# Presentation: 208096

## Atmospheric Correction of Satellite Ocean-Color Imagery

- Robert Frouin
*Scripps Institution of Oceanography*
*La Jolla, California*

- OCRT Meeting, Newport, RI, 11 April 2006

## Collaborators

- Pierre-Yves Deschamps, University of Lille
- Lydwine Gross-Colzy, Capgemini, Toulouse
- Bruno Pelletier, University of Montpellier

## Approaches to Atmospheric Correction

-     1. Linear Combination of Observations (Frouin et al., JO, in press)
-     2. Decomposition in Principal Components (Gross and Frouin, SPIE, 2004)
-     3. Fields of Nonlinear Regression Models (Frouin and Pelletier, RSE, in revision)

## 1. Linear Combination of Observations

- -Perturbing signal expressed as a polynomial or a linear combination of orthogonal components
- -TOA Reflectance in selected spectral bands linearly combined to eliminate perturbing signal
- -“Progressive” atmospheric correction from near-infrared to visible

## 2. Decomposition in Principal Components

- -TOA reflectance decomposed in principal components
- -Components sensitive to the ocean signal combined to retrieve the principal components of marine reflectance, allowing reconstruction of the marine reflectance.

## 3. Fields of Non-Linear Regression Models

- To estimate marine reflectance *******w** *from top-of-atmosphere reflectance *******TOA* and angular variables ***t*** without knowing the other variables ***x*** that influence the radiative transfer in the ocean-atmosphere system

## Methodology

- Explanatory variables (*******TOA*) are considered separately from the conditioning variables (***t***).
- An inverse model is attached to each ***t***, and the attachment is continuous, i.e., the solution is represented by a continuum of parameterized statistical models (a field of non-linear regression models) indexed by ***t***:
*******w** = ******t****(********TOA**) + *******
- where******is the residual of the modeling.

## Methodology (cont.)

- Ridge functions, selected for their approximation properties, especially density, are used to define the statistical models explaining *******w* from* ********TOA* and ***t***:
*****t****j**(********TOA**) = ****i = 1, ..., n** c**ij**h(****a****i**.********TOA** + b**i**)*
***wj** = ******t****j**(********TOA**) + ****j*
- where ***a****i**(****t****),* *b**i**(****t****),* and *c**ij**(****t****)* are the model parameters.

## Simulated Data Sets

- 62,000 joint samples of *******TOA* and *******w* split in two data sets, *D**0**e* and *D**0**v*, for construction and validation. Noisy versions *D**1**e*, D1v, D2e, and *D**2**v* generated, by adding 1 and 2% of noise to *******TOA*. The noise is defined by:
***TOAj**’ = ****TOAj** + ****c****TOAj** + ****uc**j****TOAj*
- where ***c* and ***uc**j* are random variables uniformly distributed on the interval [*-****/200, ****/200*], where ** is the total amount of noise in percent.

## Simulated data Sets (cont.)

*******TOA* simulated in SeaWiFS spectral bands using radiative transfer code of Vermote (1997).
- Marine reflectance assumed to be isotropic and to depend only on chlorophyll-a concentration (Case 1 waters).
- Wide range of aerosol optical thickness and models, including absorbing aerosols, wind speed, chlorophyll-a concentration and sun and viewing angles considered.

## Function Field Construction

- The free parameters of the field, i.e., the maps ***a****i**(****t****),* *b**i**(****t****)*, and *c**ij**(****t****)*, are estimated by multi-linear interpolation on a regular grid covering the range of ***t***.
- The adjustment is considered in the least-square sense, and minimization of the mean squared error is carried out using a stochastic gradient descent algorithm.

## Function Field Construction (cont.)

- A sufficient number of *n = 15* basis functions was selected via simulations, and three fields of this kind, ***0*, ***1*, and ***2* were constructed for 0, 1, and 2% of noise.
- Since the components *****t****j* take their values in the same vector space (the vector space spanned by the linear combinations of ridge functions), the approach is not equivalent to separate retrievals on a component-by-component basis.

## Table 1. Root Mean Squared error (RMS) and Root Mean Squared Relative error (RMSR) for the models 0, 1, and 2 evaluated on the construction and validation data sets (D0e and D0v) and on 1% and 2% noisy versions of them (D1e, D1v, D2e, and D2v).

- Table 1. Root Mean Squared error (RMS) and Root Mean Squared Relative error (RMSR) for the models ***0*, ***1*, and ***2* evaluated on the construction and validation data sets (*D**0**e* and *D**0**v*) and on 1% and 2% noisy versions of them (*D**1**e*, *D**1**v*, *D**2**e*, and *D**2**v*).

## Figure 1. Estimated versus expected marine reflectance for model 0 adjusted on non-noisy data.

- Figure 1. Estimated versus expected marine reflectance for model ***0* adjusted on non-noisy data.

## Figure 2. Estimated versus expected marine reflectance for model 1 adjusted on 1% noisy data.

- Figure 2. Estimated versus expected marine reflectance for model ***1* adjusted on 1% noisy data.

## Figure 3. Conditional quantiles (of order 0.1, 0.25, 0.5, 0.75, and 0.9) of the residual w error distributions as a function of aerosol optical thickness at 550nm for model 1 applied to 1% noisy data. 

- Figure 3. Conditional quantiles (of order 0.1, 0.25, 0.5, 0.75, and 0.9) of the residual ***w* error distributions as a function of aerosol optical thickness at 550nm for model ***1* applied to 1% noisy data.

## Figure 4. Conditional quantiles (of order 0.1, 0.25, 0.5, 0.75, and 0.9) of the residual w error distributions at 412 and 555 nm as a function of the fraction of one aerosol model in a mixture of two for model 1 applied to 1% noisy data. 

- Figure 4. Conditional quantiles (of order 0.1, 0.25, 0.5, 0.75, and 0.9) of the residual ***w* error distributions at 412 and 555 nm as a function of the fraction of one aerosol model in a mixture of two for model* ****1* applied to 1% noisy data.

## Figure 5. w(443)/w(555) and w(490)/w(555) as a function of [Chl-a] for theoretical w, for w estimated by 0 from non-noisy data, and for w estimated by 1 from 1% noisy data.

- Figure 5. ***w**(443)/****w**(555)* and ***w**(490)/****w**(555)* as a function of *[Chl-a]* for theoretical ***w*, for ***w* estimated by ***0** *from non-noisy data, and for ***w* estimated by ***1* from 1% noisy data.

## Figure 6. Estimated [Chl-a] using w(443)/w(555) and w(490)/w(555) obtained by 0 on D0 and 1 on D1 versus expected [Chl-a].

## Application to SeaWiFS Imagery

- Function field methodology tested on SeaWiFS imagery acquired on day 323 of year 2002 over Southern California.
*****t****2* gives large differences in ***w* compared with SeaDAS values, resulting in 78% difference in chlorophyll-a concentration on average.

## Application to SeaWiFS Imagery (cont.)

- Differences may be explained by large noise level on *******TOA** *(e.g., 14% at 412 nm), due to RT modeling uncertainties.
- Noise distribution estimated on 2,000 randomly selected pixels of the imagery, and introduced during the execution of the stochastic fitting algorithm, yielding function field *****t******.

- Figure 7. Marine reflectance ***w* estimated by ***** for SeaWiFS imagery acquired on day 323 of year 2002 over Southern California.

- Figure 8. Marine reflectance ***w* estimated by SeaDAS for SeaWiFS imagery acquired on day 323 of year 2002 over Southern California.

## Figure 9. Histograms of marine reflectance w retrieved by SeaDAS and *. 

- Figure 9. Histograms of marine reflectance ***w* retrieved by SeaDAS and ****.*

## Figure 10. Marine reflectance spectra retrieved by SeaDAS and *.

## Figure 11. [Chl-a] retrieved by SeaDAS and *, fractional difference, and histograms for SeaWiFS imagery acquired on day 323 of year 2002 over Southern California. Average difference is 19.6%.

- Figure 11. *[Chl-a]* retrieved by SeaDAS and *****, fractional difference, and histograms for SeaWiFS imagery acquired on day 323 of year 2002 over Southern California. Average difference is 19.6%.

## Conclusions

- Fields of non-linear regression models emerge as solutions to a continuum of similar statistical inverse problems. They match well the characteristics of the remote sensing problem, allowing separation of the explanatory variables (*******TOA*) from the conditioning variables (***t***).
- The inversion is robust, with good generalization, and computationally efficient. The retrievals of *******w* are accurate, with an error uniform over the entire range of *******w* values. Situations of absorbing aerosols are handled well.

## Conclusions (cont.)

- For noise levels up to a few percent, a general noise scheme may be appropriate, but for large noise levels, the noise distribution needs to be estimated. A plug-in approach may be reasonable.
- Extension of the methodology to atmospheric correction over optically complex waters is possible.