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    • 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
Numerical integration of gridded values
  • The basics
  • Newton-Cotes techniques
  • Rectangular rule (also called the mid-point rule)
  • 1) Trapezoid rule. 2) Summary of Trapezoid rule.
  • 1) Simpson's rule  - note its only valid for even number of intervals (odd number of points)! 2) Summary of 1/3 Simpson's rule. 3) Summary of 3/8 Simpson's rule.
  • How to use Simpson's rule for odd number of intervals - combine 1/3 and 3/8 Simpson's rules!
Numerical integration of analytical functions
  • Introduction
  • 1) Richards extrapolation. 2) Summary of Richard's extrapolation. 3) Definition of Richard's extrapolation.
  • 1) Romberg integration. 2) Summary of Romberg integration. 3) Good explanation of Romberg integration. 4) This is an example of sequence (series) acceleration.
  • Gaussian quadrature
  • Improper integrals
  • Advanced material - adaptive integration. Example with MPI FORTRAN code.
Random numbers 
  • Random numbers
  • Typical pseudo-random number generator equation - Linear congruential generator. What is a modulus operator? How to compute the modulus.
  • A popular pseudo-random number generator -- the Mersenne Twister. The recurrence equation is complicated, but has a VERY LONG period.
  • Random number generation in Excel. Summary of Excel function RAND
  • Tests for true randomness: 1) DIEHARD (like the movie). 2) NIST Test suite.
  • Online random number generator
Monte Carlo simulations
  • Monte Carlo simulations
  • The classic dice example
  • Example: Collision-coalescnce process
  • Example: Light scattering
  • Example: Tornado risk assessment
Spreadsheet example, random number
  • Random walk. Spreadsheet example. 
  • Adding "noise" to a dataset. Spreadsheet example of a line with "noise" added.
  • Create a randomized list. Create a random normal distrbution. Spreadsheet examples.
Spreadsheet examples, integration
  • Area under curve. Spreadsheet examples.
  • Monte Carlo integration. Spreadsheet examples.
Miscellaneous random number code from different computer languages
  • DIEHARDER: contains both DIEHARD and NIST Test Suite
  • R routine version of DIEHARDER
  • Mersenne twister, FORTRAN routine
  • Linear Congruent Method, FORTRAN routine
  • Random number generation, Matlab
  • Random number generator, Mathematica
  • Random walk, FORTRAN routine
Miscellaneous numerical integration routines from different computer languages
  • Matlab trapezoid rule function. Matlab routine.
  • Matlab Simpson rule function. Matlab routine.
  • Matlab Romberg function. Matlab routine.
  • Matlab built-in functions for integration.
  • Mathematica functions.
  • R routine for integration.
  • FORTRAN trapezoid rule subroutine. Driver program.
  • FORTRAN Simpson rule subroutine .
  • FORTRAN adaptive integration. Driver program.
  • FORTRAN Monte Carlo integration. Driver program. 
Recorded lectures
  • Numerical integration one - Audio only
  • Numerical integration two - Audio only
  • Random numbers - 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