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A Bayesian model is developed to match aerospace ocean color observation to field measurements and derive the spatial variability of match-up sites. The performance of the model is tested against populations of synthesized spectra and full and reduced resolutions of MERIS data. The model derived the scale difference between synthesized satellite pixel and point measurements with R^{2} > 0.88 and relative error < 21% in the spectral range from 400 nm to 695 nm. The sub-pixel variabilities of reduced resolution MERIS image are derived with less than 12% of relative errors in heterogeneous region. The method is generic and applicable to different sensors.

Consistent and accurate matching of aerospace observation data to field measurements are necessary calibration and validation steps towards creating reliable products of inherent optical properties (IOPs). The scientific procedure to match satellite observations to field measurements can generally be divided into three steps:

Matching procedures for ocean color sensors were addressed by many researchers. For instance, Harding

In this paper we introduce a complete scheme to quantify the scale difference between a satellite pixel and a point (field) measurement. We used Bayesian inference method [

In this study we will use the model of Gordon _{w}_{1}, _{2} are subsurface expansion coefficients due to internal refraction, reflection and sun zenith; _{w}_{1} = 0.0949, _{2} = 0.0794, _{w}_{b}_{b}_{phy}_{dg}_{spm}

Our basic assumption is that each pixel in aerospace ocean color data represents the mean of an unknown theoretical probability distribution function (PDF) over a pixel area. In this respect, any field measured radiance is a sample drawn from this PDF [

The radiometric mismatch, between aerospace and field sensors, is attributed to the scale difference and is represented as an upper and lower bounds with a 1−_{±} is the upper “+” and lower “−” bounds of

The plausible range of the IOPs can be estimated from the upper and lower radiometric bounds by simply inverting (1) for the upper and lower radiometric bounds. A first estimate of the IOPs standard deviation is then derived form their plausible range using the method of Salama and Stein [

The above-mentioned theoretical derivation of the Bayesian inference are summarize in the following algorithm:

use (2) to estimate the upper and lower bounds of spectra;

derive the IOPs ranges from these spectra by inverting (1);

use the method of section 3 to estimate the standard deviation of IOPs from their range;

generate the prior PDF of IOPs using log-normal distribution;

initiate the posterior using log-normal distribution;

maximize (3) to obtain an new estimate of the posterior;

update the posterior;

iterate between steps (5) and (6) till convergence;

permutate the resulting PDF of IOPs;

forward the sets of IOPs using (1) to obtain the empirical PDF of spectra.

The proposed model was applied on [

Note that reflectance values are also log-normally distributed.

The Ansari-Bradley test shows that wavelength 495 nm is more probable to produce the sought variability regardless of the position of _{dg}_{spm}_{phy}^{2} values derived from model-II regression [^{2} larger than 0.88. The RMSE values almost increase towards the median and start decreasing to reach a minimum value of 0.17 for 0.95 quantile. Close quantiles to the median (0.5) reproduce PDF with small dispersions.

The stability of the model to uncertainties in atmospheric correction and sensor noise was tested by perturbing the mean,

We used full (FR) and reduced (RR) resolutions of MERIS images acquired over the North Sea [

Our method is formulated based on two steps. First we estimate the plausible range of IOPs. Second, we derive the posterior PDF of IOPs. In the first step we use the method of Bates and Watts [

Validation with IOCCG data set shows that the root-mean of squared relative error is less than 15% for all possible field measurements. Moreover, derived values of variability are linearly related to the known values on a log scale with R^{2} > 0.88. Derived variability values from the green band, centered at ∼ 495 nm, are more probable and are invariant to the position of

The proposed model is further tested with full and reduced resolution MERIS products covering part of the Dutch coastal waters. The highest errors in derived values of sub-pixels variability are in spatially homogenous areas. In these areas all quantile values are close to the mean and thus little information can be derived. This can also be observed in

Since atmospheric correction is a significant issue in water remote sensing [

In this paper we developed and applied a Bayesian approach to address the scale variability between point and aerospace measurements above water. The model used the differences between the field and aerospace observed spectra to derive prior information on the IOPs. We then applied Bayesian inference to derive the optimum posterior distribution of IOPs by maximizing the joint entropy of the prior-posterior. Our approach provided information about the sub-scale variability of match-up pixel on the IOPs and radiometric levels. We, further, showed that match-up sites for radiometric quantity could be inhomogeneous and preferably located on the edge of the turbidity zone. Information on the sub-scale variability of geo-biophysical processes will facilitate planning of calibration and validation of future sensors, resolving the critical scale of variability of an observed feature and improving the assimilation of EO products into model grid and field data. Although the approach was developed for radiometric quantities in a match-up pixel, it has the potential to be applied on bio-geophysical properties using prior knowledge on their plausible ranges. In addition, we believe that our methodology is general and applicable to land surface studies. The same principle applies: utilizing prior knowledge about geo-biophysical quantities to derive sub-scale variability of satellite pixel. However, the proposed model needs a more extensive validation with different data sets on land parameters.

The authors would like to thank the European Space Agency (ESA) for supplying MERIS data, Wim Timmermans from ITC, the Netherlands for providing technical assistance related to the Eagle 2006 data set, the Management Unit of the North Sea Mathematical Models (MUMM, Belgium) for maintaining the Oostende site. Anonymous reviewers are acknowledged for their comments.

The absorption and scattering coefficients of water molecules, _{w}_{w}_{phy}_{0}(λ) and _{1}(λ) are empirical coefficients. The absorption effects of detritus and dissolved organic matter are combined due to the similar spectral signature [^{−1}. The scattering coefficient of SPM _{spm}_{b}

Observed remote sensing reflectance can be approximated as being the sum of the model best-fit _{m}_{m}_{m}

Salama and Stein [_{u}_{obs}_{l}

Random normal fluctuations are assumed to be wavelengths dependent with zero mean and standard deviation calculated from the theoretical PDF, see

Empirical PDFs generated from field spectra corresponding to predefined quantiles: 0.05 “doted line”, 0.25 “dashed line”, 0.5 “full line”, 0.75 “plus”, 0.9 “square”, 0.95 “circles” for six wavelengths. The theoretical PDF is illustrate as gray area.

Known

Relative values of fluctuations added to the aerospace mean for six wavelengths.

RMS-RE values in estimated

Probability of empirical PDF having the same variability of the theoretical PDF.

Quantile values
| ||||||
---|---|---|---|---|---|---|

band nm/IOPs | 0.05 | 0.25 | 0.5 | 0.75 | 0.9 | 0.95 |

440 | 0 | 0.26 | 0 | 0 | 0.31 | 0 |

495 | 0.59 | 0.19 | 0.17 | 0.2 | 0.11 | 0.13 |

550 | 0 | 0.01 | 0 | 0.03 | 0.06 | 0.57 |

| ||||||

_{phy} |
0 | 0 | 0.09 | 0.17 | 0 | 0.18 |

_{dg} |
0 | 0.93 | 0.09 | 0.06 | 0.37 | 0.58 |

_{spm} |
0.33 | 0.21 | 0.34 | 0.51 | 0.05 | 0.3 |

Relative errors (%) in derived log

Quantile values
| ||||||
---|---|---|---|---|---|---|

band nm | 0.05 | 0.25 | 0.5 | 0.75 | 0.9 | 0.95 |

400 | 9.08 | 6.24 | 10.39 | 0.77 | −13.12 | −1.29 |

440 | 9.78 | 1.67 | 6.71 | −1.65 | −9.29 | −1.16 |

495 | 7.29 | −3.91 | 0.45 | −7.32 | −9.34 | −5.74 |

550 | 15.35 | 4.57 | 8 | −1.52 | −3.06 | −1.53 |

675 | 10.05 | 14.81 | 16.11 | 6.4 | 0.61 | −0.4 |

695 | 13.77 | 20.7 | 20.99 | 11.12 | 3.84 | 2.32 |

| ||||||

RMS-RE | 12.5 | 14.6 | 14.8 | 7.6 | 5.7 | 2.7 |