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  • Synoptic lab
    • Syllabus
    • Take-home exams
    • Homework
    • Tropical Cyclones
      • Tropical cyclone climatology and overview
      • Tropical cyclone life cycle and motion
      • Tropical cyclone structure
      • Hurricane forecasting learning material
      • Hurricane forecasting tools
      • Storm surge
      • TUTTs and LA/MS flood event of 2016
    • Ocean applications
      • Waves
      • Tides
      • Miscellaneous ocean products
    • Streamlines
    • Regression, MOS, and NBM
    • Forecasting baroclinic systems and other products
      • Analysis
      • Model guidance
      • Model Output Statistics and National Blend of Models
      • Useful forecast products
  • Synoptic class
    • Syllabus
    • Exam Information
    • Homework
    • Stability
      • The Basics
      • The SkewT and related diagnostic tools
      • POP, air mass thunderstorms, sea breeze thunderstorms
      • Severe Weather
    • Map analysis
      • Upper-level synoptic charts
      • Contouring
      • Surface analysis and fronts
      • Vertical structure
      • Jet Streaks
    • Dynamics
      • Review of dynamics
      • Dynamics applications
      • QG Theory and the Omega equation
      • Cyclogenesis and baroclinic instability
    • Modeling
    • Fog
    • Winter Weather
  • Intro Dynamics
    • Syllabus
    • Homework
    • Sample exam questions
    • Introduction (Chap. 1, HH)
    • Basic equations of meteorology (Chap. 2, HH)
    • Imbedded processes in equations of meteorology (Chap. 3, HH). Exam 2 material
    • Imbedded processes in equations of meteorology (Chap. 3, HH). Exam 3 material
    • Planetary boundary layer
    • Vorticity (Chap.4, HH)
    • Pertubation method and atmospheric waves (Chap 5, HH)
  • Numerical methods
    • Syllabus
    • Homework
    • Number Series
    • Interpolation
    • Basic matrix math
    • Filters And Fourier Analysis
    • Numerical derivatives
    • Numerical integration, random numbers, and Monte Carlo
    • Numerical solutions of differential equations and atmospheric modeling
    • Parameterization, data assimilation, and overview on WRF model
    • Final computer exercises
  • Dashboard
Trigonometric nomenclature for periodic waves
  • The basics. But note that in a discretized domain, L is the horizontal spatial scale of the domain, and x=n delta x.
  • Spreadsheet example of discretized waves in periodic domain. 
Digital filters
  • Recursive and nonrecursive filters
  • An example of a nonrecursive low-pass filter used by meteorologists - the Shapiro filter
  • Spreadsheet example of 1-2-1 Shapiro filter
  • Spreadsheet example of 1-8-28-56-186 Shapiro filter. Why is the result almost the same as the original? What has been removed?
  • The classic Shapiro papers. Paper one. Paper two.
  • Other nonrecursive common filters. Written by Ken Turkowki of Apple Computer. Notice that in electronic nomenclature, "decimation" represents the weight values,  the frequency response is written such that no damping effect has a gain of 0 dB, and damping is represented by negative gain.
  • Gibbs phenomenon. 
  • The classic Lanczos paper. Written by Claude Duchon. The Lanczos filter can perform lowpass or highpass fitlering, and is designed to reduce Gibbs phenomenon.
  • Optional material - More on recursive exponential smoothing
  • Optional material - Univariate time series methods
Running mean (moving average)
  • Nonrecursive moving average, also known as a running mean. Smooths data, but does not remove any frequencies since all weights are equal. See Kalnay lecture for more details.
  • Example application - the Oceanic Nino Index (ONI)
  • Example application - Running mean of global temperatures since 1880
Comparison of moving average, the one-two-one filter, and the Lanczos filter
  • Lecture by Eugenia Kalnay
Fourier (Harmonic) analysis
  • The basics (from "ask a mathematician")
  • The mathematics (from Lamar University)
  • The fundamental equations - a Fourier series; computing harmonic coefficient; computing signal power; computing explained variance
  • Fourier series with a nonzero background average
  • Trig identities used in Fourier series derivation. Youtube example for integrating sin(3x)sin(6x)dx 
  • Orthogonal relationships in Fourier series
  • Example: decomposing two superimposed wave solutions, analytical solution
  • Spreadsheet example: the same two superimposed waves decomposed with a background average. Power and explained variance also computed.
  • Example application - ocean tide prediction
          1) Tidal analysis and prediction
          2) Example - Bridgeport, CT
          3) Example - San Diego, CA
          4) More realistic example - Seattle, WA
  • Supplemental material - Signal power from Fourier analysis
  • Supplemental material - Explained variance by Fourier analysis
  • Supplemental material - another good summary from thefouriertransform.com
  • Optional material - 1) The classic paper on spherical harmonics by Baer. 2) Advanced graduate school material from CSU professor David Randall.
  • Optional material - more on time series analysis and some issues with Fourier analysis
  • Optional material - creating mp3 files with assistance from Fourier analysis
Computer language examples
  • ncl filters
  • Matlab Lanczos filter 
  • Matlab digital filter
  • Signal smoothing with Matlab
  • Designing a digital filter with Matlab
  • Shapiro filter for ROMS model output in FORTRAN
  • Online calculator, running mean
  • Matlab moving average
  • Matlab Fourier analysis in one and two dimensions
  • Fourier analysis with ncl
  • Fourier analysis with IDL
  • Software for harmonic analysis sea level data
Recorded lectures
  • Filters and running means. Audio only.
  • More on filters. Audio only.
  • Fourier series. Audio only.
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  • Home
  • Contact information
  • Synoptic lab
    • Syllabus
    • Take-home exams
    • Homework
    • Tropical Cyclones
      • Tropical cyclone climatology and overview
      • Tropical cyclone life cycle and motion
      • Tropical cyclone structure
      • Hurricane forecasting learning material
      • Hurricane forecasting tools
      • Storm surge
      • TUTTs and LA/MS flood event of 2016
    • Ocean applications
      • Waves
      • Tides
      • Miscellaneous ocean products
    • Streamlines
    • Regression, MOS, and NBM
    • Forecasting baroclinic systems and other products
      • Analysis
      • Model guidance
      • Model Output Statistics and National Blend of Models
      • Useful forecast products
  • Synoptic class
    • Syllabus
    • Exam Information
    • Homework
    • Stability
      • The Basics
      • The SkewT and related diagnostic tools
      • POP, air mass thunderstorms, sea breeze thunderstorms
      • Severe Weather
    • Map analysis
      • Upper-level synoptic charts
      • Contouring
      • Surface analysis and fronts
      • Vertical structure
      • Jet Streaks
    • Dynamics
      • Review of dynamics
      • Dynamics applications
      • QG Theory and the Omega equation
      • Cyclogenesis and baroclinic instability
    • Modeling
    • Fog
    • Winter Weather
  • Intro Dynamics
    • Syllabus
    • Homework
    • Sample exam questions
    • Introduction (Chap. 1, HH)
    • Basic equations of meteorology (Chap. 2, HH)
    • Imbedded processes in equations of meteorology (Chap. 3, HH). Exam 2 material
    • Imbedded processes in equations of meteorology (Chap. 3, HH). Exam 3 material
    • Planetary boundary layer
    • Vorticity (Chap.4, HH)
    • Pertubation method and atmospheric waves (Chap 5, HH)
  • Numerical methods
    • Syllabus
    • Homework
    • Number Series
    • Interpolation
    • Basic matrix math
    • Filters And Fourier Analysis
    • Numerical derivatives
    • Numerical integration, random numbers, and Monte Carlo
    • Numerical solutions of differential equations and atmospheric modeling
    • Parameterization, data assimilation, and overview on WRF model
    • Final computer exercises
  • Dashboard