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6311 results in Earth and Environmental Science: general interest

Page 36 of 316

  • 24 - Some Non-Rotating Turbulent Flows

  • from Part V - Non-Rotating Flows
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 242-250

  • Summary

    In this chapter we consider some non-rotating turbulent flows that are a bit more complicated than the flow of water down a slope considered in § 23.6. These include turbulent katabatic winds driven by thermal buoyancy (§ 24.1), avalanches driven by snow suspended in air (§ 24.2) and cumulonimbus clouds driven by the release of latent heat as water vapor condenses (§ 24.3).

    Turbulent Katabatic Winds

    In § 22.2.5, we investigate the katabatic wind down a slope in the case that the flow is laminar and, using typical parameter values, found that the flow is very likely turbulent rather than laminar. When flow is turbulent, the diffusivity coefficients for momentum and heat are not constant, but instead vary linearly with elevation. In this section, we will revisit the problem formulated in § 22.2.5, but with variable diffusivities, using Reynolds analogy to set the turbulent thermal diffusivity equal to the turbulent diffusivity of momentum.1

    The governing equations now are

    where z is elevation above the ground, u is the downslope speed, is the dimensionless perturbation temperature, is the reduced gravity, s the down-slope thermal gradient, is the temperature contrast, is the turbulent diffusivity, is a small dimensionless parameter is the velocity scale and is the roughness scale. As before, these equations are to be solved on the domain subject to the conditions u And as before, we can combine the two equations into a single complex equation, although the scalings are somewhat different. In the present case

    while the complex equation for W= T ∗ −iu/U is

    where is the dimensionless vertical distance and is the scaled boundary roughness, subject to conditions W(0) = 1 and W(∞) = 0.

    The problem for the turbulent katabatic winds is a bit more challenging than for the laminar winds because our complex ordinary differential equation now has a variable coefficient. We can get this equation in “standard” form by introducing a new independent variable; let

  • 35 - Lava Flows

  • from Part VII - Silicate Flows
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 367-378

  • Summary

    Commonly andesitic and rhyolitic magmas rise up discrete conduits, whereas basaltic magma often – at least during the early stages of an eruption phase – rises to the surface through a magmatic dike (also called a rift or fissure): a linear crack extending from a magma chamber at depth to the surface. Typically as magma erupts from a dike, outgassed volatiles (mainly water) produce a linear fire fountain composed of incandescent blobs of lava.

    After magma reaches the surface, its subsequent behavior depends largely whether the eruption is effusive or explosive. A discussion of the possible styles of lava flow is found in Griffiths (2000). Effusive eruptions lead to the production of lava which flows relatively gently downslope, initially in the form of a planar sheet. We will consider a simple model of lava sheet flow in the following section.

    On the other hand, explosive eruptions display a variety of behaviors depending on the speed, mass flow and composition of the erupting magma. Some of these behaviors are explored in § 35.2.

    Lava Sheet Flow

    In this section we will develop a simple model of subaerial (that is, on land, as opposed to under the sea) lava sheet flow: a sheet of hot basaltic lava flowing steadily down a slope (in the x direction) from a magmatic dike.

    Our simple model will ignore

  • • variations in topography;

  • • horizontal variations in the lava surface;

  • • fluid inertia;

  • • loss of heat to the ground;

  • • radiative cooling;

  • • variations in the ambient temperature;

  • • effects of rain;

  • • the dynamic effect of crystals suspended in the liquid lava;

  • • buoyancy effects; and

  • • variations of thermal conductivity with temperature.

    • In order to simplify the analysis further, we will assume that the cool upper layer of lava is rafted along with the flow, rather than being anchored to the ground at the sides. The weight of this rafted layer is borne by the liquid beneath, so that there is a vertical hydrostatic balance. This rafted layer may deform as needed in order to adjust to the flow of the liquid beneath.

  • Part III - Waves in Non-Rotating Fluids

  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 101-102

  • Summary

    Our investigation of waves is divided into two parts depending whether rotation of the Earth has a part to play. In the present part we limit our attention to those waves unaffected by rotation of Earth, with those waves that are influenced by this rotation investigated in Part IV. We begin our investigation in the following chapter with an introduction to – and brief survey of – the types of waves that occur on and within Earth. The simplest type of wave, the sound wave, is quantified and discussed in § 9.3. The most common type of elastic wave is the seismic wave; these waves occur continuously as the brittle portions of Earth's crust rub against one another. Seismic waves that are unaffected by Earth's surface, called body waves, are considered in § 10.1 and § 10.2, while those that affected the surface are considered in § 10.3 and § 10.4. While body waves in fluids (we are thinking of air and water) are essentially sound waves, edge waves in fluids (particularly water) are distinctly different from elastic edge waves. We investigate deep-water waves from a mathematical perspective in Chapter 11 and in § 11.5 discuss how these waves occur and interact on the oceans. The rich variety of waves occurring in shallow water is investigated in Chapters 12 and 13. Our survey of non-rotating waves concludes in Chapter 14 with an investigation of capillary, interfacial and internal waves.

  • 10 - Multivariate Statistics

  • Alan D. Chave, Woods Hole Oceanographic Institution, Massachusetts
  • Book:
    Computational Statistics in the Earth Sciences
    Published online:
    06 October 2017
    Print publication:
    19 October 2017, pp 281-343

  • Summary

    This chapter provides a survey of some of the major techniques provided for multivariate random variables. The multivariate Gaussian distribution is characterized, and its key properties are elucidated. The sample mean vector and sample covariance matrix are specified, and shown to have analogous properties to their univariate counterparts. The treatment is extended to the complex multivariate Gaussian distribution. The multivariate counterpart to the Student's t test, Hotelling's T squared test is described and used to obtain simultaneous confidence intervals on multiple parameters. Multivariate analysis of variance is outlined, leading to the multivariate generalization of the F distribution, Wilks' lambda distribution. Three types of hypothesis tests on the covariance matrix are defined. Multivariate linear regression is specified, along with tools for its assessment. Canonical correlation for multiple response and predictor variables is outlined. Empirical orthogonal function (also called principal component) analsysis is defined and illustrated using global temperature data.

  • 8 - Resampling Methods

  • Alan D. Chave, Woods Hole Oceanographic Institution, Massachusetts
  • Book:
    Computational Statistics in the Earth Sciences
    Published online:
    06 October 2017
    Print publication:
    19 October 2017, pp 214-246

  • Summary

    In the real world, departures from parametric statistical models frequently occur, including mixtures of distributions, outliers and noncentrality. This has led to the development of estimators that are based on resampling of the available data. The most widely used of these is the bootstrap that is based on resampling with replacement from a data set, and may be used to characterize parameter and their bias, along with confidence intervals on them, and testing hypotheses about them. A Monte Carlo approach to remove biass on the Kolmogorov-Smirnov statistic when the distribution parameters are estimated from the data is elucidated. Exact permutation tests based on sampling without replacement from a given data set are defined, and their high power is established. Implementations for one and two sample tests for a location parameter, a two sample test for paired data and a two sample test for dispersion are given. Finally, the jackknife that is a linear approximation to the bootstrap is covered, and is useful for computing estimator variance for very large data sets.

  • 4 - Characterization of Data

  • Alan D. Chave, Woods Hole Oceanographic Institution, Massachusetts
  • Book:
    Computational Statistics in the Earth Sciences
    Published online:
    06 October 2017
    Print publication:
    19 October 2017, pp 48-85

  • Summary

    This chapter covers myriad ways to statistically characterize data as a first step in their analysis. The concept of an estimator is introduced, and the common estimators of location (e.g, the sample mean and median), dispersion (e.g., the sample variance), shape (e.g., skewness), direction and association (e.g., correlation coefficient) are described, and related to their implementation in Matlab. The important limit theorems of statistics, the Laws of Large Numbers and the central limit theorems, are elucidated. Exploration data analysis tools are defined. These include the histogram, empirical cumulative distribution function, kernel density estimator, percent-percent plot and quantile-quantile plot, all of which are illustrated using data. Sampling distributions are defined, and the important central and non-central chi square, Student's t, F and correlation coefficient cases are elaborated. The distribution of order statistics is treated, leading to the distribution of the sample median and interquartile range. The joint distribution of the sample mean and sample variance is derived using Cochran's Therore, and their independence for Gaussian data is established.

  • 5 - Point, Interval, and Ratio Estimators

  • Alan D. Chave, Woods Hole Oceanographic Institution, Massachusetts
  • Book:
    Computational Statistics in the Earth Sciences
    Published online:
    06 October 2017
    Print publication:
    19 October 2017, pp 86-141

  • Summary

    The choice of an estimator depends on criteria for "best estimator". The key estimator properties of consistency, unbiasedness, and robustness are defined. The mean squared error is introduced and used to characterize different unbiased estimators. The Cramér-Rao lower bound on the variance of an estimator as the inverse sample Fisher information is derived. The concept of a sufficient statistic is introduced, and its properties are defined, leading to the minimal and complete forms. Statistical decision theory is outlined, leading to risk functions and admissible estimators.The method of moments that has wide applicability but limted optimality is introduced. The maximum likelihood estimator is derived, and is shown to be asymptotically consitent, asymptotically efficient and asymptotically Gaussian, so its covariance is given by the inverse Fisher information. The confidence interval is defined for discrete and continuous random variables, and the Bonferroni method to control the tail probability when multiple parameters are estimated is elucidated. Estimators for the ratio of two random variables are examined.

  • 11 - Compositional Data

  • Alan D. Chave, Woods Hole Oceanographic Institution, Massachusetts
  • Book:
    Computational Statistics in the Earth Sciences
    Published online:
    06 October 2017
    Print publication:
    19 October 2017, pp 344-390

  • Summary

    Compositional data are multivariate data whose components sum to a constant. They are very common in the earth sciences, and are typified by the weight percent oxides in major rock analyses. Statistical tools for their analysis have only become available over the past thirty years. Compositional data are closed, and subcompositions may be obtained by applying the closure operation to a subset of a larger data set. Their sample space is the simplex, of which the simplest example is ternary. A vector space for compositional data is obtained by defining the perturbation and powering operators that are analogous to addition and multiplication in a real vector space, along with an inner product and distance measure. Transformation between the simplex and a real vector space is facilitated by centered and isometric log ratio transformations. Transformed data may be analyzed using standard tools in a real space and then transformed back to the simplex. All of these concepts are illustrated using rock composition data.

  • 9 - Linear Regression

  • Alan D. Chave, Woods Hole Oceanographic Institution, Massachusetts
  • Book:
    Computational Statistics in the Earth Sciences
    Published online:
    06 October 2017
    Print publication:
    19 October 2017, pp 247-280

  • Summary

    The linear regression or least squares methodl is introduced first by example and then theoretically based on four simple and widely applicable assumptions. The theory explicitly pertains to predictor variables that are random variables, as is usually the case with actual data. The hat matrix is defined and its use to detect leverage is specified. Statistical inference in linear regression is outlined, incluing least squares analysis of variance, assessing the regression parameters and assessing the randomness and correlation of the residuals. Use of the bootstrap and jackknife to obtain confidence intervals is specified. The importance of assessing the least squares model is emphasized and illustrated using several data sets. Robust and bounded influence estimators that remove bias caused by outliers and leverage points are derived and implemented with Matlab. Methods that reduce the bias caused by noise in the predictor variables are described. Shrinkage estimators for variable selection are elucidated. Finally, logistic regression that pertains when the response variable is categorical is described.

  • 3 - Statistical Distributions

  • Alan D. Chave, Woods Hole Oceanographic Institution, Massachusetts
  • Book:
    Computational Statistics in the Earth Sciences
    Published online:
    06 October 2017
    Print publication:
    19 October 2017, pp 18-47

  • Summary

    This chapter describes the characteristics of the major probability distributiions used in the sciences, and serves as a compendium of information about them for reference purposes.It also introduces their implementation in Matlab. The treatment is separated into discrete and continuous cases. The binomial distribution that characterizes the number of successes in a fixed number of trials is defined. The negative binomila, multinomial and hypergeometric distributions are reviewed. The Poisson distribution is derived from the binomal, and characterizes rare discrete events. The important Gaussian distribution is treated thorougly, and its genearalization through the stable distribution familiy is defined. The Rayleigh, lognormal, Weibull, beta and extreme value distributions are described, and the gamma distribution family and its important exponential member are elucidated.The bivariate Gaussian distribution is derived. Finally, the von Mises and Fisher distributions of direction on the unit circle and sphere are obtained from first principles.

  • 1 - Probability Concepts

  • Alan D. Chave, Woods Hole Oceanographic Institution, Massachusetts
  • Book:
    Computational Statistics in the Earth Sciences
    Published online:
    06 October 2017
    Print publication:
    19 October 2017, pp xi-xiv

  • Summary

    This chapter introduces the probability concepts of experiments, events, distributions, random variables and expectation through Matlab examples. It then outlines set theory and the three actions complement, union and intersection using Venn diagrams. The Kolmogorov axioms that constitute the basis for probability theory are described, and used to derive the General Law of Addition that is applied throughout statistics. Permutation and combination are introduced, leading to the concepts of sampling with and without replacement. The probability of a union of events is derived, yielding marginal and joint probabilities. Conditional probability links the joint and marginal probabiilities. Independent events yield a joint probability that is the product of the marginal probabilities. Finally, Bayes’ Theorem is derived, leading to the concepts of prior and posterior distributions in Bayesian statistics.

  • 6 - Hypothesis Testing

  • Alan D. Chave, Woods Hole Oceanographic Institution, Massachusetts
  • Book:
    Computational Statistics in the Earth Sciences
    Published online:
    06 October 2017
    Print publication:
    19 October 2017, pp 142-168

  • Summary

    This chapter discusses the theory of hypothesis testing under the Neyman-Pearson Lemma. The concepts of null and alternate hypottheses and distributions, type 1 error, power and p-values are introduced, including several cautionary notes about the latter. The commonly used parametric hypothesis tests (z, t, chi square, F and Bartlett's M) are defined and illustrated with data, including power calculation. The distribution of the correlation coefficient is derived and implemented with Matlab. One-way analysis of variance as a method to compare the means of two or more groups is elucidated. The relationship between hypothesis tests and confidence intervals is established, Likelihood ratio tests for simple and composiite hypotheses are defined, and shown to be widely optimal. The Wald and score tests that are simple, but asymptotically equivalent to a likelihood ratio test, are introduced. The simultaneous testing of multiple hypotheses is then described, including the seminal Benjamini-Hochberg approach.

  • 2 - Statistical Concepts

  • Alan D. Chave, Woods Hole Oceanographic Institution, Massachusetts
  • Book:
    Computational Statistics in the Earth Sciences
    Published online:
    06 October 2017
    Print publication:
    19 October 2017, pp 1-17

  • Summary

    The probability density function for discrete and continuous random variances is defined, and related to the probability concepts from Chapter 1. The cumulative distribution function is introduced and its properties are outlined. The inverse cumulative distribution function is explained, leading to the quantiles of a distribution. The characteristic function and conditional probability distribution are defined, and their applications are explained. The important concepts of independent and exchangeable random variables are specified; the latter underlies the nonparametric and resampling estimators described in later chapters. Transformation of a distribution between functions of a random variable is illustrated . Key population measures of location, dispersion, shape, direction and association are described. The conditional expected value and variance are defined, leading to the Laws of Total Expectation, Variance and Covariance. The Markov, Chebyhev and Hoeffding's probability inequalities used throughout statistics are derived, and the concept of convergence of random variables is elucidated.

Page 36 of 316