SeaWinds Error Bars Overview

1. SeaWinds Error Bars Overview Jan 17,2006 Bryan Stiles and R. Scott Dunbar

2. Overview • Error Bars previously in product were computed: • using a few objective functions values in the neighborhood of each ambiguity • Assuming the probability distribution was Gaussian both speed and direction • Two major problems with this technique • 1) Probability is often not even close to Gaussian in direction • 2) The immediate neighborhood in the the objective function is insufficient to compute the error bars • Goal: Create a more accurate estimate with low impact on code • 1) Does not need to exhaustively compute entire 2-D probability distribution • 2) Makes use of intermediate quantities computed during wind retrieval

3. Error Bars:Algorithm Design Decisions • Error bars for each ambiguity are conditional on that ambiguity being the nearest to the truth. • No error bars computed for DIRTH vectors. • (Could be computed by transforming selected vectors according to the relative performance of selected and DIRTH with buoys.)

4. Probability Model • Objective Function: • Conditional Probability of 0 given wind • Using Bayes Theorem: • and assuming • constant effective width of the best speed ridge • uniform priors • We get

5. Error Bars • Compute the expected mean square direction error by integrating over all possible true directions like so: • For now skill=1; so that error bars are computed the same for all ambiguities. • Compute the expected mean speed error by: • kskill is a skill coefficient applied to weight directions within the neighborhood of the selected ambiguity higher than those outside. • Expected speed error computation is separated into portions along the best speed ridge and orthogonal to it. (See next slide)

6. Speed Error Bars • To compute speed error bars accurately, the width of the best speed ridge must be taken into account. • Using three points on the objective function we estimate a quadratic in speed. • We compute 2(u|) by assuming a Gaussian probability in speed so that: • Here (u0,f0) is the point at the peak of the objective function and (u1,f1) is some other point on the objective function.

7. Impact on Code • DIRTH produces 3 intermediate values useful for computing error bars • Direction Intervals • Direction Probabilities • Best speed values as a function of direction. • A minor modification of current code yields Probabilities that are: • 0.1 to 0.2 m/s from best speed ridge • On both sides of the ridge • For each 8 degrees directional bin • This is all we need to compute the error bars.

8. Error bars: Status • Tasks completed • Algorithm developed and tested for computing directional and speed error bars. • Algorithm was compared to brute force computation that uses full 2-D probability distribution in speed and direction, to compute expected speed and direction error. (See next two slides) • Black lines are full-up computation • Red solid lines are the approximation we employs • Green solid lines are a more accurate approximation we decided we do not need. • Dotted lines are RMS difference between full-up computation and other techniques. • Code snippet and description given to Scott