1. INTRODUCTION

Ocean color remote sensing is an important component of climate system observations [1], which requires an accurate quantification of uncertainties in data products [2].

Besides recent exceptions [3], most ocean color retrieval algorithms assume an infinite surface of homogeneous reflectance equal to that of the water target element [4,5]. While this approximation is acceptable in open ocean, where surface reflectance has a slow spatial variation, it may induce significant inaccuracies in the proximity of the coast. In fact, in coastal areas the radiance reflected by the land and propagated by the atmosphere into the sensor field-of-view may introduce significant spectral perturbations. These kind of perturbations, due to cross-contaminations between areas with different reflectance properties overlaid by a scattering atmosphere, are usually termed adjacency effects [6]. More specifically, the adjacency radiance $L_{\mathrm{adj}}$ is used to define the difference in the background radiance [7] (also called environmental radiance [8]) between the case accounting for the non-uniformity of the underlying reflecting surface and the case assuming an infinite uniform surface [9]. As such, $L_{\mathrm{adj}}$ can vary from negative to positive.

A previous theoretical analysis of adjacency effects in typical ocean color observations of northern Adriatic waters highlighted significant adjacency contributions to the total signal at the sensor [9]. The analysis was performed along a study transect extending from the coast and intercepting the Aqua Alta Oceanographic Tower (AAOT, 45.31N, 12.51E) validation site [10], and included observation conditions representative for the Sea-Viewing Wide Field-of-view Sensor (SeaWiFS), the Moderate Resolution Imaging Spectroradiometer (MODIS), the Medium Resolution Imaging Spectrometer (MERIS), and in principle the Sentinel-3 Ocean and Land Color Instrument (OLCI).

That study is now extended to evaluate biases induced by adjacency effects in satellite-retrieved radiometric products (i.e., the water-leaving radiance). A further investigation of adjacency effects on derived products, such as chlorophyll concentration Chl, is out of the scope of this paper.

The analysis accounts for atmospheric multiple scattering, off-nadir illumination and observation geometries, sea surface reflectance anisotropy, and coastal morphology. It further considers the range of atmospheric conditions typical of the area, as well as the seasonal variability of solar illumination, and land and seawater optical properties. The latter have been extracted from multi-year databases of satellite and in situ data, respectively.

How adjacency effects impact the accuracy of derived remote sensing products depends on the specific procedure applied in the retrieval process. Two main categories of procedures for the removal of the atmospheric effects are here considered: those not deriving (AC-1) or alternatively deriving (AC-2) the atmospheric contributions from the remote sensing data themselves. Examples of AC-1 schemes are those based on look-up tables of forward-simulated atmospheric contributions. Conversely, examples of AC-2 schemes are the SeaWiFS Data Analysis System (SeaDAS) by the National Aeronautics Space Administration Ocean Biology Processing Group (NASA OBPG) [11,12] and the Optical Data Processor of the European Space Agency (ODESA [http://earth.eo.esa.int/odesa/]), both determining the atmospheric properties from near-infrared (NIR) wavelengths. In the case of AC-1 schemes an exact knowledge of the atmospheric contributions is assumed. Conversely, AC-2 schemes are supposed to accurately determine the water-leaving radiance at NIR wavelengths.

Considering that operational retrieval schemes nest several procedural steps that may interfere with each other, the study is complemented with an analysis investigating possible compensations of biases.

The study acquires general relevance when considering that measurement sites of the Ocean Color component of the Aerosol Robotic Network (AERONET-OC) [10], established to support the validation of satellite ocean color products (i.e., normalized water-leaving radiance and aerosol optical thickness), rely on fixed deployment platforms located in coastal regions at distances from the land varying from a few up to tens of kilometers. Because of this, regardless of the specific site considered for the study, results are expected to provide general conclusions applicable to a variety of measurement conditions and sites.

The simulation procedure and the selected test cases are illustrated in Section 2. Theoretical biases induced by adjacency effects on the retrieved water-leaving radiance are presented and discussed in Section 3. Summary and conclusions are drawn in Section 4.

2. SIMULATION PROCEDURE

A. Simulation of the Adjacency Radiance

Simulations of the adjacency radiance $L_{\mathrm{adj}}$ for typical SeaWiFS, MODIS, MERIS, and OLCI observation conditions along a study transect in the northern Adriatic Sea starting from the coast and intercepting the AAOT (Fig. 1), have been performed adopting the methodology described in [9]. Specifically, the adjacency radiance has been modeled as

 

Fig. 1. Land/sea mask utilized in the simulations: land elements are indicated in dark gray, while sea elements are in light gray. Each element is $2\times 2\mathrm{km}$ wide. The black line represents the transect (34 km long), the black circle the AAOT (45.31°N, 12.51°E).

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For the benefit of the reader, a list of the most used symbols is provided in Appendix A.

In the present exercise: (i) seawater optical properties, assumed invariant along the transect, have been extrapolated from the comprehensive dataset of in situ measurements collected at the AAOT since 1995 [15,16]; (ii) land optical properties have been inferred from the MODIS multi-year 16-day climatological snow-free aggregate database [17] of land directional hemispherical reflectance (DHR, i.e., the reflectance for incoming light from a single direction [14]) and land isotropic bihemispherical reflectance (${BHR}_{iso}$, i.e., the reflectance for an incoming isotropic light field [18]); (iii) functions $S$, $C$, and $W$ have been taken from a pre-existing dataset of values simulated in the same area utilizing the Novel Adjacency Perturbation Simulator for Coastal Areas (NAUSICAA) full 3D backward Monte Carlo (MC) code [9] and the plane-parallel Finite Element Method (FEM) numerical algorithm [19,20].

The NAUSICAA MC code fully accounts for multiple scattering, off-nadir illumination and observation geometries, sea surface roughness, actual coastal morphology, and its precision is set to meet actual ocean color sensors radiometric resolutions [9]. The FEM numerical code has been extensively benchmarked with other popular radiative transfer codes [20–22], and has been already used to perform radiative transfer simulations in realistic cases [9,22–25].

Simulations of $L_{\mathrm{adj}}$ have been performed at typical ocean color center wavelengths (namely, $\lambda =412$, 443, 490, 510, 555, 765, and 865 nm) for several test cases representing typical annual and intra-annual observation conditions.

Selected spectral values of the average remote sensing reflectance ${\overline{R}}_{\mathrm{rs}}$ are illustrated in Fig. 2, while the water signal at NIR has been assumed negligible. Notably, no significant difference in the simulated adjacency radiance has been observed when assuming a non-null NIR water-leaving radiance corresponding to the 0.5 and 0.9 quantiles of the values determined with SeaDAS from a set of 1124 SeaWiFS images acquired in correspondence of the AAOT between 1997 and 2008, and all considered suitable for match-up construction with corresponding in situ data.

 

Fig. 2. Spectral values of in situ ${\overline{R}}_{\mathrm{rs}}$ adopted in the simulations: symbols represent different annual and intra-annual periods. Error bars indicate the standard deviation $\pm {\sigma }_{R_{\mathrm{rs}}}$.

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The average land albedo has been computed as ${\overline{\rho }}_l=(1-S_E)·⟨DHR⟩+S_E·⟨BH{R}_{iso}⟩$ [18], where $S_E$ is the ratio between diffuse and direct irradiance at the surface, and where DHR and $BH{R}_{iso}$, provided at MODIS land bands (centered at $\lambda =470$, 555, 659, and 858 nm), have been spatially averaged over the considered region (see Fig. 1) and inferred at reference ocean color wavelengths assuming a cropland ecosystem (see [9] for details). Average land albedo values, illustrated in Fig. 3, have been computed for the whole year and for selected representative intra-annual periods, i.e., (a) early spring (March 5–20); (b) late spring (May 24–June 8); (c) mid-summer (July 27–August 11); (d) early autumn (September 29–October 14); and (e) mid-winter (December 18–31).

 

Fig. 3. Spectral values of climatological ${\overline{\rho }}_l$ adopted in the simulations: symbols represent different annual and intra-annual periods. Error bars indicate standard deviations $\pm {\sigma }_{{\rho }_l}$.

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It is recalled that the anisotropy of sea surface reflectance is accounted for in term $W$ of Eq. (1) through the bidirectional reflectance distribution function of a wind-generated rough sea surface [9,26]. Conversely, the sea albedo ${\rho }_{\mathrm{sea}}$, approximated by ${\rho }_{\mathrm{sea}}\approx \pi R_{\mathrm{rs}}+0.04$ (where 0.04 represents a typical sea surface irradiance reflectance for the propagation from air to water [13]), is used in Eq. (1) only to model multiple reflections at the sea surface. Acknowledging the qualitative relevance of ${\rho }_{\mathrm{sea}}$, its approximate expression allows an easy comparison between sea and land albedos. It specifically allows evidencing that the land is definitely more reflecting than the sea at red and NIR wavelengths, while sea and land albedos become more similar toward the blue, with ${\rho }_l$ exceeding ${\rho }_{\mathrm{sea}}$ only in spring.

For the considered intra-annual periods, average atmospheric and meteorological parameters encountered during clear sky conditions in the northern Adriatic Sea [15,27] have been assumed (i.e., Ångström coefficient $\alpha =0.05$, Ångström exponent $\nu =1.7$, single-scattering albedo ${\omega }_0=0.99$, and wind speed $W_s=3.3{\mathrm{ms}}^{-1}$). For the simulation of yearly average adjacency effects, additional extreme atmospheric observational conditions have been accounted for (i.e., $\alpha =0.02$ and 0.08, $\nu =1.4$ and 1.9, $W_s=1$ and $6{\mathrm{ms}}^{-1}$).

For each test case, simulations have been performed for a complete set of representative observation geometries, i.e., sensor viewing angle $\theta v=5°$, 20°, and 50°; sensor azimuth angle ${\phi }_v=\pm 75°$ and $\pm 100°$; solar azimuth angle ${\phi }_0=\pm 160°$. It is noted that sensor viewing angles are determined with respect to the local vertical, while sensor azimuth angles are counted clockwise from the north direction (as generally adopted in satellite geolocation). The geometric parameters have been selected on the basis of a comprehensive multi-annual database of the actual observation and illumination conditions in the selected area. Solar zenith angles ${\theta }_0=25°$ and 65° have been adopted for simulations related to the summer and winter intra-annual periods, respectively; ${\theta }_0=45°$ has been assumed in all other cases.

B. Simulation of the Atmospheric Optical Quantities

For the same set of test cases, the plane-parallel FEM numerical algorithm [19,20] has been used to simulate atmospheric optical quantities such as (i) the diffuse transmittance $t$ [28]; (ii) the path radiance $L_{\mathrm{path}}$ describing the radiance due to atmospheric scattering along the optical path, and to specular reflection by the sea surface of atmospherically scattered light [5] (often indicated as $L_{\mathrm{atm}}$ [4] in ocean color remote sensing); (iii) the path radiance components separately due to gas molecules and aerosol scattering; and (iv) the diffuse and direct downward irradiances.

3. RESULTS AND DISCUSSION

In ocean color remote sensing the radiance $L_t$ reaching a space sensor looking at a water element out of the region of sunglint and in the absence of whitecaps is traditionally modeled as [4,5]

By indicating with $\widehat{x}$ the quantity $x$ affected by land perturbations, the bias induced by adjacency effects on $x$ is defined as

The statistical analysis, over all test cases and for each sea element along the study transect, is performed on biases ${\psi }_{t{L}_w}$ of the water-leaving radiance retrieved at visible wavelengths at the top of the atmosphere (TOA). The latter choice is justified by the fact that in AC-2 schemes the determination of the transmittance $t$ is affected by adjacency effects, too. Results for an AC-1 scheme are illustrated in Section 3.A, while those for an AC-2 scheme are presented in Section 3.B.

A. Perturbations Induced by Adjacency Effects in AC-1 Schemes

Equation (2) is strictly valid only for open-ocean observations, where it is feasible to assume a homogenous underlying water surface. In the vicinity of the coast, Eq. (2) more correctly becomes

Yearly average values ${\overline{\psi }}_{t{L}_w}$, are illustrated in Fig. 4 at representative wavelengths and as a function of the distance along the transect moving away from the coast. It is noted that, due to the complex coastal pattern, the above distance does not correspond to the geometric distance from the coast.

 

Fig. 4. Annual average biases ${\overline{\psi }}_{t{L}_w}$ at representative wavelengths for an AC-1 scheme. Here and in the following figures, results are presented as a function of the distance along the study transect and error bars indicate the standard deviation $\pm {\sigma }_{\overline{\psi }}$ (black, not visible in the present plots) and the sample variance (gray). $L_w$ is assumed constant all along the transect. The horizontal dotted lines indicate $\pm 5\%$, while the vertical dashed line identifies the position of the AAOT site.

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As expected, biases monotonically decrease with distance from the coast. A minor non-monotonicity occurs in correspondence of an isolated group of few land elements regarded as a small island in the grid applied for the MC simulations (see Fig. 1). Mean biases ${\overline{\psi }}_{t{L}_w}$ are almost within $\pm 5\%$ throughout the transect in the spectral range 490–555 nm, and hence comparable to the target accuracy of satellite-derived water-leaving radiance [4], as well as to estimated average uncertainties of in situ measurements [29]. At blue and red wavelengths, where the water signal is low (see Fig. 2), ${\overline{\psi }}_{t{L}_w}$ shows larger values. Specifically, in proximity to the coast ${\overline{\psi }}_{t{L}_w}$ is up to $-21\%$ and $+34\%$ at 412 and 670 nm, respectively. Significant is the large variance, particularly toward the blue and red spectral regions when approaching the coast.

The dependence of biases on atmospheric parameters in the considered range of variation is not significant, except in the right vicinity of the coast at 670 nm where $\alpha =0.08$ leads to larger overestimates of the water-leaving radiance. As well, biases do not show sensitivity to the wind speed in the selected range of values. For the considered illumination and observation geometries, biases also do not show appreciable sensitivity to the sensor location with respect to land. It is nevertheless recalled that other mutual orientations of land, sun, and sensor might consistently affect adjacency effects and hence biases of satellite primary products [9].

Conversely, ${\overline{\psi }}_{t{L}_w}$ appreciably varies with the viewing angle and across the different intra-annual periods, as illustrated in Figs. 5 and 6. The highest seasonal dispersion of values is noted at blue and red wavelengths and for slanted observations. Biases nearly double when the viewing angle varies from 5° to 50°. Values for ${\theta }_v=20°$ (not shown here) are instead very similar to those for ${\theta }_v=5°$. Misestimates are the highest in mid-winter at blue wavelengths (where the water-leaving radiance is heavily underestimated toward the coast), and in late spring at red wavelengths (where conversely the water signal is largely overestimated). This is consistent with values of seawater and land reflectances adopted for the simulations. Indeed, in winter the sea reflectance is at its maximum (Fig. 2) while the land albedo is rather low (Fig. 3), thus suggesting negative $L_{adj}$ contributions [see Eq. (1)]. Conversely, in late spring the sea reflectance is at its minimum (Fig. 2) and the land albedo is the highest (Fig. 3), suggesting positive $L_{adj}$ contributions. In Figs. 5 and 6, error bars displayed in gray represent the sample variance. Darker error bars represent the standard deviation $\pm {\sigma }_{\overline{\psi }}$, mainly due to uncertainties in water leaving radiance and land albedo.

 

Fig. 5. Intra-annual values of ${\overline{\psi }}_{t{L}_w}$ at representative wavelengths for an AC-1 scheme and for a sensor viewing angle ${\theta }_v=5°$. Symbols are as in Fig. 3. Gray error bars represent the sample variance, while darker error bars represent the standard deviation $\pm {\sigma }_{\overline{\psi }}$. The horizontal dotted lines indicate $\pm 5\%$, while the vertical dashed line identifies the position of the AAOT site.

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Fig. 6. As in Fig. 5, but for ${\theta }_v=50°$.

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It is recalled that the water-leaving radiance has been assumed constant along the transect. An increase of water turbidity toward the coast, as often observed in realistic conditions, would likely lead to a decrease of biases. Considering that an analysis performed at 670 nm assuming an exponential increase of $L_w$ toward the coast up to values threefold those observed at the AAOT did not show appreciable changes in $L_{adj}$ [9], the decrease of ${\overline{\psi }}_{t{L}_w}$ at 670 nm toward the coast for increasing water turbidity is expected to be proportional to the increase of $L_w$ [see Eq. (6)].

Simulations in correspondence of the AAOT indicate an average annual underestimate of the water-leaving radiance toward the blue and an overestimate toward the red, with annual ${\overline{\psi }}_{t{L}_w}$ values only exceeding $\pm 5\%$ at 412 and 670 nm for slanted observations. In mid-winter the water-leaving radiance is underestimated throughout the spectrum with ${\overline{\psi }}_{t{L}_w}$ exceeding $-5\%$ at 412, 443, and 670 nm. In the other periods ${\overline{\psi }}_{t{L}_w}$ varies within $\pm 5\%$ up to 555 nm, but is well above $+5\%$ at 670 nm.

It is important to underline that where adjacency effects are low, the accuracy in the modeling of the propagating system becomes more critical. This mainly happens where land and sea albedos are very similar (i.e., at blue wavelengths) and when ${\theta }_0$ is high (i.e., in winter). As such, simulated data for winter acquisitions at blue wavelengths are likely very sensitive to a correct modeling of the land albedo, of the sea surface anisotropy, and of the atmospheric optical properties. For this reason, and even more when recalling that the land albedo at blue wavelengths has been inferred from climatological data at 470 nm, the confidence on simulated results in this spectral region (and particularly in winter) is lower.

Results presented so far indicate that for a correction scheme that does not derive the atmospheric properties from the remote sensing data, the land perturbations in the retrieved water-leaving radiance may be significant at 412 and 670 nm only.

While recalling that the present study focuses on biases at visible wavelengths, it is remarked that relative biases at NIR wavelengths are much higher than those quantified at shorter wavelengths. Specifically, even for the highest selected values of $L_w$ in the NIR (i.e., the 0.9 quantiles of the values determined by SeaDAS at the AAOT from the set of 1124 SeaWiFS images), ${\overline{\psi }}_{t{L}_w}$ at the AAOT reaches 35% and 66% at 765 and 865 nm, respectively. By assuming an exponential increase of $L_w$ toward the coast up to threefold the value at the AAOT, ${\overline{\psi }}_{t{L}_w}$ at the coast would reach 74% and 150% at 765 and 865 nm, respectively (Fig. 7). If $L_w$ instead is assumed constant along the whole transect, biases at the coast exceed 200% and 400% at 765 and 865 nm, respectively.

 

Fig. 7. Annual average biases ${\overline{\psi }}_{t{L}_w}$, determined at 765 and 865 nm for the considered AC-1 scheme, assuming an exponential increase toward the coast up to threefold the 0.9 quantile of satellite-derived $L_w$ at the AAOT. Gray error bars represent the sample variance, while darker error bars represent the standard deviation $\pm {\sigma }_{\overline{\psi }}$. The horizontal dotted line indicates $+5\%$.

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B. Perturbations Induced by Adjacency Effects in an AC-2 Scheme Deriving the Atmospheric Contribution from the Remote Sensing Signal in the NIR

For an AC-2 scheme determining the atmospheric optical properties from the radiometric signal in the NIR, where $L_w$ is assumed a priori known, the adjacency effects introduce at each NIR wavelength ${\lambda }_N$ a perturbation on the derived atmospheric contribution:

The value of $\mathrm{\Delta }{L}_{\mathrm{path}}({\lambda }_V)$ depends on the specific procedure applied to extrapolate the atmospheric properties from the NIR wavelengths. The theoretical analysis of adjacency-induced biases is here performed making reference to the algorithm proposed by Gordon and Wang [4] and implemented in the SeaDAS processing scheme [11,12]. In this algorithm the path radiance is modeled as

It is worthwhile noticing that taking into consideration Eq. (12), Eq. (11) can be alternatively written as

Equation (13) indicates the presence of two sources of land perturbation at each wavelength: one induced by adjacency radiance at the wavelength itself, $L_{\text{adj}}({\lambda }_V)/t{L}_w({\lambda }_V)$ (whose analysis has been extensively performed in Section 3.A), and one induced by adjacency radiance at NIR wavelengths, $-\mathrm{\Delta }{L}_A({\lambda }_V)/t{L}_w({\lambda }_V)$.

Term $\mathrm{\Delta }{L}_A$ in Eq. (11) is modeled as ${\psi }_{L_A}·L_A$ and it is estimated (i) using simulated $L_A$ values and (ii) computing the adjacency-induced bias on the atmospheric radiance ${\psi }_{L_A}$ in single-scattering approximation. Specifically, by applying the single-scattering approximation [4],

The single-scattering approximation is generally considered appropriate for an aerosol optical thickness at 865 nm, ${\tau }_a(865)$, up to 0.2 [4], and it is consequently applicable for the range of values considered in this study (i.e., $0.06<{\tau }_a(865)<0.12$). It is moreover noticed that the single-scattering approximation is here only applied for the evaluation of ${\psi }_{L_A}$, while all other terms, $L_A$ included, are computed carefully accounting for multiple scattering.

The determination of the atmospheric properties from the signal at the sensor tends to amplify the land perturbations in satellite data products, as indicated by the spatial plots of ${\overline{\psi }}_{t{L}_w}$ illustrated in Fig. 8 at sample wavelengths. In comparison with values from AC-1 schemes (see Fig. 4), biases are now all negative and generally larger, with ${\overline{\psi }}_{t{L}_w}$ nearby the coast reaching $-59\%$, $-28\%$, $-16\%$, and $-74\%$ at 412, 490, 555, and 670 nm, respectively. Notably, uncertainties on ${\overline{\psi }}_{t{L}_w}$ at 412 nm are larger than in other spectral regions. The plots in Fig. 9, which illustrate the two separate components ${\overline{\psi }}_{t{L}_w}[L_{\mathrm{adj}}({\lambda }_V)]$ and ${\overline{\psi }}_{t{L}_w}[L_{\mathrm{adj}}({\lambda }_N)]$ [Eq. (13)], show that while at blue wavelengths perturbations due to $L_{\mathrm{adj}}({\lambda }_V)$ are equivalent in sign and absolute value to those induced by $L_{\mathrm{adj}}({\lambda }_N)$, in the red ${\overline{\psi }}_{t{L}_w}$ predominantly originates from the term ${\overline{\psi }}_{t{L}_w}[L_{\mathrm{adj}}({\lambda }_N)]\equiv -\mathrm{\Delta }{L}_A({\lambda }_V)/t{L}_w({\lambda }_V)$, which is negative and therefore leads to an overall underestimate of the signal from the water.

 

Fig. 8. Annual average values of ${\overline{\psi }}_{t{L}_w}$ at representative wavelengths for the considered AC-2 scheme. Gray error bars represent the sample variance, darker error bars represent the standard deviation $\pm {\sigma }_{\overline{\psi }}$. The horizontal dotted lines indicate $\pm 5\%$, while the vertical dashed line identifies the position of the AAOT site.

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Fig. 9. Annual average values of the two sources of land perturbation in ${\overline{\psi }}_{t{L}_w}$ [see Eq. (13)] at representative wavelengths for the considered AC-2 scheme. Filled circles refer to ${\overline{\psi }}_{t{L}_w}[L_{\mathrm{adj}}({\lambda }_V)]$, while empty circles refer to ${\overline{\psi }}_{t{L}_w}[L_{\mathrm{adj}}({\lambda }_N)]$. Gray error bars represent the sample variance, darker error bars represent the standard deviation $\pm {\sigma }_{\overline{\psi }}$. The horizontal dotted lines indicate $\pm 5\%$, while the vertical dashed line identifies the position of the AAOT site.

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Bias variation as a function of the different atmospheric optical conditions appears not very significant, besides at 670 nm in the right vicinity of the coast for the sole parameter $\alpha$. Conversely, biases show an even stronger dependence on the seasonal variation of observation conditions (Figs. 10 and 11) when compared to results for AC-1 schemes. Values still nearly double when the viewing angle increases from 5° to 50° (results for ${\theta }_v=20°$ are not shown, but indicate values very similar to those determined for ${\theta }_v=5°$), while the highest misestimates now occur in mid-summer when the vegetation is mature and the land albedo in the NIR is at its maximum (Fig. 3). Conversely, biases are the lowest in early spring, when the land albedo in the NIR is low and the water reflectance is higher than the average. The large standard deviations are mainly induced by the variance of the land albedo in the NIR and by the standard deviation of the water-leaving radiance at the considered wavelength.

 

Fig. 10. Intra-annual values of ${\overline{\psi }}_{t{L}_w}$ at representative wavelengths for the considered AC-2 scheme and for ${\theta }_v=5°$. Symbols are as in Fig. 3. Gray error bars represent the sample variance, darker error bars represent the standard deviation $\pm {\sigma }_{\overline{\psi }}$. The horizontal dotted lines indicate $\pm 5\%$, while the vertical dashed line identifies the position of the AAOT site.

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Fig. 11. As in Fig. 10 but for ${\theta }_v=50°$.

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It is finally recalled that the land albedo at 765 nm was obtained through interpolation from climatological data at 659 and 858 nm assuming a “typical” cropland spectral signature [9]. Cropland vegetation is characterized by a steep reflectance slope around 700 nm (the so-called red edge) that consistently varies with crop type, phenological state and moisture content, and with soil contribution. To account for this variability and in agreement with cropland signatures from the Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) [30] and the United States Geological Survey (USGS) [31] spectral libraries, ${\rho }_l(765)$ was obtained assuming a ratio $\mathfrak{R}({\rho }_l)={\rho }_l(765)/{\rho }_l(865)$ [hereafter $\mathfrak{R}(x)=x(765)/x(865)$] ranging between 0.7 and 0.95 with an average value of 0.83. Figure 12 illustrates how remarkable is the sensitivity of ${\overline{\psi }}_{t{L}_w}$ on $\mathfrak{R}({\rho }_l)$. Biases drastically decrease when the land albedo in the NIR is spectrally steeper, while they increase when it is spectrally flatter. Such a strong dependence of ${\overline{\psi }}_{t{L}_w}$ on $\mathfrak{R}({\rho }_l)$ is discussed in Subsection 3.B.1.

 

Fig. 12. Annual average values of ${\overline{\psi }}_{t{L}_w}$ at representative wavelengths for the considered AC-2 scheme by assuming $\mathfrak{R}({\rho }_l)=0.75$ (empty circles), 0.83 (stars), 0.95 (filled circles). Gray error bars represent the sample variance, darker error bars represent the standard deviation $\pm {\sigma }_{\overline{\psi }}$. The horizontal dotted lines indicate $\pm 5\%$, while the vertical dotted line identifies the position of the AAOT site.

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1. Adjacency-Induced Biases on the Atmospheric Radiance

Figure 13 shows the spatial trend of the average bias induced by adjacency effects on the atmospheric radiance at 865 nm, ${\overline{\psi }}_{L_A}(865)$, which highlights an increasing overestimate when approaching the coast.

 

Fig. 13. Values of annual average ${\overline{\psi }}_{L_A}$ at 865 nm for the considered AC-2 scheme. Gray error bars represent the sample variance, darker error bars (not visible) represent the standard deviation $\pm {\sigma }_{\overline{\psi }}$. The vertical dotted line identifies the position of the AAOT site.

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Misestimates of the atmospheric radiance at 865 nm impact the retrieval of the aerosol load. A spectral variation of ${\psi }_{L_A}(\lambda )$ only occurs if adjacency effects impact the retrieval of the aerosol type, too. By applying to Eq. (15) the Taylor expansion to first-order approximation [32], the spectral ratio of ${\psi }_{L_A}$ can be expressed as (see Appendix B)

Equation (17) clearly indicates that until $\mathfrak{R}(L_{\text{adj}})\sim ϵ$, term ${\psi }_{L_A}(\lambda )$ is spectrally invariant, i.e., adjacency effects do not perturb the retrieval of $ϵ$ and hence that of the aerosol type. As soon as $\mathfrak{R}(L_{\text{adj}})$ becomes larger (smaller) than $ϵ$, ${\psi }_{L_A}$ becomes increasingly higher (lower) for decreasing wavelengths. In other words, as soon as the spectral dependence of the adjacency radiance in the NIR departs from that of the aerosol radiance, ${\psi }_{L_A}(\lambda )$ consistently varies toward the blue.

Figure 14 further indicates a correlation between simulated ${\overline{\psi }}_{L_A}(\lambda )/{\overline{\psi }}_{L_A}(865)$ and the ratio of the land albedo in the NIR, $\mathfrak{R}({\rho }_l)$. Indeed, excluding observations for very large ${\theta }_0$ (for which the contribution from the land is highly reduced) or water target elements closed to land elements located in the solar half-plane [for which the term $W$ in Eq. (1) is more significant], the adjacency radiance in the NIR (where ${\rho }_l$ is much larger than the sea albedo) can be simply expressed as [see Eq. (1)]

 

Fig. 14. Plot of ${\overline{\psi }}_{L_A}(\lambda )/{\overline{\psi }}_{L_A}(865)$ at the AAOT for the considered AC-2 scheme and for different values of $\lambda$ as a function of $\mathfrak{R}({\rho }_l)$.

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Equation (18) implies that $\mathfrak{R}(L_{\mathrm{adj}})\sim \mathfrak{R}({\rho }_l)·\mathfrak{R}(C)$. Thus Eq. (17) can be rewritten as

Equation (19) explains the correlation between ${\psi }_{L_A}(\lambda )/{\psi }_{L_A}(865)$ and $\mathfrak{R}({\rho }_l)$ observed in Fig. 14.

The previous analysis allows drawing three important general considerations:

These general considerations, determined for an atmospheric correction scheme assuming a power-law extrapolation of the aerosol radiance, are nonetheless valid for the scheme proposed by Gordon and Wang [4] that accounts for multiple scattering and determines $ϵ$ from $\mathfrak{R}(L_A)$. In this case the aerosol type retrieval is not perturbed as long as $\mathfrak{R}(L_{\mathrm{adj}})\sim \mathfrak{R}(L_A)$.

By furthermore observing that the spectral dependence of the function $C$ in the NIR is similar to that of $L_{\mathrm{path}}$, the value of $\mathfrak{R}(L_{\mathrm{adj}})$, approximated as $\mathfrak{R}({\rho }_l)·\mathfrak{R}(C)$, can be determined with an uncertainty of $\sim 7\%$ (see Fig. 15) as

 

Fig. 15. Plot of $\mathfrak{R}(L_{\mathrm{adj}})$ versus $\mathfrak{R}({\rho }_l)·\mathfrak{R}(L_{\mathrm{path}})$ for all test cases characterized by $\nu =1.7$ and ${\theta }_0<65°$, and assuming $\mathfrak{R}({\rho }_l)=0.75$, 0.83, and 0.95. rmsd is the root mean square deviation, $\psi$ and $|\psi |$ represent the average bias and the average absolute bias of data all in percent.

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By inserting Eq. (20) in Eq. (19) and by further recalling Eq. (14),

Equation (21) indicates that ${\psi }_{L_A}(\lambda )\sim {\psi }_{L_A}(865)$ when $\mathfrak{R}({\rho }_l)\sim \mathfrak{R}(L_A/L_{\mathrm{path}})$. For the considered test cases $\mathfrak{R}(L_A/L_{\mathrm{path}})\sim 0.86$, and notably in Fig. 14 ${\overline{\psi }}_{L_A}(\lambda )\sim {\overline{\psi }}_{L_A}(865)$ for $\mathfrak{R}({\rho }_l)\sim 0.86$. As a rule of thumb Eq. (21) suggests that whenever $\mathfrak{R}(L_A/L_{\mathrm{path}})$ resembles $\mathfrak{R}({\rho }_l)$, adjacency effects are foreseen to not affect the retrieval of the spectral dependence of the atmospheric radiance, and thus the selection of the aerosol type. With such a condition, the adjacency effects only affect the retrieval of the aerosol load and induce a spectrally invariant atmospheric bias. Conversely, when $\mathfrak{R}({\rho }_l)$ diverges from $\mathfrak{R}(L_A/L_{\mathrm{path}})$, the atmospheric bias becomes spectrally dependent. When $\mathfrak{R}({\rho }_l)$ exceeds $\mathfrak{R}(L_A/L_{\mathrm{path}})$ the bias increases toward the blue and the adjacency effects are likely to induce a large overestimate of the atmospheric radiance at blue wavelengths. In the opposite case, the bias decreases for decreasing wavelengths and misestimates of the atmospheric radiance might even become negligible. The approximate relationship (21) also provides a rough estimate of ${\psi }_{L_A}$ once ${\psi }_{L_A}(865)$ and $\mathfrak{R}({\rho }_l)$ are known. This is shown in Fig. 16 that illustrates the spectral values of ${\overline{\psi }}_{L_A}$ quantified at the AAOT site via simulations and additionally through Eq. (21) for $\mathfrak{R}({\rho }_l)=0.75$, 0.83, and 0.95.

 

Fig. 16. Spectral plot of ${\overline{\psi }}_{L_A}(\lambda )$ at the AAOT for the considered AC-2 scheme. Empty circles, stars, and filled circles refer to simulated data for $\mathfrak{R}({\rho }_l)=0.75$, 0.83, and 0.95, respectively. Crosses represent the corresponding values obtained with Eq. (21).

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2. Simulated Biases at the AAOT

Annual average biases at the AAOT are theoretically quantified up to approximately $-15\%$ at 412, 443, and 670 nm (Fig. 17 upper panel). Notably, average absolute differences $\overline{|d|}=⟨|\mathrm{\Delta }t{L}_w|⟩$ are significantly lower at 670 nm (Fig. 17 lower panel) where the water-leaving signal is lower, but the lower confidence on results at blue wavelengths (see Section 3.A) needs also be recalled.

 

Fig. 17. Annual average spectral values of (upper panel) ${\overline{\psi }}_{t{L}_w}$ and (lower panel) $\overline{|d|}$ $[{\mathrm{Wm}}^{-2}{\mathrm{\mu m}}^{-1}{\mathrm{sr}}^{-1}]$ at the AAOT for the considered AC-2 scheme. Black bars represent the standard deviation $\pm {\sigma }_{\overline{\psi }}$, while gray bars are the sample variance. The horizontal dotted lines in the upper panel indicate $\pm 5\%$.

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Biases quantified for specific intra-annual periods are given in Fig. 18 for ${\theta }_v=5°$ and ${\theta }_v=50°$. For quasi-nadir observations ${\overline{\psi }}_{t{L}_w}$ at 412 spans from $-7\%$ (early spring) to $-18\%$ (mid-summer). For the same periods values at 670 nm vary from $-4\%$ to $-27\%$. When the viewing angle is increased to 50°, ${\overline{\psi }}_{t{L}_w}$ doubles. It must be noted that the standard deviation for larger values is also pronounced.

 

Fig. 18. Intra-annual spectral values of ${\overline{\psi }}_{t{L}_w}$ at the AAOT for the considered AC-2 scheme, for ${\theta }_v=5°$ (upper panel) and 50° (lower panel). Bars represent the standard deviation. Symbols are as in Fig. 3. The horizontal dotted lines indicate $\pm 5\%$.

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A recent evaluation of the average biases between in situ and MODIS (on board both Aqua and Terra platforms), and SeaWiFS satellite-retrieved water-leaving radiance at the AAOT for the period 2002–2010, showed values ranging from $-14\%$ to $+4\%$ at 412; $-4\%$ and $+1\%$ at 490 nm; $-5\%$ to $-2\%$ at 555 nm, and from $-26\%$ and $-15\%$ at 670 nm (or equivalent) [33]. A direct comparison of results from the previous and this study is possible assuming that biases from match-ups between satellite and in situ water-leaving radiances are equivalent at surface or sensor level. Biases determined in this study (see Fig. 17 upper panel) are consistent with the mean experimental values reported at 555 and 670 nm, but they appear larger at 412 and 490 nm. Differences might be explained by the uncertainties on the actual values of ${\rho }_l$ (see Fig. 12) and approximations and simplifications applied in the present study, but also by possible additional mechanisms of compensation within the operational correction code.

Simplifications applied in the present study are not addressed. They comprise (i) neglecting the wind direction (which might lead to different directional reflectance properties of the sea surface); (ii) assuming an isotropic land surface (which might affect simulations of adjacency effects toward the NIR, as shown in Fig. 22 of [9]); and (iii) adopting the single-scattering approximation in the spectral extrapolation of adjacency perturbations in the derived atmospheric radiance.

A possible mechanism of compensation, which is hereafter investigated, could be triggered by the NIR correction algorithm [34] included in SeaDAS. This algorithm iteratively determines a non-null remote sensing reflectance $R_{\mathrm{rs}}$ at NIR wavelengths from retrieved values of Chl and $R_{\mathrm{rs}}$ at 443, 555, and 670 nm [34]. In the analysis of biases due to adjacency effects it was assumed an a priori exact knowledge of the water-leaving radiance in the NIR. It is likely that misestimates of $R_{\mathrm{rs}}$ (and hence of $L_w$) in the NIR may mask adjacency perturbations.

The possible occurrence of misestimates of the NIR water-leaving radiance has been investigated by selecting a set of 163 SeaWiFS images acquired in correspondence of the AAOT between 1997 and 2008, all qualified for match-up construction with corresponding in situ data indicating clear water. The empirical criterion $n{L}_w(670)<0.1{\mathrm{Wm}}^{-2}{\mathrm{\mu m}}^{-1}{\mathrm{sr}}^{-1}$ has been assumed to indicate null water-leaving radiance at NIR wavelengths. Figure 19 illustrates the normalized frequency histograms of ${\zeta }_w=(t{L}_w/L_{\mathrm{tot}})·100$ as retrieved by SeaDAS at 765 and 865 nm. Assuming null the true reference value, the plotted data clearly indicate an overestimate of $L_w$ at NIR in $\sim 85\%$ of the analyzed cases, which suggests that SeaDAS might indeed misinterpret part of the adjacency radiance as water-leaving radiance.

 

Fig. 19. Normalized frequency histogram distribution of ${\zeta }_w=(t{L}_w/L_{\mathrm{tot}})\cdot 100$ derived with SeaDAS in correspondence of the AAOT at 765 (upper panel) and 865 nm (lower panel) for clear water conditions verified with in situ data. The total number of cases is $N_{\mathrm{tot}}=163$. The black bar corresponds to cases for which ${\zeta }_w=0\%$ ($N_{{\zeta }_w=0}=24$ at 865 nm, with 23 corresponding nihil values at 765 nm), the gray bars correspond to cases for which ${\zeta }_w>0\%$.

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A comparison between ${\overline{\psi }}_{t{L}_w}$ values simulated by assuming an exact knowledge of $L_w$ at NIR (as in the upper panel of Fig. 17) and alternatively the average overestimate of $L_w$ observed in Fig. 19 (i.e., $t{L}_w\sim 0.3\%$ and $\sim 0.2\%$ of $L_{\mathrm{tot}}$ at 765 and 865 nm, respectively) shows an appreciable decrease of adjacency-induced biases in the latter case, particularly significant at blue and red wavelengths (see Fig. 20). This suggests that mechanisms of compensation in the operational correction scheme might indeed mask biases induced by adjacency effects.

 

Fig. 20. Spectral values of ${\overline{\psi }}_{L_w}$ at the AAOT for the considered AC-2 scheme assuming an exact knowledge of $L_w$ in the NIR (empty stars), and assuming an overestimate of $t{L}_w$ equal to 0.3% and 0.2% of $L_{\mathrm{tot}}$ at 765 and 865 nm, respectively (filled stars). Error bars represent the standard deviation $\pm {\sigma }_{\overline{\psi }}$ (black) and the sample variance (gray). The horizontal dotted lines indicate $\pm 5\%$.

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Results displayed in Figs. 12 and 20 suggest that accounting for the uncertainties in ${\rho }_l$ at NIR wavelengths and for possible mechanisms of compensation in the operational correction code, theoretical adjacency-induced biases might become consistent with differences observed between satellite and in situ data.

To further investigate consistency between biases from theoretical and match-up analysis, the intra-annual trend of ${\overline{\psi }}_{t{L}_w}$ and $\overline{d}=⟨\mathrm{\Delta }t{L}_w⟩$ at the AAOT has been analyzed (Fig. 21). Departures $\mathrm{\Delta }{\overline{\psi }}_{t{L}_w}$ and $\mathrm{\Delta }\overline{d}$ from simulated average biases and absolute differences are shown in Fig. 21 for selected intra-annual periods (upper panels) and solar zenith angles (lower panels). Results are consistent with trends provided and discussed in [33].

 

Fig. 21. Spectral values of $\mathrm{\Delta }{\overline{\psi }}_{L_w}$ and $\mathrm{\Delta }\overline{d}[{\mathrm{Wm}}^{-2}{\mathrm{\mu m}}^{-1}{\mathrm{sr}}^{-1}]$ at the AAOT for the considered AC-2 scheme. In the upper panels differences are between intra-annual and yearly average simulated data (with symbols as in Fig. 3). Error bars represent $\pm \sigma$. In the lower panels differences are between results for ${\theta }_0=25°$, 45°, 65° and average simulated data.

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4. SUMMARY AND CONCLUSIONS

Accurate simulations of the total signal reaching an ocean color satellite sensor observing coastal areas under realistic observation conditions [9] have been utilized to estimate biases induced by adjacency effects in the retrieval of the water-leaving radiance.

Simulations have been performed for a transect in the northern Adriatic Sea crossing the Venice Lagoon and intercepting the AAOT site used since 1995 for the validation of ocean color products. The set of selected test cases represent observation conditions typically encountered in the region by MODIS-Aqua, MODIS-Terra, SeaWiFS, MERIS sensors and, in principle by the OLCI sensor, too. The optical properties of the water have been extracted from a long record of in situ measurements, while those of the land from a climatological record assuming a cropland ecosystem. Yearly average values, in the presence of mean and extreme atmospheric conditions, as well as representative intra-annual values have been selected. The influence of different aerosol profiles, including the presence of volcanic aerosols or thin cirrus clouds was not accounted for, but it will be matter of future detailed investigations.

The impact of adjacency effects on the accuracy of derived remote sensing products strictly depends on the retrieval procedure. Two main categories of atmospheric correction schemes have been considered: AC-1 and AC-2, identifying methods not relying and alternatively requiring the determination of the atmospheric optical properties directly from the remote sensing signal. Among the latter methods, the analysis specifically focused on an AC-2 scheme deriving the atmospheric contributions from NIR wavelengths.

For both schemes, biases display a steep decrease with increasing distance from the coast. Perturbations do not show a significant sensitivity to the atmospheric parameters in their selected range of variation; conversely, they exhibit a considerable dependence on the viewing angle (values nearly double when it varies from 5° to 50°) and a large seasonal spread (particularly at blue and red wavelengths). Nevertheless, the amplitude, sign, as well as seasonal dependence of biases is consistently different in the two analyzed typologies of correction schemes.

Misestimates for AC-1 schemes are only due to adjacency effects at the considered wavelength, and the water signal is on average underestimated up to 510 nm, and overestimated at longer wavelengths. Mean biases are approximately within $\pm 5\%$ throughout the study transect in the spectral range 490–555 nm, while at blue and red wavelengths (where the water-leaving radiance is low) they show larger values, reaching $-21\%$ and $+33\%$ at 412 and 670 nm at the coast, respectively. The largest underestimates occur at blue wavelengths in winter, while the largest overestimates are observed at red wavelengths in late spring.

In AC-2 schemes, misestimates of the atmospheric contributions induced by adjacency effects at NIR wavelengths sum up with misestimates observed for AC-1 schemes. Biases become all negative and more pronounced toward the blue and red ends of the spectrum, where the water signal is lower. For the specific case of an AC-2 scheme adopting a power-law spectral extrapolation of the atmospheric contributions from NIR wavelengths, as for the single-scattering approximation of the Gordon and Wang algorithm [4], simulated average biases ${\overline{\psi }}_{t{L}_w}$ nearby the coast reach $-59\%$, $-28\%$, $-16\%$, and $-74\%$ at 412, 490, 555, and 670 nm, respectively. Misestimates are the highest in summer when the vegetation is mature and the land albedo in the NIR is at its maximum. Conversely, the lowest biases occur in early spring when the land albedo in the NIR is low and the water reflectance is higher than the average.

Obviously, the impact of adjacency effects at each single wavelength will depend on the radiometric sensitivity of the sensor at that wavelength.

For AC-2 schemes, biases induced by adjacency effects on the atmospheric radiance ${\psi }_{L_A}$ show a relevant sensitivity to the ratio $\mathfrak{R}(L_{\mathrm{adj}})$ between the adjacency radiance at those wavelengths from which the atmospheric properties are inferred (i.e., 765 and 865 nm, or equivalent). The analysis shows that until $\mathfrak{R}(L_{\mathrm{adj}})$ resembles the spectral dependence of the aerosol radiance ($ϵ$ in single scattering), the determination of the aerosol type is not affected, and ${\psi }_{L_A}$ is spectrally invariant. As soon as $\mathfrak{R}(L_{\mathrm{adj}})$ exceeds (is less than) $ϵ,{\psi }_{L_A}$ increases (decreases) toward the blue wavelengths. Results additionally show that (excluding low sun elevations and water target elements close to land in the solar half-plane) there is a correlation between $\mathfrak{R}(L_{\mathrm{adj}})$ and the ratio $\mathfrak{R}({\rho }_l)$ between corresponding land albedos.

In correspondence of the AAOT and for an AC-1 scheme, ${\overline{\psi }}_{t{L}_w}$ is estimated to slightly exceed $\pm 5\%$ only for slanted observations at 412 and 670 nm. For the considered AC-2 scheme ${\overline{\psi }}_{t{L}_w}$ is instead quantified lower than $-5\%$ only at 555 nm, and rising up to $\sim -15\%$ at blue and red wavelengths.

Theoretical biases at 555 and 670 nm for the considered AC-2 scheme are consistent with differences observed at the AAOT between in situ and MODIS-Aqua, MODIS-Terra, and SeaWiFS data for the period 2002–2010 [33]. Conversely, the larger biases at 412 and 490 nm might be explained by the lower confidence on results at blue wavelengths (where adjacency effects are low and its simulation is more sensitive to approximations in the modeling of the propagating system), by the uncertainties in the land albedo at NIR wavelengths, and by the simple power-law spectral dependence of the aerosol radiance adopted in modeling ${\psi }_{L_A}$. Nonetheless, mechanisms of compensation in the operational correction code could also apply, as documented by overestimates of the water-leaving radiance in the NIR by SeaDAS. This might also explain why a previous study on the variation of satellite-derived products along transects starting from the coast and intercepting selected AERONET-OC sites did not provide firm evidence of appreciable adjacency effects [27].

Notably, the intra-annual variation of simulated ${\overline{\psi }}_{t{L}_w}$ is consistent with that determined from the analysis of in situ and satellite match-ups.

Results presented in this manuscript for an AC-1 scheme are representative for mid-latitude coastal regions characterized by a deciduous vegetation type in the absence of snow. In the presence of non-deciduous vegetation, biases would likely show a more limited seasonal variation and higher average values. The presence of snow would lead to radically different results, likely characterized by consistent positive biases in the blue end part of the spectrum. It is emphasized that results obtained for the considered AC-2 scheme are instead valid only for a vegetation cover whose albedo slope at NIR reference wavelengths, $\mathfrak{R}({\rho }_l)$, ranges between 0.75 and 0.95, and for an aerosol characterized by an Ångström exponent between 1.4 and 1.9. Other $\mathfrak{R}({\rho }_l)$ values in combination with other aerosol spectral shapes might lead to different results.

To conclude, overall results suggest that observed biases between in situ and satellite-retrieved water-leaving radiance, as well as their seasonal trends, might be at least partly explained by perturbations from the nearby land. This suggests the need to include an evaluation of adjacency effects in the validation process of satellite products through in situ radiometric measurements performed at coastal sites. It additionally indicates the urgent need to account for adjacency perturbations in the retrieval of the water-leaving radiance from satellite data from coastal regions. In particular, results once more underline that schemes extrapolating the atmospheric optical properties from the NIR bands are extremely sensitive to any perturbation affecting the radiometric signal at these wavelengths [32]. Hence care should be put (i) to avoid misinterpreting land contributions as turbid water ones, and (ii) to accurately quantify land albedos at NIR wavelengths. This would suggest considering the determination of land products at ocean color NIR wavelengths when coastal applications are regarded as relevant.

APPENDIX A

List of Most Used Symbols

View Table

APPENDIX B

With reference to Eq. (14), the term $\mathrm{\Delta }{L}_A({\lambda }_V)$ can be written as

(B1)$$\mathrm{\Delta }{L}_A({\lambda }_V)\equiv {\widehat{L}}_A({\lambda }_V)-L_A({\lambda }_V)\sim {\widehat{ϵ}}^{\beta }{\widehat{L}}_A(865)-{ϵ}^{\beta }{L}_A(865),$$

and further accounting for Eq. (16),

(B2)$$\begin{gathered} \mathrm{\Delta }{L}_A({\lambda }_V)\sim {\left[\frac{L_A(765)+L_{\mathrm{adj}}(765)}{L_A(865)+L_{\mathrm{adj}}(865)}\right]}^{\beta }[L_A(865)+L_{\mathrm{adj}}(865)]-{ϵ}^{\beta }{L}_A(865) \\ ={[L_A(765)+L_{\mathrm{adj}}(765)]}^{\beta }[L_A(865)+L_{\mathrm{adj}}(865){]}^{1-\beta } \\ -{ϵ}^{\beta }{L}_A(865). \end{gathered}$$

By using the Taylor series expansion ${(x+a)}^n=x^n+n{x}^{n-1}a+(n-1)x^{n-2}{a}^2+\cdots$ in the first-order approximation (i.e., neglecting terms in $L_{\mathrm{adj}}$ of second or higher order), Eq. (B2) becomes

(B3)$$\mathrm{\Delta }{L}_A({\lambda }_V)\sim [L_A{(765)}^{\beta }+\beta L_A{(765)}^{\beta -1}{L}_{\mathrm{adj}}(765)][L_A{(865)}^{1-\beta }+(1-\beta )L_A{(865)}^{-\beta }{L}_{\mathrm{adj}}(865)]-{ϵ}^{\beta }{L}_A(865).$$

Simple computation leads to

(B4)$$\mathrm{\Delta }{L}_A({\lambda }_V)\sim (1-\beta ){ϵ}^{\beta }{L}_{\mathrm{adj}}(865)+\beta {ϵ}^{\beta -1}{L}_{\mathrm{adj}}(765),$$

which divided by $L_A({\lambda }_V)={ϵ}^{\beta }{L}_A(865)$, gives

(B5)$${\psi }_{L_A}({\lambda }_V)\sim {\psi }_{L_A}(865)[1+\beta ·(\frac{\mathfrak{R}(L_{\mathrm{adj}})}{ϵ}-1)],$$

with ${\psi }_{L_A}(865)=\frac{L_{\mathrm{adj}}(865)}{L_A(865)}$.

Funding

Joint Research Centre (JRC).

Acknowledgment

The authors wish to acknowledge the three anonymous reviewers for their valuable comments.

REFERENCES

1. Global Climate Observing System, “Systematic observation requirements for satellite-based data products for climate 2011 Update GCOS-154,” (World Meteorological Organization, 2011).

2. J. Yang, P. Gong, R. Fu, M. Zhang, J. Chen, S. Liang, B. Xu, J. Shi, and R. Dickinson, “The role of satellite remote sensing in climate change studies,” Nat. Clim. Change 3, 875–883 (2013). [CrossRef]  

3. V. Kiselev, B. Bulgarelli, and T. Heege, “Sensor independent adjacency correction algorithm for coastal and inland water systems,” Remote Sens. Environ. 157, 85–95 (2015). [CrossRef]  

4. H. W. Gordon and M. Wang, “Retrieval of water-leaving radiance and aerosol optical thickness over the oceans with SeaWiFS: a preliminary algorithm,” Appl. Opt. 33, 443–452 (1994). [CrossRef]  

5. D. Antoine and A. Morel, “A multiple scattering algorithm for atmospheric correction of remotely sensed ocean colour (MERIS instrument): principle and implementation for atmospheres carrying various aerosols including absorbing ones,” Int. J. Remote Sens. 20, 1875–1916 (1999). [CrossRef]  

6. J. Otterman and R. S. Fraser, “Adjacency effects on imaging by surface reflection and atmospheric scattering: cross radiance to zenith,” Appl. Opt. 18, 2852–2860 (1979). [CrossRef]  

7. P. Y. Deschamps, M. Herman, and D. Tanre, “Definitions of atmospheric radiance and transmittances in remote sensing,” Remote Sens. Environ. 13, 89–92 (1983). [CrossRef]  

8. A. Sei, “Analysis of adjacency effects for two Lambertian half-spaces,” Int. J. Remote Sens. 28, 1873–1890 (2007). [CrossRef]  

9. B. Bulgarelli, V. Kiselev, and G. Zibordi, “Simulation and analysis of adjacency effects in coastal waters: a case study,” Appl. Opt. 53, 1523–1545 (2014). [CrossRef]  

10. G. Zibordi, F. Mélin, J. Berthon, B. Holben, I. Slutsker, D. Giles, D. D’Alimonte, D. Vandemark, H. Feng, and G. Schuster, “AERONET-OC: a network for the validation of ocean color primary products,” J. Atmos. Ocean. Technol. 26, 1634–1651 (2009). [CrossRef]  

11. B. A. Franz, S. W. Bailey, P. J. Werdell, and C. R. McClain, “Sensor-independent approach to the vicarious calibration of satellite ocean color radiometry,” Appl. Opt. 46, 5068–5082 (2007). [CrossRef]  

12. Z. Ahmad, B. A. Franz, C. R. McClain, E. J. Kwiatkowska, J. Werdell, E. P. Shettle, and B. N. Holben, “New aerosol models for the retrieval of aerosol optical thickness and normalized water-leaving radiances from the SeaWiFS and MODIS sensors over coastal and open oceans,” Appl. Opt. 49, 5545–5560 (2010). [CrossRef]  

13. C. D. Mobley and C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).

14. J. V. Martonchik, C. J. Bruegge, and A. H. Strahler, “A review of reflectance nomenclature used in remote sensing,” Remote Sens. Rev. 19, 9–20 (2000). [CrossRef]  

15. J. F. Berthon, G. Zibordi, J. P. Doyle, S. Grossi, D. van der Linde, and C. Targa, “Coastal atmosphere and sea time series (CoASTS), part 2: date analysis,” NASA Technical Memorandum—SeaWIFS Postlaunch Technical Report Series, no. 20 (2002), pp. 1–25.

16. J. Berthon, F. Mélin, and G. Zibordi, “Ocean colour remote sensing of the optically complex European seas,” in Remote Sensing of the European Seas (Springer, 2008), pp. 35–52.

17. E. G. Moody, M. D. King, C. B. Schaaf, and S. Platnick, “MODIS-derived spatially complete surface albedo products: spatial and temporal pixel distribution and zonal averages,” J. Appl. Meteor. Climatol. 47, 2879–2894 (2008). [CrossRef]  

18. B. Pinty, A. Lattanzio, J. V. Martonchik, M. M. Verstraete, N. Gobron, M. Taberner, J.-L. Widlowski, R. E. Dickinson, and Y. Govaerts, “Coupling diffuse sky radiation and surface albedo,” J. Atmos. Sci. 62, 2580–2591 (2005). [CrossRef]  

19. V. B. Kisselev, L. Roberti, and G. Perona, “Finite-element algorithm for radiative transfer in vertically inhomogeneous media: numerical scheme and applications,” Appl. Opt. 34, 8460–8471 (1995). [CrossRef]  

20. B. Bulgarelli, V. Kisselev, and L. Roberti, “Radiative transfer in the atmosphere-ocean system: the finite-element method,” Appl. Opt. 38, 1530–1542 (1999). [CrossRef]  

21. B. Bulgarelli and J. Doyle, “Comparison between numerical models for radiative transfer simulation in the atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transfer 86, 315–334 (2004). [CrossRef]  

22. B. Bulgarelli, G. Zibordi, and J. Berthon, “Measured and modeled radiometric quantities in coastal waters: toward a closure,” Appl. Opt. 42, 5365–5381 (2003). [CrossRef]  

23. B. Bulgarelli and G. Zibordi, “Remote sensing of ocean colour: accuracy assessment of an approximate atmospheric correction method,” Int. J. Remote Sens. 24, 491–509 (2003). [CrossRef]  

24. B. Bulgarelli and F. Mélin, “SeaWiFS-derived products in the Baltic Sea: performance analysis of a simple atmospheric correction algorithm,” Oceanologia 45, 655–677 (2003).

25. G. Zibordi and B. Bulgarelli, “Effects of cosine error in irradiance measurements from field ocean color radiometers,” Appl. Opt. 46, 5529–5538 (2007). [CrossRef]  

26. V. Kisselev and B. Bulgarelli, “Reflection of light from a rough water surface in numerical methods for solving the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 85, 419–435 (2004). [CrossRef]  

27. G. Zibordi, J. F. Berthon, F. Mélin, D. D’Alimonte, and S. Kaitala, “Validation of satellite ocean color primary products at optically complex coastal sites: Northern Adriatic Sea, Northern Baltic Proper and Gulf of Finland,” Remote Sens. Environ. 113, 2574–2591 (2009). [CrossRef]  

28. H. Yang and H. R. Gordon, “Remote sensing of ocean color: assessment of the water-leaving radiance bidirectional effects on the atmospheric diffuse transmittance,” Appl. Opt. 36, 7887–7897 (1997). [CrossRef]  

29. G. Zibordi, F. Mélin, and J. F. Berthon, “Intra-annual variations of biases in remote sensing primary ocean color products at a coastal site,” Remote Sens. Environ. 124, 627–636 (2012). [CrossRef]  

30. A. M. Baldridge, S. J. Hook, C. I. Grove, and G. Rivera, “The ASTER spectral library version 2.0,” Remote Sens. Environ. 113, 711–715 (2009). [CrossRef]  

31. R. N. Clark, G. A. Swayze, R. Wise, K. E. Livo, T. M. Hoefen, R. Kokaly, and S. J. Sutley, USGS digital spectral library splib06a: U.S. Geological Survey, 2007, http://speclab.cr.usgs.gov/spectral.lib06, accessed 21 Apr. 2016.

32. C. Hu, K. L. Carder, and F. E. Muller-Karger, “How precise are SeaWiFS ocean color estimates? Implications of digitization-noise errors,” Remote Sens. Environ. 76, 239–249 (2001). [CrossRef]  

33. G. Zibordi, F. Mélin, and J. F. Berthon, “Trends in the bias of primary satellite ocean-color products at a coastal site,” IEEE Geosci. Remote Sens. Lett. 9, 1056–1060 (2012). [CrossRef]  

34. S. W. Bailey, B. A. Franz, and P. J. Werdell, “Estimation of near-infrared water-leaving reflectance for satellite ocean color data processing,” Opt. Express 18, 7521–7527 (2010). [CrossRef]