Aquarius Level 3 Processing

National Aeronautics and Space Administration

Aquarius Level 3 Processing

J. M. Lilly and G. S. E. Lagerloef

Earth and Space Research

March 18—20, 2008 GSFC

2 of

March, 18—20, 2008AquariusLilly & Lagerloef

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 of


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 of


Aquarius sampling patterns 1/2

5 of


Aquarius sampling patterns 2/2

Sampling is dense but inhomogeneous

6 of


First try --- Smooth with 75 km Gaussian

0.02 psu global RMS

7 of


Higher Errors in Curved Regions

Simple smoothing performs less well in high curvature regions

8 of


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 of


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 of


Temperature Decorrelation Scale

Gyre-scale decorrelation conflicts with 150 km mission requirement

11 of


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 of


Example of smoothing splines

From Kim and Gu, 2004

13 of



Splines automatically vary effective smoothing radius

[From Silverman (1984)].

Probably not what we want.

14 of



Shape of asympototic effective smoothing function

[From Silverman (1984)]

15 of


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 of


Constant vs. linear fit, noisy flat surface

17 of


Constant vs. linear fit, curving surface

18 of


Constant vs. linear fit, curving surface

19 of


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 of


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 of


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 of


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 of


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 of


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 of


Mapping possibilities

• Smoothing variants (simple, inhomogeneous)

• Exact interpolation (bilnear, bicubic)

• Penalized least squares / smoothing spline

• Optimal interpolation

• Spatio-spectral localization

26 of


Method comparison

Method name Pros Cons

Explicit filter Control of filter width

Known statistics

Too easy; cant be right

Exact interpolation Easy, smooth, fast Imprint of grid

Uniform data weighting

Smoothing splines Highly flexible

Statistical framework

Expensive (global)

Need to specify

“smooth in what sense?”

Optimal interpolation Given statistics, equals best answer

Expensive (global)

Need prior information (!)

27 of


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 of


Extra equation

29 of


Extra equation

30 of


Extra equation

31 of


A Very Simple Interpolation

0.01 psu global RMS (~50% less if ocean is smoothed)

Aquarius Level 3 Processing

Documents

Transcript of Aquarius Level 3 Processing