SIO 221B:  Analysis of Physical Oceanographic Data


 
Course Outline
I.
Principles of ocean instruments (2)
A.
How are sea water properties, velocity, air-sea fluxes, and surface waves measured?
1.
How do instruments work?
2.
How are observations made?
3.
What does data look like?
B.
Some simple observational problems
1.
Drake Passage transport
2.
Observing the Ekman spiral
II.
Randomness and statistics (6)
A.
The origin of ``randomness'' in dynamical systems.
1.
The concepts of dynamical degrees of freedom and unpredictability based on a simple chaos model.
2.
Relevance to scale ranges in the ocean/atmosphere system.
B.
Basic probability.
1.
Probability density functions (PDFs) and joint probability density functions.
2.
Averages and moments.
3.
Averages from PDFs.
4.
Scatter plots, covariance and correlation.
5.
Conditional probability and the approach to determinism.
6.
Correlation of independent events.
7.
PDFs of functions.
C.
Discrete random walks.
1.
Central limit theorem.
2.
Serially correlated discrete random walks.
3.
Continuous random walks (Taylor diffusion).
4.
The diffusion equation from random walk and central limit theorem.
III.
Decomposition of signals (1)
A.
The philosophy of signal vs. noise decompositions.
1.
The algebraic problem: Inverse theory.
2.
The statistical problem: Statistical Estimation.
B.
Some examples.
1.
Function fitting.
2.
Fourier analysis of time series.
IV.
Inverse problems (9)
A.
Examples of oceanographic inverse problems.
1.
Beta spiral.
2.
Control volumes.
B.
Least-squares problems.
1.
``Over-'' and ``under-'' determined problems.
2.
Constraints.
3.
Simultaneous minimization of misfit to data and solution size.
C.
A practical review of linear algebra.
D.
Singular value decomposition.
1.
Relationship to the simultaneous minimization problem.
E.
Resolution and error as measures of goodness.
V.
Applying probability concepts to data (3)
A.
Construction ``ensembles'' for statistical treatment of observations.
1.
What stationarity really means.
2.
Ergodicity.
B.
Sampling errors of mean and variance.
1.
$1/\sqrt{N}$ convergence.
2.
Bias, mean-square error and probable error of sample estimates.
3.
Estimating variance: an introduction to statistical ``beauty'' principles.
4.
Effect of serial correlation on sampling errors.
VI.
Statistical estimation (6)
A.
Regression models.
1.
Joint-normal distributions.
2.
Statistical forecasting.
3.
Improving persistence forecasts.
B.
Objective mapping as multivariate regression.
1.
Unbiased estimates and the mean.
2.
Mixing observations of different types.
3.
Imposing constraints.
4.
Model testing from mapped fields vs. statistical tests.
VII.
Efficiency of representations (3)
A.
Principal axes.
B.
Review of Fourier spectra.
C.
Empirical Orthogonal Functions (EOFs).
1.
Relation of EOFs to Fourier analysis.