Bayesian Methodology for Atmospheric Correction of PACE Ocean-Color Imagery

Transcription

1 A Proposal to the National Aeronautics and Space Administration Submitted in Response to NASA Solicitation NNH13ZDA001N - A.25 PACE Science Team Bayesian Methodology for Atmospheric Correction of PACE Ocean-Color Imagery Principal Investigator: Dr. Robert Frouin Climate, Atmospheric Science, and Physical Oceanography Division Scripps Institution of Oceanography, University of California San Diego 9500 Gilman Drive, La Jolla, CA rfrouin@ucsd.edu, (voice), (fax) Co-Investigator: Prof. Bruno Pelletier Department of Mathematics UMR CNRS 6625 Université Rennes II Place du Recteur le Moal, Rennes, France Bruno.Pelletier@univ-rennes2.fr, (voice), (fax) Period: 01 October September 2017

2 TABLE OF CONTENTS SUMMARY PART I: PROJECT DESCRIPTION 1.0 INTRODUCTION OBJECTIVES TECHNICAL PLAN Introduction 3 Formulation of Atmospheric correction 3 Ill-posed nature of atmospheric correction Bayesian methodology 4 General Presentation 4 Inverse model in practice 5 Connection with the classical approach 5 Illustration on SeaWiFS data Improvements with PACE 9 Using spectral information 9 Using polarization and multi-angle information 9 Modeling of the TOA signal 10 Specification of noise and prior distributions 11 Practical implementation EXPECTED RESULTS AND SIGNIFICANCE MANAGEMENT PLAN Personnel and project responsibilities Work schedule REFERENCES 16

3 SUMMARY The PACE mission will carry into space a spectrometer measuring at 5 nm resolution in the UV to NIR and at lower resolution in spectral bands in the NIR and SWIR and, eventually, a multispectral, multi-angle polarimeter measuring in the UV to SWIR. These instruments have great potential for improving estimates of marine reflectance in the post-eos era. In view of this, the proposal objectives are as follows. The first objective is to evaluate, using the Bayesian approach to inverse problems, the gain in marine reflectance accuracy expected by 1) including observations in the UV and SWIR and 2) further including polarimetric and directional observations in selected spectral bands. This for the PACE threshold aggregate bands with respect to the standard MODIS set of bands used to generate ocean color products. The second objective is to assess, also in a Bayesian context, the utility of hyper-spectral information for improving atmospheric correction in the aggregate bands, and to quantify the accuracy of the atmospheric correction at 5 nm resolution for separating ocean constituents and characterizing phytoplankton communities. To achieve these objectives, the TOA signal measured by the PACE spectrometer and the eventual polarimeter will be simulated for a variety of realistic atmospheric and oceanic conditions. Typical prior distributions for the aerosol, water reflectance, and surface parameters, suitable for utilization at a global scale, will be used, as well as noise distributions. The noise will encapsulate all the sources of uncertainties in the radiative transfer (RT) modeling and include sensor noise. The inverse models will be constructed based on several considerations, i.e., computational cost, convenience to approximate the conditional covariance (a second order quantity), and detection of abnormal values (due to limitations of the forward model). Ways to improve performance by specifying prior distributions from independent information about regional and temporal variability (e.g., from output of numerical transport models) will be investigated, and practical implementation of the Bayesian methodology will be outlined for routine application. The investigation will provide a Bayesian methodology for atmospheric correction of the PACE spectrometer data. The methodology makes it possible to incorporate known constraints of the marine reflectance (i.e., correlation between components) and to account for the varied sources of uncertainty (i.e., measurement noise, RT modeling errors). Importantly, it allows the construction of reliable multi-dimensional confidence domains of the retrieved marine reflectance. Specifically, the mean and covariance of the posterior distribution are computed. These quantities provide, for each pixel, an estimate of the marine reflectance and a measure of its uncertainty. Situations for which observation and forward model are incompatible are also identified. Thus the methodology will offer the means to analyze and interpret PACE ocean-color imagery in view of confidence limits and model adequacy, on a pixel-by-pixel basis. By evaluating via theoretical studies the accuracy of the atmospheric correction of PACE oceancolor radiometry and the expected improvements with respect to current ocean-color sensors, by identifying optimum sets of spectral bands, and by providing an inverse methodology adapted to the problem, which can be viewed as a generalization of the standard algorithm, the investigation responds directly to the PACE Science Team Announcement of Opportunity, which seeks methods and approaches that will maximize the new capabilities of the PACE mission for understanding global ocean ecology in a changing climate.

4 PART I: PROJECT DESCRIPTION 1.0 INTRODUCTION Atmospheric correction of satellite ocean-color imagery is the process of extracting water-leaving radiance or, equivalently, marine reflectance from top-of-atmosphere (TOA) measurements of the radiance reflected by the ocean-surface-atmosphere system. This process is inherently difficult to achieve with sufficient accuracy, since only a small fraction (10% or less) of the measured signal in the visible, the spectral range of interest, may originate from the water body (Gordon, 1997; IOCCG, 2010). The inverse problem is ill posed, i.e., many combinations of atmospheric and oceanic parameters (aerosol and hydrosol optical properties) may yield the same TOA signal (Frouin and Pelletier, 2007, 2014). Furthermore, the wavy surface and atmospheric constituents, especially aerosols, exhibit high spatial and temporal variability. Their characteristics, therefore, cannot be specified from climatology. The classic atmospheric correction scheme (e.g., Gordon. 1997), routinely applied to secondgeneration ocean-color sensors such as SeaWiFS, MODIS, and MERIS, consists of (i) estimating the aerosol reflectance in the red and near infrared (NIR) where the ocean can be considered black (i.e., totally absorbing), and (ii) extrapolating the estimated aerosol reflectance to shorter wavelengths. The marine reflectance is then retrieved by subtraction. Variants and improvements to the classic scheme have been made over the years, especially to deal with non-null reflectance in the red and near infrared, a general situation in estuaries and the coastal zone. Depending on the application context, the retrieved marine reflectance may then related to chlorophyll-a concentration using a bio-optical model, semi-analytical or empirical (e.g., O Reilly et al., 1998), or used in inverse schemes of varied complexity to estimate optical properties of suspended particles and dissolved organic matter (IOCCG, 2006). The classic approach to atmospheric correction, however, suffers a number of limitations. Measurements in two spectral bands in the red and NIR are not sufficient to determine the aerosol influence. Sensitivity to aerosol absorption, in particular, is negligible in this spectral range. The aerosol model is selected within a sub-ensemble of possibilities, such as models from AERONET observations at coastal and island sites (Ahmad et al., 2010), but there are no guaranties that the selected model is the actual one. Furthermore, no measure of uncertainty is associated to the retrievals. Uncertainty is only evaluated in comparisons between estimated and measured marine reflectance, and using too few match-ups for quantitative assessment over the expected range of atmospheric and oceanic regimes. A different approach to satellite ocean-color inversion is to determine simultaneously the key properties of aerosols and water constituents by minimizing an error criterion between the measured reflectance and the output of a radiation transfer (RT) model (e.g., Chomko and Gordon, 1988; Stamnes et al., 2007; Kuchinke et al., 2009). This belongs to the family of deterministic solutions to inverse problems. Through systematic variation of candidate aerosol models, aerosol optical thickness, hydrosol backscattering coefficient, yellow substance absorption, and chlorophyll-a concentration, or a subset of those parameters, a best fit to the spectral TOA reflectance (visible and NIR) is obtained in an iterative manner. The advantage of this approach, compared with the twostep approach, resides in its ability to handle both Case 1 and Case 2 waters. It also can handle both 1

5 weakly and strongly absorbing aerosols, even if the vertical distribution of aerosols, an important variable in the presence of absorbing aerosols, is not varied in the optimization procedure. Another route is to cast atmospheric correction as a statistical inverse problem and to define a solution in a Bayesian context (e.g., Frouin and Pelletier, 2010; 2014). This route is natural in view of the ill-posed nature of the problem. The Bayesian approach to inverse problem consists of first specifying a probability distribution, called the prior distribution, on the input parameters (atmospheric and oceanic) of the RT model. As the name implies, the prior distribution reflects prior knowledge that may be available before the measurement of the TOA reflectance. A probabilistic modeling of any perturbation of the TOA radiance is also typically considered, in the form of an additive random noise. The solution to the inverse problem is then expressed as a probability distribution which, in the present context of atmospheric correction, measures the likelihood of encountering values of marine reflectance given the TOA reflectance (i.e., after it has been observed). The posterior distribution is a very rich object, and its complete reconstruction and exploration can rapidly become prohibitive from the computational side. Instead, one may reduce the ambition to extracting useful quantities, like its expectation and covariance. The expectation provides an estimate of the marine reflectance, while the covariance allows a quantification of uncertainty in the marine reflectance estimate. The current global ocean-color sensors, however, provide limited spectral information and do not provide any directional or polarized information. Extending the TOA observations to the ultraviolet (UV), where aerosol absorption is effective, and to the shortwave infrared (SWIR), where even the most turbid waters are black and sensitivity to the aerosol coarse mode is higher, and measuring at hyper-spectral resolution (e.g., in the oxygen A-band to estimate aerosol altitude) would allow, at least in principle, a more accurate atmospheric correction. Multi-angle and polarized measurements, sensitive to aerosol properties (e.g., size distribution, index of refraction), would further help to specify the aerosol model. The PACE mission, with a hyper-spectral sensor measuring at 5 nm resolution in the UV to NIR and with additional, lower resolution spectral bands in the NIR and SWIR, and with eventually a multi-spectral, multi-angle polarimeter measuring in the UV to SWIR, has great potential for improving estimates of marine reflectance in the post-eos era. 2.0 OBJECTIVES In view of the PACE mission capability for improving atmospheric correction and, therefore, yielding more accurate retrievals of marine reflectance, the proposal objectives are as follows. The first objective is to evaluate, using the Bayesian approach to inverse problems, the gain in marine reflectance accuracy expected by 1) including observations in the UV and SWIR and 2) further including polarimetric and directional observations in selected spectral bands. This for the PACE spectrometer threshold aggregate bands, as defined in the PACE SDT report, with respect to the standard MODIS set of bands used to generate ocean color products. The second objective is to assess, also in a Bayesian context, the utility of hyper-spectral information for improving atmospheric correction in the aggregate bands, and to quantify the accuracy of the atmospheric correction at 5 nm resolution for separating ocean constituents and characterizing phytoplankton communities. To achieve these objectives, the TOA signal measured by the PACE spectrometer and the eventual polarimeter will be simulated for a variety of realistic atmospheric and oceanic conditions. 2

6 Typical prior distributions for the aerosol, water reflectance, and surface parameters, suitable for utilization at a global scale, will be used, as well as noise distributions. The noise will encapsulate all the sources of uncertainties in the RT modeling and include sensor noise. The inverse models will be constructed based on several considerations, i.e., computational cost, convenience to approximate the conditional covariance (a second order quantity), and detection of abnormal values (due to limitations of the forward model). Ways to improve performance by specifying prior distributions from independent information about regional and temporal variability (e.g., from output of numerical transport models) will be investigated, and practical implementation of the Bayesian methodology will be outlined for routine application. 3.0 TECHNICAL PLAN 3.1 Introduction Formulation of atmospheric correction We formalize the inverse problem of atmospheric correction as follows. Let ρ be a reflectance derived from a TOA measurement of the outgoing radiation flux (ρ can be the result of one or multiple pre-processing operations, like correction for know gaseous absorption and molecular scattering). Consider a forward modeling of the TOA signal leading ρ being expressed as ρ = ρ a + T aρ w + ε, (1) 1 S a ρ w where ρ a, T a and S a represent the contribution, transmittance, and spherical albedo of the atmosphere, ρ w is the marine reflectance, and ε is an additive noise. All the variables in (1) are functions of the wavelength λ. The observed data is finite-dimensional and is a vector y = (y 1,..., y d ) of measurements of ρ in spectral bands centered at wavelengths λ 1 < < λ d, i.e., y i = ρ(λ i ), for i = 1,..., d. With these notations, atmospheric correction refers to the process of estimating ρ w from y and without knowledge of ρ a, T a and S a in (1). Note that since only y is observed, atmospheric correction is only one part of the complete inverse problem of estimating both the atmospheric and oceanic parameters from ρ, i.e., (ρ a, T a, S a ) and ρ w, even if in atmospheric correction, interest is only in ρ w. Ill-posed nature of atmospheric correction Atmospheric correction is an ill-posed inverse problem: even without noise, different states of the atmosphere and of the ocean may correspond to very close values of the satellite signal. To see this using model (1), denote by X a and X w two sets of values for the triple (ρ a, T a, S a ) and the water reflectance ρ w, respectively. Consider a reflectance ρ, fix a threshold δ > 0, and search for all the possible combinations of (ρ a, T a, S a ) and ρ w in the sets X a and X w such that ρ a +T a /(1 S a ρ w ) is at a distance no more than δ from ρ. This will lead to the set of pre-images of a ball of radius δ by the operator mapping ((ρ a, T a, S a ), ρ w ) to ρ that is, to the set of simultaneous conditions of the ocean and the atmospheric body than can correspond to TOA reflectance differing from ρ by no more than δ. One example of pre-images is provided in Figure 1 using a reflectance evaluated in 8 spectral bands from the visible to the near infra-red. It is apparent in this case that quite different marine reflectance spectra are mapped to very close TOA reflectance spectra. As a consequence, in the presence of noise on the measurements, the uncertainty on the retrieved marine reflectance may be large. Naturally, the size of the set of pre-images depends on the choice of the threshold δ and on the search spaces X a and X w. 3

7 Figure 1: Example of pre-images. Actual values of ρ w, ρ, T a and ρ a are displayed in red, and the pre-images at a distance no more than δ = (see text) are displayed in black. The search spaces for the pre-images include NOMAD and AERONET-OC data sets and maritime, continental, and urban aerosols in various proportions and amount. 3.2 Bayesian methodology General presentation Let x a,i = (ρ a (λ i ), T a (λ i ), S a (λ i )) and x w,i = ρ w (λ i ) for all i = 1,..., d, and let x a = (x a,1,..., x a,d ) X a R 3d and x w = (x w,1,..., x w,d ) X w R d. The subsets X a and X w in the above equations are constraint sets for the atmospheric parameters and the marine reflectance components respectively. The forward model is written as y = Φ(x a, x w ) + ε, (2) where ε is a random vector in R d, and where Φ : X a X w R d is the map with components (Φ i ) 1 i d defined by Φ i (x a, x w ) = ρ a (λ i ) + T a (λ i )ρ w (λ i )/[1 S a (λ i )ρ w (λ i )]. In the Bayesian approach to inverse problem, x a, x w, and y in (2) are treated as random variables. This defines a probabilistic model, where any vector of measurements y obs is considered as a realization of the random vector y in (2). The model is specified by (2) together with the distributions of ε and of (x a, x w ). Fix a distribution P ε for the random noise ε, and assume that the noise ε is independent from (x a, x w ). The distribution of (x a, x w ), called the prior distribution, describes in a probabilistic manner the prior knowledge one may have about x a and x w before the acquisition of the data y. Since there is no particular reason to expect that the atmospheric parameters and the marine reflectance should be correlated, the prior distribution can be taken as a product measure P xa P xw, where P xa and P xw are probability measures on R 3d and R d, respectively. The Bayesian solution to the inverse problem of retrieving (x a, x w ) from y is defined as the conditional distribution P (xa,x w) y of (x a, x w ) given y. It is called the posterior distribution. Hence, given the observation y obs, the solution is expressed as the probability measure P (xa,x w) y=y obs. Generally one is interested only in certain relevant characteristics of the posterior distribution: importantly, the mean of the posterior distribution, which gives an estimate of the parameters to 4

8 be retrieved (x a and x w ), and its covariance matrix, which provide an accompanying measure of uncertainty. Note that the posterior distribution varies with the TOA measurements y; its mean and covariance are the conditional expectation E[x w y] and conditional covariance Cov(x w y) of x w given y. Inverse modeling in practice To process a satellite image for instance, E[x w y] and Cov(x w y) have to be evaluated for each data y in the image. One option would be to sample from the posterior distribution using a Monte Carlo procedure. However, to keep the computational cost of the procedure low, it is more efficient to define a model of the function y E[x w y] and y Cov(x w y). These maps can be approximated by simulation using any technique from nonparametric regression and with theoretical guarantees, as long as it is universally consistent in L 2 (i.e., converges to the unknown function as the number of simulated data goes to infinity, and this for any function in L 2.) The choice is large; see Gyorfi et al. (2002). The method selected in Frouin and Pelletier (2014) is based on models constructed on a partition of the space of TOA reflectance. These models allow one to keep the execution time low. In addition, it is possible to define an additional quantity, called a p-value, which gives the probability that y takes a value at least as extreme as the one which has been observed. Formally, if y obs denotes the reflectance to be inverted, then the p-value at y obs is defined by P(f y (y) f y (y obs )), where f y is the probability density of y in the probabilistic model (2). Since the whole procedure consists of inverting a forward model (a component of which is a RT model), the p-value allows one to detect situations for which the forward model is unlikely to explain the observed data. Connection with the classical approach The general Bayesian scheme can be related to the classical approach as follows. Consider the conditional expectation E[x w y]. Since E[x w y] = E[E[x w y, x a ] y], we see that E[x w y, x a ] can be modeled first, and then averaged conditionally on y in a second time. This correspond to inverting y assuming that the atmosphere is in the state x a, and then averaging the results according to the distribution of x a given y. So, compared with the classical approach, instead of picking an aerosol model from the TOA reflectance (roughly, inferring one value of x a from y), and then inverting y assuming the atmosphere is in the state x a, the Bayesian methodology amounts to placing a probability distribution on x a, depending on y, inverting y for each x a, and then averaging the results accordingly. Illustration on SeaWiFS data The Bayesian inverse methodology has been applied to actual SeaWiFS imagery (Frouin and Pelletier, 2014). Figure 2 displays, for imagery acquired on 14 February 1999 around South Africa, in the region of the Agulhas and Benguela Currents, Agulhas retro-reflection, and South Atlantic Current, the marine reflectance retrieved by the Bayesian inversion scheme in spectral bands centered on 412, 443, 490, 510, 555, and 670 nm. Clouds, masked using SeaDAS flags, are displayed in white. Near the coast, where upwelling brings nutrients to the surface, values are relatively low at 412, 443, and 490 nm and high at 555 and 670 nm near the coast (typically at 443 nm and 555 nm), indicating productive waters. In the turbulent Agulhas retro-reflection zone (center of the image), marine reflectance in the blue is lower than in the surrounding regions of anti-cyclonic circulation associated with the Benguela drift (to the left) and the Agulhas return current (to the right), with typical values of instead of 0.02 to Near clouds, and in regions of broken cloudiness, the spatial features of marine reflectance exhibit 5

9 continuity with respect to adjacent clear sky regions. No significant correlation exists between marine reflectance and wind speed, which exhibits a strong gradient from the center to the bottom of the image (not shown here), and between marine reflectance and TOA reflectance at 865 nm. At the edge of the cloud system on the left of the image, for example, mesoscale eddies of relatively higher marine reflectance are revealed, with spatial features apparently not affected by the large gradient of TOA reflectance (i.e., aerosol optical thickness) across the edge. Figure 2: Estimated ρ w and associated uncertainty ρ w at 412, 555, and 670 nm by the Bayesian methodology applied to SeaWiFS imagery acquired on 14 February 1999 over South Africa. The absolute uncertainty associated with the marine reflectance estimates is provided in Figure 3 for each spectral band on a pixel-by-pixel basis. This uncertainty was calculated as the covariance of the conditional (posterior) distribution of ρ w given ρ. Values remain within ±0.003 in the blue, ±0.002 in the green, and ± in the red for most pixels. Larger uncertainties, i.e., ±0.005 to ±0.01 in the blue, are encountered near clouds (imperfect cloud masking or processes not accounted for in the modeling) and where the TOA reflectance at 865 nm is large. The amplitude of the uncertainty is generally larger when the marine reflectance is higher, and it represents a larger percentage of the marine reflectance when marine reflectance is lower, ranging between 40% (coastal regions) and 10% (outside the retro-reflection zone). The Bayesian technique retrieves not only marine reflectance, but also aerosol optical thickness τ a and the atmospheric term ρ a (see Equation 1) at all wavelengths. Figure 3 displays τ a and ρ a at 865 nm and their associated uncertainties. The τ a values range from 0.02 to 0.2 and the ρ a values from to High values are encountered near the edge to the cloud system off the West coast of South Africa and in the immediate vicinity of clouds, and they correspond to high TOA reflectance values at 865 nm. As expected the fields of τ a and ρ a are well correlated, since ρ a is essentially proportional to τ a. The uncertainty on τ a and ρ a is generally within ±0.01 and ±0.0005, with higher values in regions of relatively high aerosol optical thickness (e.g., upper left part of the 6

10 Figure 3: Estimated τ a and ρ a and associated uncertainties τ a and ρ a at 865 nm, and p-value for SeaWiFS imagery acquired on 14 February 1999 over South Africa. image). The p-value associated to each pixel of (or retrieval in) the image is also displayed in Figure 3. This parameter quantifies how likely is the observation ρ with respect to the model. A low p- value (i.e., < 0.01 or 0.05) indicates that model and observation are incompatible. The p-value in Figure 3 is above 0.05 almost everywhere, except in the vicinity of clouds, where some processes may not be accounted for in the RT modeling (e.g., adjacency effects due to the high reflectance contrast between clouds and the environment, large optical thickness). Low p-values are also encountered where winds are especially strong, i.e., in the region between the Cape of Good Hope and the Southeastern part of the cloud system in the upper left part of the image, where wind speed exceeds 12 m/s. Such conditions are outside the atmospheric parameter space used for approximating the forward operator. The performance of the Bayesian technique has been evaluated experimentally in comparisons with in situ measurements of marine reflectance. The measurements were taken from the NOMAD data set (Werdell and Baily, 2005) and matched with the satellite data, within ±3 hours of overpass. The closest 3 3 pixel box was selected for processing by the inversion scheme, and the marine reflectance retrieved for each of the 9 pixels was interpolated to the geographic location of the in situ measurements. The cases for which some of the pixels in the box did not pass the SeaDAS cloud-screening flags were eliminated. The TOA radiance was not corrected for vicarious calibration adjustment (only temporal calibration changes taken into account). Note that the NO- MAD data can be used in the comparisons because they were only used in the model specification to define the support of the prior distribution on the marine reflectance. (The match-up data set is not used in the construction of the models.) The marine reflectance match-up data sets covered the period from September 1997 to March 7

11 Figure 4: Estimated versus measured ρ w for NOMAD match-up data. Left: Scatter plot including uncertainty on the Bayesian estimates (vertical bars); Right: Comparison statistics (NOMAD) and consisted of 690 pairs of estimated and measured values, respectively. These included 132, 144, 144, 113, 129, and 28 pairs at 412, 443, 490, 510, 555, and 670 nm Sun zenith angle varied from 3 to 58 degrees, view zenith angle from 22 to 58 degrees, and relative azimuth angle from 75 to 180 degrees, i.e., the match-up data encompassed a wide range of geometry conditions. Most of the points are located between 60S and 60N in the Atlantic and Pacific Oceans, and in coastal regions of the United States, but the Indian Ocean and the Mediterranean Sea are also sampled. Oligotrophic (e.g., Tropical Pacific) to productive (e.g., Patagonia shelf, Benguela current) biological provinces, Case 1 and Case 2 waters are represented in the match-up data, as well as various types of aerosols, (e.g., maritime in the open ocean, continental and pollution-type in coastal regions). Figure 4 displays scatter plots of estimated versus measured marine reflectance for the NO- MAD data sets, and gives the comparison statistics in terms of coefficient of determination r 2, bias (estimated minus measured values) and RMS difference. In the scatter plots, the uncertainties associated with the marine reflectance retrievals are also displayed. The biases (higher Bayesian values) are or 7.8% at 412 nm, or 5% at 443 nm, and smaller at the other wavelengths (< at 670 nm). They represent a small component of the RMS errors, which decrease from at 412 nm to at 555 nm and to at 670 nm. These RMS errors are comparable with those obtained for the SeaDAS algorithm by the NASA Ocean Biology Processing Group and available from their web site using a much larger match-up data set sampling a wider range of conditions (4577 points), i.e., at 412 nm, at 555 nm, and at 670 nm. They are larger, however, than those obtained for the SeaDAS algorithm at the BOUSSOLE site, i.e., at 412 nm, at 555 nm, and at 670 nm (Antoine et al., 2008), but in this case the sampling is limited to a single site. One cannot conclude, however, based on the analysis of such limited match-up data, whether or not the Bayesian technique performs better, in terms of accuracy, than the SeaDAS algorithm. The lack of comprehensive evaluation data set emphasizes the importance of generalization in develop- 8

12 ing inversion schemes for global application, i.e., in our Bayesian approach, proper approximation of the forward operator, and of associating uncertainties to marine reflectance estimates on a pixelby-pixel basis. 3.3 Improvements with PACE Using spectral information The PACE mission will provide TOA reflectance information at 5 nm resolution in the nm range and at lower spectral resolution in bands centered on 865, 1240, 1640, and 2130 nm. To address threshold ocean science questions, retrievals may employ spectral bands typically 15 nm wide by aggregating the 5 nm resolution data. The observations in the UV, where coupling between aerosol absorption and molecular scattering is effective, have the potential to improve atmospheric correction in the presence of absorbing aerosols, a serious difficulty with sensors like MODIS. First, the PACE aggregate spectral bands for threshold science requirements will be considered. The Bayesian methodology will be applied to TOA reflectance data in those bands and in the MODIS bands used in the standard atmospheric correction scheme. Improvements in the accuracy of marine reflectance retrievals obtained with the PACE aggregate bands will be quantified. Second, the 5 nm bands will be considered (the Bayesian methodology is not restricted to a particular set or number of bands), and the accuracy of the atmospheric correction at that spectral resolution will be quantified. It is expected that retrieval errors will be larger in spectral regions affected by gaseous absorption. Such regions may be eventually excluded in the atmospheric correction scheme, but keeping the TOA observations in the oxygen A-band around 763 nm, sensitive to aerosol altitude, will be important to improve retrieval accuracy when aerosols are absorbing. In the presence of such aerosols, the atmospheric reflectance after correction of molecular effects, ρ a, depends on the aerosol vertical distribution, especially at smaller wavelengths (UV and blue). The accuracy of the atmospheric correction at 5 nm resolution will be compared with the accuracy obtained by approximating ρ a and T a determined in the aggregate bands by a polynomial with a few terms, since these atmospheric functions vary smoothly with wavelength. Simulated TOA reflectance data for a wide range of geophysical conditions and solar and viewing angles and including various noise configurations, that will account for spatial resolution (case of OCI/OG, see PACE SDT report), will be used (see details in sections below). Fine sampling, and on an extended spectral domain, of the radiation field can provide additional information compared with the measurements obtained with an instrument like MODIS. How much is gained by this increase in the number of measurements will also be assessed through simulation. Indeed, the measurements are expected to be highly correlated, so that the dimension of the data could be significantly reduced. This can be achieved, for instance, by fitting a linear model first in each aggregate band. Note that for this purpose, increasing the number of measurements remains useful since it allows one to filter out the noise. Then, the methodology developed in Frouin and Pelletier (2014) is directly applicable. Using polarization and multi-angle information Multi-directional and polarized measurements provide additional/complementary information on aerosol characteristics (Deschamps et al., 1994; Diner et al., 2005; Dubovik et al., 2010; Hasekamp et al., 2011). Measurements in different directions allow one to sample the aerosol phase function at different scattering angles. Measurements 9

13 of the degree and angle of polarization of TOA light are sensitive to aerosol type (size distribution, index of refraction, shape). The impact of having, in addition to spectral information by the spectrometer, directional and polarized information in selected spectral bands by a polarimeter like 3MI will be evaluated. 3MI will measure in 10 to 14 directions per ground pixel and 13 optical channels from the visible to SWIR, 8 of them equipped with polarizers to provide the first three components of the Stokes vector, i.e., not only specific intensity, but also degree of linear polarization and polarization direction (polarization is mostly linear for prevailing atmospheric conditions). The TOA measurements in the aggregate bands of the spectrometer, augmented by the directional and polarized measurements of the polarimeter, will be used as input in the Bayesian inversion scheme. It will be assumed that the polarized bands of the polarimeter have the same spectral characteristics as the corresponding aggregate bands of the spectrometer. In other words, the set of TOA measurements to invert will consist of 1) the first component of the Stokes vector (i.e., total radiance or reflectance after normalization) in all the aggregate bands of the spectrometer for the viewing direction of the spectrometer, and 2) the first three components of the Stokes vector in the aggregate bands corresponding to the polarized bands of the polarimeter, this for all the viewing directions of the polarimeter. Typical viewing configurations of an ocean surface target will be considered. The utility of polarization information in the SWIR bands (included in 3MI essentially to handle polarization of land surfaces in aerosol retrievals) will also be assessed, by comparing marine reflectance estimates obtained with and without the polarization measurements in those bands. Modeling of the TOA signal Constructing the inverse models requires multiple simulations of the TOA signal, total and polarized. This will be accomplished using an accurate radiative transfer code that fully accounts for multiple scattering and interactions between scattering and absorption, the LBLADM code (Dubuisson et al., 1996; Duforet et al., 2005, 2007). The essential features of the code are summarized in the following. First, gaseous absorption is calculated at high spectral resolution, using a line-by-line (LBL) code originally developed for the long-wave spectral region (Scott, 1974) and extended to the solar spectrum (Dubuisson et al., 1996). The HITRAN 2004 database (Rothman et al., 2005) is used to define the spectroscopic parameters for oxygen absorption lines (spectral position, intensity, and width). Second, the radiative transfer equation is solved at each wavelength step of the LBL code, assuming a vertically heterogeneous atmosphere stratified into plane and parallel layers. The Adding- Doubling Method (ADM) (De Haan et al., 1987) is used to account for scattering processes and interactions between scattering and absorption. For each layer, the ADM requires the absorption optical thickness (calculated with the LBL code for gaseous absorption), as well as scattering parameters, i.e., scattering phase matrix, single scattering albedo, and scattering optical thickness for the atmospheric constituents (i.e., molecules and aerosols). In the mathematical developments, the Stokes parameters and the phase matrix, azimuth-dependent quantities, are expanded in a Fourier series, reducing the number of variables treated at one time and simplifying the radiative transfer computations. Third, the atmosphere is bounded at the bottom by the surface, and interactions between the atmospheric radiance field and surface reflection are accurately taken into account. The surface can 10

14 be flat or wavy, and reflection isotropic or anisotropic (see Duforet et al., 2005 for the treatment of a wind-ruffled ocean surface). The LBLADM code has been successfully compared with a successive-orders-of-scattering reference code and it has been used to simulate POLDER measurements, showing agreement with respect to uncertainties on aerosol properties (Duforet et al., 2005). It has also been used to develop and evaluate a technique to estimate aerosol scale height/altitude from MERIS and POLDER measurements in the oxygen A-band (Duforet et al., 2007; Dubuisson et al., 2007). Various aerosol profiles (type, amount) will be considered in the simulations. These include aerosols confined in the boundary layer and aerosols located in altitude. Typical aerosol models in various proportions will be used, i.e., maritime, continental, and urban, as well as dust-like, biomass burning, and volcanic, and varied aerosol amounts. Their characteristics, i.e., size distribution, refractive index, will be taken from WMO (1983), Dubovik et al. (2002), and other references. Aerosol type will be assumed homogenous vertically. The marine reflectance will be simulated using the Hydrolight code (Mobley, 1989). This code is general, allowing simulations of complex waters containing phytoplankton, yellow substances, and suspended inorganic material, including bottom effects. Only optically deep waters will be considered. Available data on the inherent properties of the various oceanic constituents (e.g., Loisel and Morel, 1998; Morel and Maritorena, 2002; Babin et al., 2003, etc.) will be used in the simulations. Fresnel reflection at the wavy air-sea interface and effects of whitecaps will be specified as a function of wind speed. Atmospheric pressure will also be varied. Polarization properties of the hydrosols, mostly unknown, will be ignored. This assumption is not dramatic for accuracy analyses involving TOA polarized measurements, since for observations from space the scattering angle in water is mostly large, i.e., the contribution of polarized light to the total water signal is expected to be small. In practice, after correction for gaseous absorption (except in the oxygen A-band, see above) the LBLADM code will be run twice, once with aerosols and non-null water signal and once with only molecules and null water signal, and the second output will be subtracted from the first to yield the signal of interest, i.e., ρ in the case of TOA measurements of total radiance. The computations of total and polarized radiance will be made for a wide range of solar and viewing zenith angles. This range will cover conditions generally encountered in ocean-color remote sensing, including observations in the Sun glint. For each geometry the forward operator will be quantized, i.e., a data base stored on disk will be generated, where the corrected signal (e.g., ρ) is evaluated at selected sampling points in the parameter space of atmospheric and oceanic variables. To evaluate the corrected signal at an arbitrary point, i.e., for an arbitrary state of the ocean and the atmosphere, the operator will be approximated by interpolation. Naturally there is a price to pay for this strategy, which results in an approximation error of the theoretical forward operator, i.e., the differences between actual computations with the radiative transfer code and the results from the interpolation procedure. However, with properly chosen quantization points, located on a fine enough grid, the approximation error of the forward operator will be small compared with the other sources of uncertainties, and accounted for via the noise term ε. Specification of noise and prior distributions The additive noise term ε in (1) is intended to encapsulate all the sources of uncertainties in the forward modeling. These include, in particular, measurement uncertainties, modeling uncertainties of the various radiative transfer processes, as well as the approximation error associated with the reconstruction of the forward operator by in- 11

15 terpolation of the discrete (simulated) data. For the theoretical analysis, it will be assumed that ε is a gaussian random vector with mean zero and spherical covariance matrix, i.e. E[ε] = 0 and Cov(ε) = σ 2 I. An ellipsoidal covariance matrix Cov(ε) = diag(σ 1,..., σ d ) will also be considered, to allow for different noise levels in each bands. In this model, the parameter σ i plays the role of noise level in a deterministic setting. This assumption is reductive. Indeed, modeling errors on the forward operator, radiometric calibration errors, etc. are likely to have a non-zero average (i.e., yielding a bias) and/or depend in magnitude on the input parameters (atmosphere and ocean states). For the theoretical analysis, PACE instrument requirements in terms of signal-to-noise will be used to define the σ i s, which will provide a lower bound for the effect of noise. Larger noise levels will also be introduced, that may represent modeling, approximation, and calibration errors, to examine the sensitivity of retrieval accuracy to noise. Since the objective is to develop an atmospheric correction algorithm valid at the global scale, a suitable prior distribution would have to reflect realistically the frequencies of occurrence for parameters like marine reflectance or aerosol optical thickness at the global scale (in space and time). In addition, since the prior distribution is one of the elements defining the Bayesian solution to the inverse problem, any information used to specify it must not originate from inversions of satellite observations, but instead from separate, independent field campaigns or numerical models. For the theoretical analysis, the prior distributions for individual parameters will be specified according to Frouin and Pelletier (2014). They will be uniform over the range of possible values for atmospheric pressure, wind speed, marine reflectance, aerosol scale height, and aerosol model proportions, and log-normal for aerosol optical thickness. Naturally, since at a global scale most of the oceans are Case I waters, while only a small proportion corresponds to more optically complex waters, it may seem desirable to favor the first type of waters or, in other words, that the prior distribution on ρ w places more weight on Case I ρ w than others. How to do so in an objective way is a non-trivial matter at a global scale, for this would require extensive field campaigns to estimate reliably the frequencies of the marine reflectance. As indicated in Section 3.1, assuming that the atmospheric parameters and marine reflectance are not correlated, the prior distribution on the parameter space X a X w will be the product P xa P xw. Practical implementation The definitions of the Bayesian solution and of the inverse applications require a prior distribution on the oceanic and atmospheric parameters, and a noise distribution. In practice, without assumption on the distribution of ε, this latter cannot be inferred since ε is never observed (and nor are the variables x a, x w, and y measured simultaneously). It is therefore reasonable to impose a parametric assumption on the distribution of ε. To determine a value of σ, considering a common noise level, one can extract N TOA reflectance y 1,..., y N from various images acquired by the satellite sensor. Next, for each j = 1,..., N, the minimum distance from y j to Φ(X a, X w ) can be determined. Then, a value ˆσ 2 of σ 2 is defined by averaging the δj 2 s, i.e., ˆσ 2 = (1/N) N j=1 δ2 j. For the prior distributions, one can use those for the theoretical analysis. They have provided, with the noise distribution above, good results in application to SeaWiFS imagery (Frouin and Pelletier, 2014). However, independent information from coupled physicalbiogeochemical ocean circulation models (e.g., NEMO in ORCA2/LIM2/PISCES configuration) and atmospheric transport models (e.g., GOCART) may be used to specify more realistic, regionally and seasonally varying, distributions for marine reflectance and aerosol models, an aspect that will be investigated. 12

16 Once the noise and prior distributions are determined, the Bayesian inverse models can be constructed on the angular grid. Using partition-based models is fast in application, as demonstrated on SeaWiFS imagery in Frouin and Pelletier (2014). The partition is hierarchical, with hierarchy induced by a perfect binary tree, which drastically reduces the computational cost of determining membership. The parameters of the inverse models, defined for each point of the angular grid, need to be stored (like look-up tables for aerosols in the standard scheme), but storage space is not an issue, even though the data set may be large. The methodology allows for periodic re-processing of the satellite imagery, which may occur during mission lifetime as radiometric calibration coefficients are adjusted and more information on the distribution of atmospheric and oceanic parameters becomes available to specify prior distributions. This will require re-constructing the inverse models using updated noise and prior distributions, an operation not especially expensive computationally that can be standardized. In sum, the Bayesian methodology is applicable operationally, has re-processing capability, and can be easily implemented (without special requirements) into the standard NASA/GSFC OBPG processing system. 4.0 EXPECTED RESULTS AND SIGNIFICANCE The investigation will quantify, using the Bayesian approach to inverse problems and RT simulations, the expected accuracy of the atmospheric correction of ocean-color imagery from the PACE spectrometer, at both 5 nm resolution in the UV to NIR (i.e., in the hyper-spectral bands used to discriminate water constituents) and in the threshold aggregate bands. The gain in accuracy with respect to the accuracy provided by the current generation of ocean-color sensors will be assessed, and the benefits of using directional and polarized information from the eventual PACE polarimeter will be determined. The utility of including information at 5 nm resolution for atmospheric correction in the aggregate bands will be specified, and optimum sets of bands will be identified. The analysis will be performed for various noise configurations, typical geometries, and a wide range of geophysical conditions, including absorbing aerosols and optically complex waters. The investigation will provide a Bayesian methodology for atmospheric correction of the PACE spectrometer data. The methodology makes it possible to incorporate known constraints of the marine reflectance (i.e., correlation between components) and to account for the varied sources of uncertainty (i.e., measurement noise, RT modeling errors). Importantly, it allows the construction of reliable multi-dimensional confidence domains of the retrieved marine reflectance. Specifically, the mean and covariance of the posterior distribution are computed. These quantities provide, for each pixel, an estimate of the marine reflectance and a measure of its uncertainty. Situations for which observation and forward model are incompatible are also identified. Thus the methodology will offer the means to analyze and interpret PACE ocean-color imagery in view of confidence limits and model adequacy, on a pixel-by-pixel basis. This is a definite advantage compared with standard atmospheric correction techniques, which rely essentially on evaluation against too few in situ measurements for accuracy assessments. Details about practical implementation for routine application, including the specification of prior and noise distributions, will be provided. By evaluating via theoretical studies the accuracy of the atmospheric correction of PACE oceancolor radiometry and the expected improvements with respect to current ocean-color sensors, by identifying optimum sets of spectral bands, and by providing an inverse methodology adapted to the problem, which can be viewed as a generalization of the standard algorithm, the investigation 13

17 responds directly to the PACE Science Team Announcement of Opportunity, which seeks methods and approaches that will maximize the new capabilities of the PACE mission for understanding global ocean ecology in a changing climate. 5.0 MANAGEMENT PLAN 5.1 Personnel and project responsibilities To address the geophysical and statistical aspects of the project, a team of two scientists with complementary expertise has been assembled. The team includes Dr. Robert Frouin from the Scripps Institution of Oceanography (SIO), University of California at San Diego and Prof. Bruno Pelletier from the University of Rennes II, France. Dr. Frouin will be the Principal Investigator (PI) leading and supervising the project. Prof. Pelletier will collaborate closely with the PI as a Visiting Researcher in SIO. Each scientist s role in the project is explained below. Dr. Frouin is a Research Meteorologist in SIO s Climate, Atmospheric Science, and Physical Oceanography Division. His expertise is in radiation transfer (atmosphere and oceans) and inverse problems related to light scattering. As a member of the NASA Ocean Color Research Team, the NASA NPP Science Team, the JAXA GLI and SGLI Science Teams, the CNES POLDER Science Team, the KARI GOCI Science Team, and a MERIS investigator, he has developed atmospheric correction schemes for ocean-color imagery, checked the calibration of ocean-color sensors, and evaluated ocean-color products. He has also developed algorithms to estimate PAR from POLDER, GLI, SeaWiFS, MODIS, GOCI, MERIS, and VIIRS data. During the 1990 s he organized and conducted experiments with the aircraft version of the POLDER instrument. In 2005 he participated in the development of PHYTOSAT, a space mission similar to PACE for observing phytoplankton species and their sensitivity to climate variability, with an advanced MERIS and an advanced POLDER (proposed to ESA, but not selected). Dr. Frouin is a member of the ACE ocean biology and ocean-aerosol working groups, and he was a member of the PACE Science Definition Team. During Dr. Frouin managed NASA s Ocean Biology Program and served as the Program Scientist for SeaWiFS and MODIS. Dr. Frouin will be responsible for the overall implementation and management of the project s activities. He will ensure that the project is accomplished on time and within allocated funding. He will coordinate near- and long-term scientific priorities and establish lines of interface with appropriate institutions and relevant research programs, including satellite ocean color project offices. Dr. Frouin will be in charge of the geophysical aspects of the project, in particular simulations of the PACE measurements, development and implementation of the bayesian methodology for PACE, and evaluation of the expected gain in accuracy for various PACE measurement configurations. Prof. Pelletier is an Associate Professor of Mathematics in the University of Rennes Department of Mathematics. His expertise and research interests are in mathematical and applied statistics, including statistics on manifolds, nonparametric statistics, statistical learning, inverse problems, and remote sensing. Some of his current work is on statistical inversion of satellite data and density-based clustering of complex, high-dimensional, and dependent data. He has developed, in collaboration with the PI, a Bayesian methodology to retrieve marine reflectance and chlorophyll concentration from satellite ocean-color imagery. For this project, Prof. Pelletier will be in charge of extending the Bayesian inversion to the PACE measurement configurations, i.e., hyper-spectral 14