Note: Front Cover -- Mathematical and Physical Fundamentals of Climate Change -- Copyright -- Contents -- Preface: Interdisciplinary Approaches to Climate Change Research -- Chapter 1: Fourier Analysis -- 1.1 Fourier Series and Fourier Transform -- 1.2 Bessel's Inequality and Parseval's Identity -- 1.3 Gibbs Phenomenon -- 1.4 Poisson Summation Formulas and Shannon Sampling Theorem -- 1.5 Discrete Fourier Transform -- 1.6 Fast Fourier Transform -- 1.7 Heisenberg Uncertainty Principle -- 1.8 Case Study: Arctic Oscillation Indices -- Problems -- Bibliography -- Chapter 2: Time-Frequency Analysis -- 2.1 Windowed Fourier Transform -- 2.2 Wavelet Transform -- 2.3 Multiresolution Analyses and Wavelet Bases -- 2.3.1 Multiresolution Analyses -- 2.3.2 Discrete Wavelet Transform -- 2.3.3 Biorthogonal Wavelets, Bivariate Wavelets,and Wavelet Packet -- 2.4 Hilbert Transform, Analytical Signal, and Instantaneous Frequency -- 2.5 Wigner-Ville Distribution and Cohen's Class -- 2.6 Empirical Mode Decompositions -- Problems -- Bibliography -- Chapter 3: Filter Design -- 3.1 Continuous Linear Time-Invariant Systems -- 3.2 Analog Filters -- 3.3 Discrete Linear Time-Invariant Systems -- 3.3.1 Discrete Signals -- 3.3.2 Discrete Convolution -- 3.3.3 Discrete System -- 3.3.4 Ideal Digital Filters -- 3.3.5 Z-Transforms -- 3.3.6 Linear Difference Equations -- 3.4 Linear-Phase Filters -- 3.4.1 Four Types of Linear-Phase Filters -- 3.4.2 Structure of Linear-Phase Filters -- 3.5 Designs of FIR Filters -- 3.5.1 Fourier Expansions -- 3.5.2 Window Design Method -- 3.5.3 Sampling in the Frequency Domain -- 3.6 IIR Filters -- 3.6.1 Impulse Invariance Method -- 3.6.2 Matched Z-Transform Method -- 3.6.3 Bilinear Transform Method -- 3.7 Conjugate Mirror Filters -- Problems -- Bibliography -- Chapter 4: Remote Sensing -- 4.1 Solar and Thermal Radiation. , 4.2 Spectral Regions and Optical Sensors -- 4.3 Spatial Filtering -- 4.4 Spatial Blurring -- 4.5 Distortion Correction -- 4.6 Image Fusion -- 4.7 Supervised and Unsupervised Classification -- 4.8 Remote Sensing of Atmospheric Carbon Dioxide -- 4.9 Moderate Resolution Imaging Spectroradiometer Data Products and Climate Change -- Problems -- Bibliography -- Chapter 5: Basic Probability and Statistics -- 5.1 Probability Space, Random Variables, and Their Distributions -- 5.1.1 Discrete Random Variables -- 5.1.2 Continuous Random Variables -- 5.1.3 Properties of Expectations and Variances -- 5.1.4 Distributions of Functions of Random Variables -- 5.1.5 Characteristic Functions -- 5.2 Jointly Distributed Random Variables -- 5.3 Central Limit Theorem and Law of Large Numbers -- 5.4 Minimum Mean Square Error -- 5.5 2-Distribution, t-Distribution, and F-Distribution -- 5.6 Parameter Estimation -- 5.7 Confidence Interval -- 5.8 Tests of Statistical Hypotheses -- 5.9 Analysis of Variance -- 5.10 Linear Regression -- 5.11 Mann-Kendall Trend Test -- Problems -- Bibliography -- Chapter 6: Empirical Orthogonal Functions -- 6.1 Random Vector Fields -- 6.2 Classical EOFs -- 6.3 Estimation of EOFs -- 6.4 Rotation of EOFs -- 6.5 Complex EOFs and Hilbert EOFs -- 6.6 Singular Value Decomposition -- 6.7 Canonical Correlation Analysis -- 6.8 Singular Spectrum Analysis -- 6.9 Principal Oscillation Patterns -- 6.9.1 Normal Modes -- 6.9.2 Estimates of Principal Oscillation Patterns -- Problems -- Bibliography -- Chapter 7: Random Processes and Power Spectra -- 7.1 Stationary and Non-stationary Random Processes -- 7.2 Markov Process and Brownian Motion -- 7.3 Calculus of Random Processes -- 7.4 Spectral Analysis -- 7.4.1 Linear Time-Invariant System for WSS Processes -- 7.4.2 Power Spectral Density -- 7.4.3 Shannon Sampling Theorem for Random Processes -- 7.5 Wiener Filtering. , 7.6 Spectrum Estimation -- 7.7 Significance Tests of Climatic Time Series -- 7.7.1 Fourier Power Spectra -- 7.7.2 Wavelet Power Spectra -- Problems -- Bibliography -- Chapter 8: Autoregressive Moving Average Models -- 8.1 ARMA Processes -- 8.1.1 AR(p) Processes -- 8.1.2 MA(q) Processes -- 8.1.3 Shift Operator -- 8.1.4 ARMA(p,q) Processes -- 8.2 Yule-Walker Equation andSpectral Density -- 8.3 Prediction Algorithms -- 8.3.1 Innovation Algorithm -- 8.3.2 Durbin-Lovinson Algorithm -- 8.3.3 Kolmogorov's Formula -- 8.4 Asymptotic Theory -- 8.4.1 Gramer-Wold Device -- 8.4.2 Asymptotic Normality -- 8.5 Estimates of Means and CovarianceFunctions -- 8.6 Estimation for ARMA Models -- 8.6.1 General Linear Model -- 8.6.2 Estimation for AR(p) Processes -- 8.6.3 Estimation for ARMA(p,q) Processes -- 8.7 ARIMA Models -- 8.8 Multivariate ARMA Processes -- 8.9 Application in Climatic and Hydrological Research -- Problems -- Bibliography -- Chapter 9: Data Assimilation -- 9.1 Concept of Data Assimilation -- 9.2 Cressman Method -- 9.3 Optimal Interpolation Analysis -- 9.4 Cost Function and Three-Dimensional Variational Analysis -- 9.5 Dual of the Optimal Interpolation -- 9.6 Four-Dimensional Variational Analysis -- 9.7 Kalman Filter -- Problems -- Bibliography -- Chapter 10: Fluid Dynamics -- 10.1 Gradient, Divergence, and Curl -- 10.2 Circulation and Flux -- 10.3 Green's Theorem, Divergence Theorem, and Stokes's Theorem -- 10.4 Equations of Motion -- 10.4.1 Continuity Equation -- 10.4.2 Euler's Equation -- 10.4.3 Bernoulli's Equation -- 10.5 Energy Flux and Momentum Flux -- 10.6 Kelvin Law -- 10.7 Potential Function and Potential Flow -- 10.8 Incompressible Fluids -- Problems -- Bibliography -- Chapter 11: Atmospheric Dynamics -- 11.1 Two Simple Atmospheric Models -- 11.1.1 The Single-Layer Model -- 11.1.2 The Two-Layer Model -- 11.2 Atmospheric Composition. , 11.3 Hydrostatic Balance Equation -- 11.4 Potential Temperature -- 11.5 Lapse Rate -- 11.5.1 Adiabatic Lapse Rate -- 11.5.2 Buoyancy Frequency -- 11.6 Clausius-Clapeyron Equation -- 11.6.1 Saturation Mass Mixing Radio -- 11.6.2 Saturation Adiabatic Lapse Rate -- 11.6.3 Equivalent Potential Temperature -- 11.7 Material Derivatives -- 11.8 Vorticity and Potential Vorticity -- 11.9 Navier-Stokes Equation -- 11.9.1 Navier-Stokes Equation in an Inertial Frame -- 11.9.2 Navier-Stokes Equation in a Rotating Frame -- 11.9.3 Component Form of the Navier-Stokes Equation -- 11.10 Geostrophic Balance Equations -- 11.11 Boussinesq Approximation and Energy Equation -- 11.12 Quasi-Geostrophic Potential Vorticity -- 11.13 Gravity Waves -- 11.13.1 Internal Gravity Waves -- 11.13.2 Inertia Gravity Waves -- 11.14 Rossby Waves -- 11.15 Atmospheric Boundary Layer -- Problems -- Bibliography -- Chapter 12: Oceanic Dynamics -- 12.1 Salinity and Mass -- 12.2 Inertial Motion -- 12.3 Oceanic Ekman Layer -- 12.3.1 Ekman Currents -- 12.3.2 Ekman Mass Transport -- 12.3.3 Ekman Pumping -- 12.4 Geostrophic Currents -- 12.4.1 Surface Geostrophic Currents -- 12.4.2 Geostrophic Currents from Hydrography -- 12.5 Sverdrup's Theorem -- 12.6 Munk's Theorem -- 12.7 Taylor-Proudman Theorem -- 12.8 Ocean-Wave Spectrum -- 12.8.1 Spectrum -- 12.8.2 Digital Spectrum -- 12.8.3 Pierson-Moskowitz Spectrum -- 12.9 Oceanic Tidal Forces -- Problems -- Bibliography -- Chapter 13: Glaciers and Sea Level Rise -- 13.1 Stress and Strain -- 13.2 Glen's Law and Generalized Glen's Law -- 13.3 Density of Glacier Ice -- 13.4 Glacier Mass Balance -- 13.5 Glacier Momentum Balance -- 13.6 Glacier Energy Balance -- 13.7 Shallow-Ice and Shallow-Shelf Approximations -- 13.8 Dynamic Ice Sheet Models -- 13.9 Sea Level Rise -- 13.10 Semiempirical Sea Level Models -- Problems -- Bibliography. , Chapter 14: Climate and Earth System Models -- 14.1 Energy Balance Models -- 14.1.1 Zero-Dimensional EBM -- 14.1.2 One-Dimensional EBM -- 14.2 Radiative Convective Models -- 14.3 Statistical Dynamical Models -- 14.4 Earth System Models -- 14.4.1 Atmospheric Models -- 14.4.2 Oceanic Models -- 14.4.3 Land Surface Models -- 14.4.4 Sea Ice Models -- 14.5 Coupled Model Intercomparison Project -- 14.6 Geoengineering Model Intercomparison Project -- Problems -- Bibliography -- Index.