Aquarius Level 3 Processing

1. J. M. Lilly and G. S. E. Lagerloef Earth and Space Research March 18—20, 2008 GSFC Aquarius Level 3 Processing

2. Overview Level 2  Level 3  Gridded Products  Objective Maps Completed • Review of known mapping methods • Choice of algorithm • Implementation of prototype code (with Gene and Joel) Next • Experiments with simulator data • Implementation of (optional) improvements • Contingency planning

3. Level 3 Requirements Level 3 requirements • 0.2 psu global RMS error for monthly product • 150 km decorrelation scale distance • 1° by 1° gridded product

4. Aquarius sampling patterns 1/2

5. Aquarius sampling patterns 2/2  Sampling is dense but inhomogeneous

6. First try --- Smooth with 75 km Gaussian  0.02 psu global RMS

7. Higher Errors in Curved Regions Simple smoothing performs less well in high curvature regions

8. Various mapping methods Gauss-Markov (aka optimal interpolation) Bretherton et al. (1976); Reynolds & Smith (1994) Smoothing splines Wahba and Wedelberger (1980); Gu (2002) Local polynomial regression (e.g. LOESS) Fan and Gijbels (1997); Cleveland and Devlin (1988) Other: spherical wavelets [Holschneider et al. (2003)] spatio-spectral localization [Simons et al. (2006)] radial basis functions [Nuss and Titley (1994)]

9. Comparision of mapping methods Mapping scattered data is about the bias / variance tradeoff More smoothing = more bias but less variance Methods differ in how this tradeoff is controlled: • OI --- Smoothing controlled by covariance functions Makes sense when you think you know these • Splines --- Control measure of smoothness (norm) and smoothing parameter (controls tradeoff) Makes sense when certain measure of smoothness is defensible (e.g. mapping the streamfunction) • Local polynomial fit --- Control order of fit (constant, linear, etc.) and weighting function (what is local?)

10. Temperature Decorrelation Scale Gyre-scale decorrelation conflicts with 150 km mission requirement

11. Smoothing spline methods Penalized least squares (Gu, 2001) Minimize error of fit  Minimizing roughness Many nice properties – highly adjustable based on choice of J and lambda; mathematical and statistical underpinnings; pre-existing code; formally equivalent to optimal interpolation

12. Example of smoothing splines From Kim and Gu, 2004

13. Smoothing spline methods Splines automatically vary effective smoothing radius [From Silverman (1984)]. Probably not what we want.

14. Smoothing spline methods Shape of asympototic effective smoothing function [From Silverman (1984)]

15. Local polynomial regression At each grid point xm, fit an order P polynomial to data points xn. Data is weighted by a decaying function Kh(x)=K(x/h)/h. The radius of the fit is controlled by the bandwidth h. Good choices for K(x) are a parabola or a Gaussian. Fitting to a constant is equivalent to smoothing data with Kh(x).

16. Constant vs. linear fit, noisy flat surface

17. Constant vs. linear fit, curving surface

18. Constant vs. linear fit, curving surface

19. Why I like local polynomial regression Basic features • Explicit control over smoothing radius (aka bandwidth) • Two “knobs” for bias/variance: order and bandwidth • Easy to understand and to quantify errors • Many possibilities for refinements Possible additional products • Estimate of bias • Estimate of variance • Estimate grad S Additional possibilies • Variable (optimal) bandwidth • Variable (optimal) order • Anisotropic smoothing

20. Next to do for Level 3 • Experiments with simulator data Statistitics of “noise” and implications for choice of smoothing • Right choice of order (constant vs. linear vs. quadratic) Expect big improvements for linear fit, quadratic maybe better • Accounting for beam differences (footprint & noise level) Sensible to make effective smoothing radius ~ constant • Include adjustment to fit cal/val data Additional parameters for least-squares fit vs. say latitude

21. Choice of Spatial Averaging • Noise statistics depend upon spatial averaging • Adjacent 150 km x 150 km cells should be mostly independent • Some overlap is desirable for smoothness The Gaussian weight shown below is therefore taken as a representative filter for the purpose of computing statistics. • 75 km standard deviation (88 km half-power point) • ~0.4 correlation coefficient, or ~15% shared variance

22. Example of Simple Smoothing Map based on one week’s sampling, gridded with simple smoothing Aquarius samples mean salinity field from Dan Jacob’s model

23. Laplacian of Mapped Field Salinity curvature shows clear imprint of sampling grid (high variance)  Sub-optimal solution to the of bias / variance tradeoff tradeoff

24. Mapping algorithm considerations • Fast enough for numerous trials • Analytically tractable error analysis • Adjustable for bias / variance tradeoff • Should not have imprint of underlying grid • Should not present features resembling physical phenomena • Should be free from extraneous assumptions

25. Mapping possibilities • Smoothing variants (simple, inhomogeneous) • Exact interpolation (bilnear, bicubic) • Penalized least squares / smoothing spline • Optimal interpolation • Spatio-spectral localization

26. Method comparison

27. Level 3+ Refinement Strategy Basic observations • Mapping data is optimizing “bias / variance” tradeoff • This depends upon noise statistics, which are unknown • Must remain flexible pending reality check Principles for development • Level 3+ processing system with multiple options • Trial simulations with incoming Level 2 • Assess performance of options for different noise scenarios Suggestion: post mapped output using simulated data, on proto-Aquarius website; solicit input from potential users.

28. Extra equation

29. Extra equation

30. Extra equation

31. A Very Simple Interpolation 0.01 psu global RMS (~50% less if ocean is smoothed)