__Map projections and relevance in numerical models__

- Basics on map projections
- Example of interpolating global models to nested grid map projections in WRF
- Example of map projection equations used in mesoscale models (MM5)

__Important equations for u and v__

- Converting wind from model coordinates to earth coordinates . Required when comparing model winds to a wind observation. Example FORTRAN subroutine for RUC
- Equations relating speed and direction to u and v . See discussion here about different computer notations of atan2. What is atan2?
- Special note: oceanographers use a direction convention that is 180 deg different from meteorology, also known as the vector azimuth angle. Mathematical details are here and here.

__Interpolating between grid points__

__Interpolating 1D data, Part 1: Linear interpolation, regression, and polynomials__

- Linear interpolation and nearest neighbor
- Basic overview on regression and correlations
- Linear regression overview with a Matlab example
- Polynomial regression overview with Matlab examples
- Polynomial regression in matrix form with spreadsheet examples
- 1) Second degree Lagrange polynomial and a Matlab routine; 2) Third degree Lagrange polynomial interpolation; 3) What is the difference between a sigma summatation and a pi summation?
- Optional material - the Lagrange polynomial technique has two drawbacks - 1) Inconvenient when the appropriate degree of polynomial is not known; 2) If additional data points, everything has to be recomputed. A solution is to rewrite the technique based on Newton Divided-Difference Interpolating Polynomials. Matlab information on this technique.

__Interpolating 1D data, Part 2: Splines__

- Origin of the spline concept goes back to shipbuilding!
- 1) Quadratic and cubic splines; 2) Detailed information on cubic spline; 3) Youtube video on cubic splines; 4) Another youtube video on cubic splines
- 1) Akima spline 2) An example of why meteorologists should never use a cubic spline. In meteorology, usually our time series and curves are smooth without wiggles. In most cases, Akima fits 1D data well, but can occasionally wiggle. Akima is easy to use and only requires two data points.
- Piecewise Cubic Hermite Interpolating Polynomial (PCHIP). Superior to Akima, but harder to implement.
- Steffen. Superior to Akima, easy to implement. Currently my "goto" spline.
- A good summary by Paul Bourke, and some additional techniques

__Online calculators__

- The xuru online regression website
- Lagrange polynomial interpolation. Another Lagrange polynomial interpolation site
- Cubic spline

__Spreadsheet examples__

- Linear interpolation using floating i, index command, and VBA code. 1) What is VBA? 2) Information on Index command
- Linear interpolation using Match command, Index commands, and VBA code. 1) Documentation on technique from book by Joseph Billo; 2) Information on Match command
- Bilinear interpolation using floating i,j
- Linear regression using built-in Excel commands. Note that the x and y have been transformed to a linear equation.
- Generalized regression in Excel using LINEST function: 1) Official Microsoft documentation; 2) Tidbits; 3) More tidbits
- Lagrange fourth order polynomial interpolation. 1) Documentation
- 2D Lagrange fourth order polynomial interpolation. 1) Documentation
- Akima spline using VBA code. 1) Documentation

__Matlab examples__

- Using polyfit for polynomial fit. 1) Documentation
- User-defined function for Lagrange interpolation polynomial. 2) Documentation
- Using interp1 for nearest neighbor, linear interpolation, PCHIP, and cubic spline. 3) Documentation

__FORTRAN examples__

- 2D bilinear interpolation, floating point technique, FORTRAN routine from Dr. Fitz
- 2D bilinear interpolation, search technique, FORTRAN routine from Dr. Fitz
- 2D bilinear interpolation, FORTRAN function from Dr. Fitz
- 1) 2D bilinear interpolation, FORTRAN subroutine from Dr. Fitz. 2) Example program calling the subroutine.
- Polynomial regression for shallow wate wave equation from Dr. Fitz. 1) Based on this figure. 2) WMO documentation
- 1D interpolation using Akima, FORTRAN subroutine from Dr. Fitz
- 1D interpolation using Steffen, FORTRAN subroutine from Dr. Fitz

__Miscellaneous interpolation routines from different computer language codes__

- 1D interpolation routines for Matlab
- 1D interpolation routines for Matlab supplied by users
- 1D interpolation using Akima for Matlab supplied by a user
- 1D interpolation using Steffen for Matlab supplies by a user
- 1D interpolation for Octave (free version that closely matches Matlab)
- 1D interpolation routines from FSU
- 2D interpolation routines for Matlab
- 2D interpolation using bicubic interpolation, FORTRAN subroutine
- 2D interpolation using ncl. All ncl interpolation routines
- 1D and 2D interpolation using maple.

__Recorded lectures__

- Map projections - Video
- Map projections - Audio only
- Equations for u and v - Video
- Equations for u and v - Audio only
- Linear interpolation - Video
- Linear Interpolation - Audio only
- Polynomial regression with Excel's Linest and Matlab's polyfit
- Polynomial regression with Excel's Linest and Matlab's polyfit - Audio only
- Lagrange polynomial - Video
- Lagrange polynomial - Audio only
- Quadratic and cubic splines - Video
- Quadratic and cubic splines - Audio only
- Monotonic splines (Akima, PCHIP, Steffen) - Video
- Monotonic splines (Akima, PCHIP, Steffen) - Audio only