Geophysical Data Assimilation

Homework 1

Find formulas for the coefficients of a line and a quadratic equation for the best fit of a line to data. You should derive equations for the best fit of a and b for the equation y = a + bx, and a, b, and c for the equation y = a + bx + cx²
Calculate the best fit for each of these two models, linear and quadratic lines for this data. The data are in two columns, the first is the average Mississippi River streamflow in May, the second is measured hypoxic area (see this page for further information).
Plot the data and your fit to the line. Please be sure not to use canned routines, but use your own derived equations to calculate the coefficients for both lines. Here is an example in python that shows how to do this using polyfit, by way of example.

Further reading: Least squares page on Wikipedia

Homework 2

Find the best estimate, s_a, of a value given a series of measurements s_n

point_estimate.py

Create a plot of the distribution of measurement (using, e.g., the 'hist' plotting command) along with the analysis value and error.
Create a new data set of measurements and errors that explicitly violates our statistical assumptions. Make a similar plot of the point distribution and analysis. Write a brief paragraph explaining how the violation of your statistical assumption corrupts your analysis.

Sample figure:

Homework 3

Using the one dimensional optimal interpolation program, linked below.

Create a plot similar to the one shown below.
Create three other plots that modify the one shown below. In particular, you should experiment with non-uniformly spaced data and violations in statistical assumptions. Write a brief paragraph explaining why you get the analysis field you do for each of these cases.

Example optimal interpolation code.

Homework 4

Modify the 1D optimal interpolation for two dimensional interpolation. Use your code to interpolate the data linkedhere. Turn in both your code, and the analyzed data interpolated to a uniform grid. Find some other interpolation method (can be from MATLAB, python, or whatever). Interpolate the same data set to the same grid. Discuss the similarities and differences between your interpolated field using OI, and your new interpolated field.

Homework 5

Derive an inverse model for the equation

u_t + c u_x - k u_xx = F

subject to the appropriate boundary conditions. You should have a set of equations similar to 1.3.1 to 1.3.6 in Bennet's "Inverse Modeling of the Ocean and Atmosphere."

Second 2D optimal interpolation test data

data