aquarius level 3 processing
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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
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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
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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
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Aquarius sampling patterns 1/2
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Aquarius sampling patterns 2/2
Sampling is dense but inhomogeneous
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First try --- Smooth with 75 km Gaussian
0.02 psu global RMS
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Higher Errors in Curved Regions
Simple smoothing performs less well in high curvature regions
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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)]
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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?)
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Temperature Decorrelation Scale
Gyre-scale decorrelation conflicts with 150 km mission requirement
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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
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Example of smoothing splines
From Kim and Gu, 2004
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Smoothing spline methods
Splines automatically vary effective smoothing radius
[From Silverman (1984)].
Probably not what we want.
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Smoothing spline methods
Shape of asympototic effective smoothing function
[From Silverman (1984)]
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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).
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Constant vs. linear fit, noisy flat surface
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Constant vs. linear fit, curving surface
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Constant vs. linear fit, curving surface
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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
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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
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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
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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
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Laplacian of Mapped Field
Salinity curvature shows clear imprint of sampling grid (high variance)
Sub-optimal solution to the of bias / variance tradeoff tradeoff
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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
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Mapping possibilities
• Smoothing variants (simple, inhomogeneous)
• Exact interpolation (bilnear, bicubic)
• Penalized least squares / smoothing spline
• Optimal interpolation
• Spatio-spectral localization
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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 (!)
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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.
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Extra equation
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Extra equation
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Extra equation
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A Very Simple Interpolation
0.01 psu global RMS (~50% less if ocean is smoothed)
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