Advances in Geophysics, Vol. 40 systematically compares many of the currently used statistical approaches to time series analysis and modeling to evaluate each method's robustness and application to geophysical datasets. This volume tackles the age-old problem of how to evaluate the relative roles of deterministic versus stochastic processes (signal vs noise) in their observations. The book introduces the fundamentals in sections titled "1.2 What is a Time Series? " and "1.3 How is a Time Series Quantified?", before diving into Spectral Analysis, Semivariograms, Rescaled-Range Analysis and Wavelet Analysis. The second half of the book applies their self-affine analysis to a number of geophysical time series (historical temperature records, drought hazard assessment, sedimentation in the context of hydrocarbon bearing strata, variability of the Earth's magnetic field). This volume explores in detail one of the main components of noise, that of long-range persistence or memory. The first chapter is a broad summary of theory and techniques of long-range persistence in time series; the second chapter is the application of long-range persistence to a variety of geophysical time series.