Zhihua Zhang

HAAR WAVELET ANALYSIS OF CLIMATIC TIME SERIES

In order to extract the intrinsic information of climatic time series from background red noise, in this paper, we will first give an analytic formula on the distribution of Haar wavelet power spectra of red noise in a rigorous statistical framework. After that, by comparing the difference of wavelet power spectra of real climatic time series and red noise, we can extract intrinsic features of climatic time series. Finally, we use our method to analyze Arctic Oscillation (AO) which is a key aspect of climate variability in the Northern Hemisphere, and discover a great change in fundamental properties of the AO, commonly called a regime shift or tripping point.

Climate and Earth System Models

Various models are used to study the climate system and its natural variability, and to simulate the interaction between the physical climate and the biosphere, and the chemical constituents of the land, atmosphere, and ocean. Models are the best tools available to test hypotheses about the factors causing climate change and to assess future Earth system developments. In this chapter, we introduce basic physical principles in energy balance models, radiative convective models, statistical dynamical models, and Earth system models consisting of atmospheric models, ocean models, land surface models, and sea ice models. Then we introduce the Coupled Model Intercomparison Project and the Geoengineering Model Intercomparison Project.

Basic Probability and Statistics

Basic probability theory and statistics have a wide application in climate change research, ranging from the mean climate state and uncertainty of climatic parameters to the dynamics of the climate system. They provide powerful tools for climatologists to explain and analyze climatic data as well as to model and predict climate change. In this chapter, we introduce the basic theory and methods in probability and statistics and their applications.

FOURIER SUPPORTS OF SCALING FUNCTIONS DETERMINE CARDINALITIES OF WAVELETS

It is well-known that the different kinds of multiresolution analysis (MRA) structures generate different wavelets. In this paper, we give two uniform formulas on the number of mother functions for various wavelets associated with MRA structures. These formulas show that the number of mother functions of wavelets is determined by the support of the Fourier transform of the scaling function in MRA structure.

Intrinsic feature extraction in the COI of wavelet power spectra of climatic signals

Since the wavelet power spectra are distorted at data boundaries (the cone of influence, COI), using traditional methods, one cannot judge whether there is a significant region in COI or not. In this paper, with the help of a first-order autoregressive (AR1) extension and using our simple and rigorous method, we can obtain realistic significant regions and intrinsic feature in the COI of wavelet power spectra. We verify our method using the 300 year record of ice extent in the Baltic Sea.

Time-Frequency Analysis

The Fourier transform of a signal can provide only global frequency information. While a time-frequency distribution of a signal can provide information about how the frequency content of the signal evolves with time. This is performed by mapping a one-dimensional time domain signal into a two-dimensional time-frequency representation of the signal. A lot of techniques have been developed to extract local time-frequency information. In this chapter, we introduce basic concepts and theory in time-frequency analysis, including windowed Fourier transform, wavelet transform, multiresolution analysis, wavelet basis, Hilbert transform, instantaneous frequency, Wigner-Ville distribution, and empirical mode decomposition.

Glaciers and Sea Level Rise

Glaciers and ice sheets cover about 10% of Earth’s land surface. Most mountain glaciers have been retreating since the end of the “Little Ice Age.” The present volume of Earth’s glacier ice, if totally melted, represents about 80 m in potential sea level rise. Sea level changes, especially in densely populated, low-lying coastal areas and on islands, have significant effects on human activities and facilities. Recent research on the current sea level rise budget indicates that the contribution from land ice has increased by 60% over the last decade. In this chapter, we introduce glacier modeling with various degrees of complexity and estimate glacial contributions to sea level rise.

Empirical Orthogonal Functions

Empirical orthogonal function (EOF) analyses are often used to study possible spatial patterns of climate variability and how they change with time. One of the important results from EOF analysis is the discovery of several oscillations in the climate system, including the Pacific Decadal Oscillation and the Arctic Oscillation. Similarly to Fourier analysis and wavelet analysis, in EOF analysis, one also projects the original climate data on an orthogonal basis. However, this orthogonal basis is derived by computing the eigenvectors of a spatially weighted anomaly covariance matrix, and the corresponding eigenvalues provide a measure of the percent variance explained by each pattern. Therefore, EOFs of a space-time physical process can represent mutually orthogonal space patterns where the data variance is concentrated, with the first pattern being responsible for the largest part of the variance, the second for the largest part of the remaining variance, and so on. Later, in order to overcome some limitations of classical EOF analysis and make the resulting patterns more physically interpretable, rotated EOFs, Hilbert EOFs, and complex EOFs are developed. In this chapter, we focus on the theory and algorithms of EOF analyses and their generalization. At the same time, we introduce singular spectrum analysis, canonical correlation analysis, and principal oscillation pattern analysis.

Chapter 4 – Remote Sensing

Compared with in situ observation, remote sensing gathers information concerning Earth’s surface by using data acquired from aircraft and satellites. The traditional and physics-based concepts are now complemented with signal and image processing concepts, so remote sensing is capable of managing the interface between the signal acquired and the physics of the surface of Earth. Remove-sensing technologies have been applied widely in meteorology, climate change detection, environmental monitoring, flood prediction, agriculture, resource explorations, mapping, and so on. In this chapter introduce solar and thermal radiation, spatial filtering, blurring, distortion correction, image fusion, supervised and unsupervised classification, and the applications in climate change.

Chapter 7 – Random Processes and Power Spectra

Climatic data and theoretical considerations suggest that a large part of climatic variability has a random nature and can be analyzed using the theory of random processes. In this chapter, we describe the random approach to the study of changes in the climate system, including stationary random processes, calculus of random processes, power spectra, spectrum estimation, and Wiener filter. Moreover, in order to extract intrinsic frequency features from climatic time series, we show how random processes do statistical significance tests by Fourier analysis and wavelet analysis.