Sea. The data analyzed are from individual lines of a Landsat TM image (25-m sample interval) and include both reflective (channel 3, 30-m resolution) and thermal (channel 6, 120-m resolution) data. The wavelet variance and covariance estimates are presented and more than half of the variance is accounted for by scales of less than 800 m. A wavelet-based technique for enhancing the lower-resolution thermal data using the reflected data is introduced. The simulated effects of poor instrument resolution on the estimated lead number density and the mean lead width are investigated using a wavelet-based smooth of the observations.

The Martian annual atmospheric pressure cycle - Years without great dust storms

A model of the annual cycle of pressure on Mars has been developed for a 2-year period chosen to include 1 year at Lander 2 and to minimize the effect of great dust storms at the 22°N Lander 1 site. The model was developed by weighted least squares fitting of the Viking Lander pressure measurements to an annual mean, and fundamental and the first four harmonics of the annual cycle. The very close agreement between the two years suggests that an accurate representation of the annual CO2 condensation-sublimation cycle can be established for such years. The two annual mean pressures are identical to 0.006 mbar out of 7.9 mbar, and the differences in amplitudes for the first five periodic components between the two years range from 0.017 to 0.001 mbar. The phase angles, primarily dependent on solar insolation determined orbital dynamics, differ by −3.0° Ls for the second harmonic (year 1 minus year 2), and drop to ≤ 0.7° for the fundamental and fourth harmonic. Although the slight year-to-year differences appear to be real, this model is proposed as a “nominal” Martian annual pressure cycle and applications are suggested. By analogy, the corresponding first years representation at Lander 2 is also proposed as the “nominal” cycle, although it has not been verified by data from a subsequent year. These models provide a method of removing low frequencies from the annual pressure cycle for spectral analyses of baroclinic, tidal, and normal mode oscillations, and for comparisons of the interannual variability.

Seasonal and regional variation of pan-arctic surface air temperature over the instrumental record

Abstract Instrumental surface air temperature (SAT) records beginning in the late 1800s from 59 Arctic stations north of 64°N show monthly mean anomalies of several degrees and large spatial teleconnectivity, yet there are systematic seasonal and regional differences. Analyses are based on time–longitude plots of SAT anomalies and principal component analysis (PCA). Using monthly station data rather than gridded fields for this analysis highlights the importance of considering record length in calculating reliable Arctic change estimates; for example, the contrast of PCA performed on 11 stations beginning in 1886, 20 stations beginning in 1912, and 45 stations beginning in 1936 is illustrated. While often there is a well-known interdecadal negative covariability in winter between northern Europe and Baffin Bay, long-term changes in the remainder of the Arctic are most evident in spring, with cool temperature anomalies before 1920 and Arctic-wide warm temperatures in the 1990s. Summer anomalies are general...

Long-Memory Processes, the Allan Variance and Wavelets

Abstract Long term memory has frequently been observed in physical time series. Statistical theory for long-memory stochastic processes is radically different from the standard time series analysis, which assumes short term memory. The Allan variance is a particular measure of variability developed for long-memory processes. This variance can be interpreted as a Haar wavelet coefficient variance, suggesting an approach towards assessing the variability of general wavelet classes. The theory is applied to a ‘time’ series of vertical ocean shear measurements for which some drawbacks with the Haar wavelet are observed.