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MIT 2 693 - INTRODUCTION TO SAMPLING THEORY AND DATA ANALYSIS

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Principles of Oceanographic Instrument Systems: Sensors and Measurements Spring 2004 Jim Irish INTRODUCTION TO SAMPLING THEORY AND DATA ANALYSIS These notes are meant to introduce the ocean scientist and engineer to the concepts associated with the sampling and analysis of oceanographic time series data, and the effects that the sensor, recorder, sampling plan and analysis can have on the results. In order to plan the optimum sampling and analysis plan, one needs to understand what information and analysis are required, and how all these factors will affect the final result. To get the most from these lecture notes, the student should do supplemental readings from the references listed below. Exercises utilizing the MATLAB software package will be assigned at the appropriate place in the lectures. An outline of this section is given below, and covered in handouts. 1. Time Series and Analysis: •Properties of a random, stochastic processes •Statistical description: mean, variance, correlation/covariance, spectra •Fourier transforms, frequency domain/time domain description of a process •Digital filtering and filters: Convolution product, filters, and filter response 2. Sampling Theory: •Sampling process, sampling theorem, and sampling effects on statistics •Aliasing and the Nyquist frequency •Power density spectra, coherence, degrees of freedom, confidence limits 3. Environmental Sampling in the real world: •Calibrations: static, dynamic •Digitizing effects, prefiltering •Sensor frequency response effects •Sensor noise limitations Suggested references and readings: Jenkins, G.M. and D.G. Watts, Spectral Analysis and its Applications, Holden-Day, San Francisco, 1968. Koopmans, L.H., The Spectral Analysis of Time Series, Academic Press, New York, 1974. Bendat, J.S. and A.G. Piersol, Random Data: Analysis and Measurement Procedures, Wiley-interscience, New York, Second Edition, 1986. Daley, R., Atmospheric Data Analysis, Cambridge University Press, New York, 1991. Cochran, W.T., et al, “What is the Fast Fourier Transform,” IEEE Trans. Audio and Electroacoustics, AU-15(2), 45-55, 1967. 1Glossary of Terms, from Blackman, R.B. and J.W. Tukey, The Measurement of Power Spectra, Dover, 1958. Bingham, C., M.D. Godfrey and J.W. Tukey, “Modern Techniques of Power Spectrum Estimation,” IEEE Trans. Audio and Electroacoustics, AU-15(2), 56-66, 1967. Welch, P.D., “The use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging of Short, Modified Periodograms,” IEEE Trans. Audio and Electroacoustics, AU-15(2), 70-73, 1967. Carter G.C., C.H. Knapp and A.H. Nutall, “Estimation of the Magnitude-Squared Coherence Function Via Overlapped Fast Fourier Transform Processing,” IEEE Trans. Audio and Electroacoustics, AU-21(4), 337-344, 1967. Background Everyone has some idea vague of what is involved in making measurements of the environment. However, few people have the background to really know how to do it properly. This is an introduction to how to measure the environment and analyze the results to obtain information for scientific studies and management decisions. As an ocean scientist or engineer you desire to make and analyze observations that will give you certain statistics describing the environment. To get these statistics, you need to design an experiment, place sensors in the field, digitize and record the results, analyze them on a computer, and finally present them in a meaningful manner. In reality, all these processes that you must go through can be thought of as a filter, or that you are looking at the ocean through “colored” glasses. In order to know what your glasses are doing to your view of the ocean, you need to know how to design an experiment to get the data that you want, select the sensors which will properly measure the environment, use recorders that will satisfactorily record the data, and utilize analysis techniques which will give the desired results. What follows is a simple introduction to the background that you will need to know in order to sample the environment properly. To simplify the discussions, much of the statistical complexity has been removed, so in order to become really professionally involved in data analysis, further course work is required to fill in this statistical information. Properties of Random variables We make the assumption that the environmental data of interest is a stationary, random, stochastic process. If this is so, then the environmental process that we wish to study can be fully described by its statistics. Random Variables - A deterministic variable is one whose value may be determined or estimated exactly. An example of a variable which can be predicted is the result from an explicit mathematical relationship, e.g. y(x) = a + bx or y(x,t) = cos(kx - ωt + θ). A random variable is one in which perfect prediction of succeeding values is impossible. Examples of random variables are the time until the next alpha particles is emitted from a radioactive source, the next direction taken by a particle in Brownian motion, or the elevation of the sea surface at a specific latitude, longitude and time. A set of observations of a random variable represents only one of many possible realizations. 2Stochastic Process - A stochastic process is a collection of random variables. One observes a stochastic process when he examines a process developing in time in a manner controlled by probabilistic laws. A single set of observations is called a "sample function" or “sample record.” A random stochastic process is described by all its possible sample functions. Repeated observations will result in sample functions that are different, or are not the same function of time, but have the same statistics. Some examples of stochastic processes are the number of particles emitted from a radioactive source, the path of a particle in Brownian motion, or the sea surface elevation variations due to surface wind waves. One can not predict exactly any succeeding values, but one can describe succeeding values statistically. Stationary processes - The assumption that a random process is stationary is the most important assumption made in time series analysis. Perhaps this assumption is bad, at best it is only approximately true. A process is stationary when its "statistics" remain constant with time.


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