1. Introduction

While satellite ocean color can rely on consolidated knowledge of the optical properties of the oceanic waters, intense investigations are still needed to correctly interpret the more complex optical properties of coastal regions. The higher level of complexity derives from the simultaneous presence of not-covarying in-water optically significant components (i.e., pigments, colored dissolved organic matter, and suspended sediments), eventual sea bottom reflectance, and adjacency effects induced by the reflectance of the nearby mainland.

Adjacency effects always occur in the presence of a scattering atmosphere over a nonuniform reflecting surface, which causes the radiance from high-reflectivity areas to spill over neighboring low-reflectivity regions, thus modifying their apparent brightness [1].

The problem of adjacency effects for satellite land observations has been extensively investigated for decades (see [2–5] and references therein), and has led to the implementation of atmospheric correction codes that take into account adjacency perturbations (e.g., [6]). Conversely, fewer studies specifically addressed adjacency effects in coastal waters [7–12], and most of the operational ocean color atmospheric correction codes assume a uniform underlying reflecting surface [13,14]. Processors for the correction of adjacency effects in coastal regions are only available at a prototypal stage, such as the Improve Contrast between Ocean and Land (ICOL) processor [9,15].

When neglected, adjacency effects become a source of spectral perturbations in satellite data. Considering that the determination of ocean color products is very sensitive to even small sources of noise [16], accurate evaluation of adjacency effects is of the greatest relevance in ocean color remote sensing of coastal waters.

Within such a frame, the main objective of this work is an extensive and highly accurate theoretical evaluation of the adjacency effects in coastal regions for realistic and representative conditions, accounting for the actual sensor radiometric resolution. Indeed, a true knowledge of the actual amount of adjacency effects and a deep understanding of their physical dependences is a fundamental and necessary prerequisite for any successive quantification of adjacency-induced perturbations in satellite-retrieved products and for any further development of simplified approaches for the operational correction of adjacency effects.

Previous theoretical evaluations of the adjacency effects in coastal waters were performed applying one or more simplifying assumptions, such as nadir observation [11], straight coastline [11], Lambertian reflecting surfaces [7,11], single scattering approximation [9], and exponential decay of the environment effects [9,11]. Additionally, the uncertainties on simulated results were not explicitly discussed, and the radiometric sensitivity of the measuring sensors, which implicitly defines the threshold for discriminating adjacency effects from noise, was not specifically taken into consideration.

On the basis of the above considerations, this work attempts the simulation of adjacency effects with an improved level of accuracy, properly accounting for the involved uncertainties. This is achieved by previously solving the radiative transfer equation (RTE) for representative test cases taking into consideration off-nadir views, the Fresnel reflectance of a wind-roughed sea surface, the actual coastal morphology, and the average radiometric resolution of ocean color sensors. The accurate evaluation of uncertainties in simulated results is performed accounting for uncertainties both in the simulation procedure and on a number of input parameters.

The selected test cases correspond to the observation conditions encountered in a littoral area of the northern Adriatic Sea hosting the Aqua Alta Oceanographic Tower (AAOT, 45.31N, 12.51E) site. Since 1995 the AAOT site has been used to validate satellite ocean color products [17], and it is the first site included in the Ocean Color component of the Aerosol Robotic Network (AERONET-OC) [18], a subnetwork established in 2006 to support the validation of satellite ocean color products through highly accurate, cross-site consistent, and globally distributed measurements performed in coastal regions at distances from the land varying from a few up to several tens of kilometers. The comprehensive multiannual record of in situ measurements performed at the AAOT [17,18] allows a precise and complete characterization of the optical properties of atmosphere and seawater, enabling the definition of realistic and seasonally dependent test cases for the analysis of the adjacency effects. Furthermore, the selected region, characterized by mid-latitude atmospheric conditions and a cropland ecosystem, is well representative of mid-latitude coastal areas covered by a deciduous vegetation type (the most diffuse in Europe) in the absence of snow. Additionally, the selected optical aerosol properties correspond to those mostly encountered at different AERONET-OC sites [19]. Results are hence expected to provide general conclusions applicable to a variety of measurement conditions and coastal regions. Finally, the extremely complex coastal pattern of the Venice Lagoon makes it optimally suitable to analyze the influence on adjacency effects of the actual coastal morphology.

It must be additionally recalled that a previous attempt to estimate adjacency effects at the AAOT site using a parametric relationship [18] showed an increase of the sea reflectance at the top-of-atmosphere (TOA) varying from a few up to more than 10% from the green to the near-infrared (NIR) center wavelengths. However, a further study on the variation of satellite-derived products along transects starting from the coast and intercepting selected AERONET-OC sites did not provide any firm evidence of appreciable adjacency effects [20]. These somehow contradictory results underline once more the need for a comprehensive investigation of the problem.

The accurate simulation of adjacency effects in the selected coastal region should provide a theoretical quantification of the adjacency contributions to TOA radiances, and allow and analysis of their dependence on different geophysical parameters, and an evaluation of the physical impact of commonly adopted approximations. Specifically, it should help in evaluating the relevance of sea surface anisotropy, actual coastal morphology, off-nadir views, and atmospheric multiple scattering within the limits established by the sensor radiometric resolution. Finally, it should provide an estimate of the horizontal range of adjacency effects taking into account the actual radiometric resolution of the measuring sensors.

The quantification of the adjacency effects is performed in terms of the adjacency radiance $L_{\mathrm{adj}}$. By identifying the target element as the sensor effective footprint given by the intersection of the sensor instantaneous field of view (IFOV) with the ground for the whole sensor integration time [21], and the background radiance $L_b$ [22] (also called environmental radiance [11]) as the radiance reflected by the background surface of such a target element and then scattered by the atmosphere in the sensor IFOV, the adjacency radiance is defined as the difference in the background radiance between the case accounting for the nonuniformity of the underlying reflecting surface and the case assuming a uniform surface.

Adjacency radiance and its contribution to the signal at the TOA are simulated here making use of (i) the newly developed Novel Adjacency Perturbation Simulator for Coastal Areas (NAUSICAA) full three-dimensional (3D) backward Monte Carlo (MC) code, whose precision is set to meet actual ocean color sensors’ radiometric resolutions, and (ii) a highly accurate plane-parallel code based on the finite element method (the FEM code) [23,24].

To reduce computing time, the adjacency radiance is modeled assuming isotropic and homogeneous land and water reflectances, allowing the decoupling of land and water optical properties from the atmospheric scattering. The anisotropic reflectance of the sea surface is instead fully accounted for. This approach, while allowing a detailed and accurate (although computer-time expensive) description of the radiance propagation through the medium properly accounting for the sea surface roughness, also offers the capability to easily vary land and water input parameters.

The simulation exercise is carried out at relevant ocean color center wavelengths along a transect crossing the Venice Lagoon and intersecting the AAOT. The study includes viewing geometries typical of ocean color sensors such as the sea-viewing wide field-of-view sensor (SeaWiFS), the moderate resolution imaging spectroradiometer (MODIS), and the medium resolution imaging spectrometer (MERIS). Simulated results are analyzed along the whole transect with specific focus on results at the AAOT site.

The applied methodology is introduced in Sections 2–4. Specifically, Section 2 presents the modeling of the adjacency radiance, Section 3 the simulation procedure, and Section 4 the selected case studies. An accurate assessment of the simulation results is given in Section 5, while results are presented and discussed in Section 6. Conclusions are drawn in Section 7.

2. Modeling of the Adjacency Radiance

In general terms, the image of an object is the sum of the convolution of the original signal $f(x,y)$ with the point-spread function $h(x,y)$ (i.e., the function describing the transmission of the original signal through the propagating system), plus the noise $n(x,y)$:

According to this formulation, the radiance received by a space sensor observing a target element located at $(x_0,y_0)$ with observation direction ${\widehat{\xi }}_v$ can be modeled as

The term $L_{\mathrm{sfc}}$ in a coastal area may be distinguished in $L_{\text{land}}$ and $L_{\mathrm{sea}}$ for land and sea elements, respectively. Indicating with the term sea the ensemble of water (designating the sole water volume) and sea surface, the term $L_{\mathrm{sea}}$ may be further separated into the radiance reflected by the sea surface, $L_{\mathrm{ss}}$, and the radiance emerging from the water, $L_w$ (the so-called water-leaving radiance), so that Eq. (2) becomes

The adjacency radiance $L_{\mathrm{adj}}$, quantified as the difference $\mathrm{\Delta }{L}_b$ in background radiance between the coastal and the open-sea cases, is obtained by subtracting Eq. (5) from Eq. (3). As such, its value can range from negative to positive:

Each land element may be further characterized by a spectral bihemispherical reflectance (BHR), defined as the ratio between the irradiance reflected by the surface and the irradiance impinging at the surface [26], often simply termed albedo. So defined, the surface albedo is not an intrinsic property of the surface but is rather determined by both the surface and the overlying atmosphere [26]. The decoupling between surface and atmospheric scattering properties can be obtained by assuming an isotropic surface reflectance. By further assuming a spatially homogeneous albedo ${\rho }_l$, $L_{\text{land}}$ can be modeled as

Analogously, by assuming that the reflectance of the water volume is isotropic and exhibits small spatial variations, $L_w$ can be modeled as

Inserting Eqs. (10) and (11), Eq. (9) finally becomes

Expression (12) allows separating the dependence of $L_{\mathrm{adj}}$ on land and water reflectance properties, decoupling these from the scattering properties of the atmosphere.

The simulation of terms $C(x_0,y_0;{\widehat{\xi }}_v)$ and $W(x_0,y_0;{\widehat{\xi }}_v)$ requires a full 3D description of the propagating system. Differently, the simulation of term $S$ can be performed with a plane-parallel radiative transfer code. The input parameters ${\rho }_l$ and $R_{\mathrm{rs}}$ can be extrapolated from satellite-derived and in situ measured data, respectively. An approximation is applied to calculate the parameter ${\rho }_{\mathrm{sea}}$, which indeed appears in Eq. (12) only to account for multiple reflections at the sea surface:

The proposed modeling additionally allows distinguishing the different contributions to the adjacency radiance. Indeed, Eq. (12) may be symbolically written as

3. Simulation Procedure

Functions $C$ and $W$ [Eq. (13)] have been computed with the NAUSICAA backward MC code, while the plane-parallel FEM numerical algorithm [23,24] has been used to simulate atmospheric optical quantities such as atmospheric spherical albedo $S$, diffuse transmittance $t$, atmospheric radiance $L_{\mathrm{atm}}$, and downward irradiance at the surface $E_d$.

The FEM numerical was extensively benchmarked with other popular RTE codes [24,28,29], and additionally used to perform radiative transfer simulations in realistic cases [29–32].

A. NAUSICAA Monte Carlo Code

The NAUSICAA MC code has been specifically developed to quantify adjacency effects in satellite ocean color data acquired in coastal regions. The code simulates fully 3D optical photon transport in a nonhomogeneous and physically realistic atmosphere bounded by a nonuniform reflecting surface. The medium is modeled on a 3D grid delimiting the largest macroscopic volumes or cells of uniform optical properties. The 3D grid is then surrounded by a background plane-parallel atmosphere bounded by a uniform reflecting surface.

In each cell of the grid, the characteristics of the optically active components (air molecules and aerosols) are specified through their optical thickness $\tau$, single scattering albedo ${\omega }_0$, and scattering phase function $p$. The latter can be either analytical [Rayleigh for air molecules and two-terms Henyey–Greenstain (TTHG) for aerosol] or given in a tabulated form. In the first case the distribution function $P$ is also expressed in analytical terms.

Each element of the underlying reflecting surface can be characterized either by a Fresnel specular reflectance (suitable for a flat sea surface) or by a bidirectional reflectance distribution function (BRDF [${\mathrm{sr}}^{-1}$]). The BRDF defines the directional reflectance properties of the surface element, describing the reflection of a parallel beam of incident light from one direction into another direction in the same hemisphere [33]. The accurate expression of the BRDF for a wind-generated rough sea surface is taken from Kisselev and Bulgarelli [34]. This is preferred to the well-known BRDF expression from Cox and Munk [35] because, although based on the same two-dimensional Gaussian sea surface wave slope distribution [35], it does not tend to infinity for angles of reflection close to the horizontal direction. The approach to model the reflectance of a wind-roughed sea surface through a BRDF is applied here to ensure an equivalent sampling of photons interacting with the land and photons interacting with the sea surface, thus avoiding the eventual occurrence of artificial biases induced by oversampling one type of photons with respect to the other.

The solar source is described by a parallel beam of monochromatic photons that originates from a far field point and uniformly impinges on the TOA.

For any given target pixel, the backward MC code releases photons from the satellite sensor in the observation direction and with initial unitary statistical weight. Time reversal is applied and photons are tracked from the detector back to the source. At first, the optical distance to the first collision point is sampled. If the photon crosses a cell boundary before interacting with the medium, a new distance to collision is calculated, and the photon is relaunched from the boundary itself. At the collision point a scattering event is forced by multiplying the photon weight by ${\omega }_0$. The type of scattering (either by molecules or aerosols) is sampled and a new propagation direction of the surviving photon is determined by retrieving the scattering angle from random sampling of the distribution function $P$. The distance to the next collision point is sampled, and the photon is launched again. If the photon reaches a specular surface element, a reflection is forced by multiplying the photon weight by the Fresnel reflection coefficient at the point of reflection, and the new direction of the photon is deterministically calculated. Otherwise, if the surface element is not specular, the photon weight is multiplied by the directional–hemispherical reflectance (DHR, i.e., the reflectance for incoming light from a single direction [33]) at the point of reflection, and the interaction is treated as in a collision event. The distance to the next collision point is sampled again, and the whole process is repeated until the statistical weight of the tracked photon falls below a preset threshold value. Alternatively, the survival of the photon packet is determined randomly through the so-called Russian roulette method [36]. At any collision, the contribution of the tracked photons to the detected signal is deterministically computed. In the presence of flat sea surface areas, the eventual contribution of the direct solar beam specularly reflected by the sea surface is accounted for. For the simulation of functions $C$ and $W$, only the contributions from photons that already interacted with the underlying reflecting surface are retained.

NAUSICAA implements ad hoc biasing techniques [36,37] to inhibit photon loss, thus reducing the computational time and keeping the statistical oscillations relatively small. Simulated results are provided with their statistical uncertainty, as determined by the selected number of initiated photons and by the threshold for photon survival [37]. An extensive analysis of the uncertainties affecting NAUSICAA computations is presented in Section 5.

B. System Modeling

Simulations are performed at typical ocean color center wavelengths $\lambda =412$, 443, 490, 510, 555, 670, 765, and 865 nm.

The solar spectral extra-atmospheric irradiance $E_0$ is taken from Thuillier et al. [38], while an average Earth–Sun distance is adopted. The atmosphere is divided into 14 plane-parallel layers resolving the vertical distributions of aerosol, ozone, and other gas molecules. Each atmospheric layer contains a variable mixture of ozone (only absorbing), other gas molecules (only scattering), and aerosol (scattering and absorbing). The spectral vertical profile of the ozone optical thickness is computed in agreement with Lacis and Hansen [39]. The ozone absorption coefficients are taken from Vigroux [40], whereas an average ozone load of 300 DU is assumed. The spectral vertical profile of the Rayleigh optical thickness is computed according to Marggraf and Griggs [41], Fröhlich and Shaw [42], and Young [43] for an average atmospheric pressure. The spectral vertical profile of the aerosol optical thickness is modeled according to Ångström [44] and Elterman [45], assuming a mean aerosol scale height of 1.2 km [45]. The molecular scattering is described by a Rayleigh phase function. The aerosol scattering phase function is approximated by a spectral TTHG analytical function whose asymmetry parameters $g_1$, $g_2$, and $a_s$ were obtained from the analysis of experimental measurements performed at the AAOT [46] as a function of $\lambda$ [nm] and of the aerosol optical thickness ${\tau }_a$ at 865 nm:

In the NAUSICAA simulations, the surface grid of nonuniform reflecting properties, centered on the AAOT and extending from 44.4° N to 46.2° N and from 11.3° E to 13.7° E, is divided into $101\times 101$ squared elements, each $2\times 2\mathrm{km}$ wide (Fig. 1). The corresponding matrix discriminating between land and sea elements [see Eq. (4)] is extracted from the operational land/sea mask used in the REMBRANDT code [47] to process SeaWiFS data from the northern Mediterranean Sea. The width of the surface grid has been defined to minimize contributions from the background along the study transect, which, starting from the coast, crosses the Venice Lagoon and intersects the AAOT [Fig. 1(b)]. A wind-roughed sea surface in the absence of whitecaps is accounted for in the MC simulations. The latter assumption is reasonable for wind speeds lower than about $7{\mathrm{ms}}^{-1}$ [48]. FEM simulations of atmospheric optical quantities do not require the presence of the sea surface.

 

Fig. 1. (a) Map of the region represented by the surface grid in the NAUSICAA simulations; the AAOT (black circle, 45.31° N, 12.51° E) is also indicated. (b) Land/sea mask: land elements are indicated in dark gray, sea elements in light gray. The black line represents the transect intersecting the AAOT (black circle).

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4. Case Studies

The methodology illustrated in the previous sections is applied here for a set of test cases representative of typical SeaWiFS, MODIS, and MERIS observation conditions for the selected region. Each test case consists of a specific combination of geometric, marine, land, and atmospheric parameters. Specifically, a standard test case has been defined to represent mean atmospheric, illumination, marine, and land conditions. Additional test cases have been considered to represent average winter (November–February), early spring (March–April), late spring–summer (May–August), and autumn (September–October) conditions, by accounting for seasonal changes in land albedo, illumination geometry, and water apparent optical properties. Finally, for mean land and water albedos, specific test cases have been defined to account for a variety of atmospheric conditions characterizing the region. For each test case, all observation geometries and all reference wavelengths are considered.

A. Geometrical Observation and Illumination Conditions

An analysis of the actual observation and illumination conditions encountered in the selected area for each considered sensor has been carried out using a comprehensive database extending over several years [49]. On the basis of this analysis, typical illumination/observation conditions have been selected (Table 1). Solar and sensor zenith angles are determined with respect to the local vertical, while solar and sensor azimuth angles are counted clockwise from the north direction (as generally adopted in satellite geolocation). The solar zenith angle ${\theta }_0$ is allowed to range between 25° and 65°. In particular, it is set to 25° for the May–August test case, to 65° for the November–February test case, and to 45° in all other cases. The space sensor viewing angle ${\theta }_v$ varies from 5° to 50° and from 5° and 20° for MODIS and MERIS, respectively. It is limited to 20°–50° for SeaWiFS to account for the sensor tilt angle. The solar azimuth angle ${\varphi }_0$ is set to $\pm 160°$, while $\pm 100°$ and $\pm 75°$ are the selected values for the satellite azimuth ${\varphi }_v$. A positive satellite azimuth indicates observations from over the sea, while a negative value indicates observations occurring from the land side. The specific illumination/observation geometries selected for the satellite sensors are indicated in Table 1.

Table 1. Parameters Defining the Illumination and Observation Geometries^a,^b

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B. Atmospheric Features

The ranges of variability for the Ångström coefficient $\alpha$ and exponent $\nu$ have been defined in agreement with the average values observed during clear sky conditions in the northern Adriatic Sea [20] (Table 2). The adopted aerosol single scattering albedo is approximately 0.99 at all wavelengths. Selected parameters correspond to an aerosol optical thickness at 555 nm ranging between 0.05 and 0.25, and equal to 0.14 for the standard case. In accordance with Eq. (16) parameter $g_1$ of the aerosol scattering phase function ranges between 0.63 and 0.68, while parameter $a_s$ ranges between 0.98 and 0.99 at 412 nm, and between 0.92 and 0.93 at 865 nm. A mean wind speed $W_s=3.3{\mathrm{ms}}^{-1}$ is assumed as representative of the average conditions encountered at the AAOT [50].

Table 2. Atmospheric Parameters Used for the Simulations^a

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C. Water Radiometric Features

Water radiometric features have been extracted from in situ measurements collected since 1995 at the AAOT [50,51]. Average values of the normalized water-leaving radiance ${\overline{L}}_{\mathrm{WN}}$, where $L_{\mathrm{WN}}=R_{\mathrm{rs}}·E_0$ are listed in Table 3 together with the related standard deviations. Seasonal spectral values of in situ ${\overline{L}}_{\mathrm{WN}}$ are shown in Fig. 2. The maximum values are found in November–February, while minima are observed in May–August. The annual mean exhibits spectral values close to those observed in September–October.

 

Fig. 2. Spectral values of in situ ${\overline{L}}_{\mathrm{WN}}$ [${\mathrm{Wm}}^{-2}{\mathrm{\mu m}}^{-1}{\mathrm{sr}}^{-1}$] at the AAOT site: symbols represent different annual and intra-annual periods; error bars indicate the standard deviation ${\sigma }_{L_{\mathrm{WN}}}$.

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Table 3. Spectral Values of in situ ${\overline{\mathsf{L}}}_{\mathsf{WN}}$ [${\mathsf{Wm}}^{-\mathsf{2}}{\mathsf{\mu m}}^{-\mathsf{1}}{\mathsf{sr}}^{-\mathsf{1}}$] at the AAOT Site and Related Standard Deviations ${\mathsf{\sigma }}_{{\mathsf{L}}_{\mathsf{WN}}}$

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D. Selected Land Reflectance Properties

Values of DHR and isotropic BHR (${\mathrm{BHR}}_{\mathrm{iso}}$) reflectances (reported as black-sky and white-sky albedos in the MODIS product suite) have been extracted from the MODIS multiyear climatological snow-free aggregate database [52] at the MODIS land center wavelengths $\lambda =470$, 555, 659, and 858 nm and for each MODIS land pixel of a subset of the region of interest close to the coast (12.0° E to 13.1° E and 44.9° N to 45.8° N). These data are the sole providing quality climatological albedo products at high spectral and spatial resolution. DHR is the reflectance that would be measured if the atmospheric scattering effects were removed, thus leading to a purely monodirectional incident radiance field. Conversely, ${\mathrm{BHR}}_{\mathrm{iso}}$ is the reflectance that would be measured if the incident radiance field could be assumed isotropic [26]. Both products are independent of the environmental conditions.

In accordance with the applied methodology, DHR and ${\mathrm{BHR}}_{\mathrm{iso}}$ data have been spatially averaged over the considered area, and the annual and intra-annual values have been computed (Fig. 3). The resulting products show their strongest seasonal dependence at 858 nm with a pronounced peak in summer. Mid-seasonal spectra look very similar in the NIR, but diverge in the visible spectral region. Annual mean spectra are very close to the September–October ones. Differences between corresponding values of time and spatially averaged $〈\mathrm{DHR}〉$ and $〈{\mathrm{BHR}}_{\mathrm{iso}}〉$ appear significant at 858 nm for the May–August and November–February periods. Therefore the “actual” average land BHR, hereafter simply termed land albedo ${\rho }_l$, has been computed for each annual and intra-annual period and at each wavelength, by weighting the two distinct surface reflectance products through the ratio ${\mathcal{S}}_E$ between diffuse and direct irradiance at the surface [26]:

 

Fig. 3. Spectral values at MODIS land center wavelengths of time and spatially averaged (a) $〈\mathrm{DHR}〉$ and (b) $〈{\mathrm{BHR}}_{\mathrm{iso}}〉$: symbols represent different annual and intra-annual periods; error bars indicate the standard deviations.

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Land albedos at the ocean color wavelengths $\lambda =490$, 555, 670, and 865 nm have been taken equal to those of the closest MODIS land center wavelengths. Land albedos at the other ocean color wavelengths (i.e., 412, 443, 510, and 765 nm) have been obtained through inter/extrapolation, assuming a cropland ecosystem. Specifically, cropland spectral signatures from ASTER [53] and USGS [54] spectral libraries have been used to define average spectral slopes at inter/extrapolation wavelengths. The slope variance, particularly high at 765 nm, has been taken into account in the uncertainty estimate of ${\rho }_l$. Indeed, the reflectance spectrum of cropland vegetation consistently varies with the cropland type, its phenological state, its moisture content, and the soil contribution. Particularly sensitive to these variations is the shape of the so-called red edge: the steep reflectance gradient around 700 nm. The assumption of a cropland ecosystem is justified by the similarity between the values of $〈{\mathrm{BHR}}_{\mathrm{iso}}〉$ displayed in Fig. 3(b) and the corresponding climatological values for the cropland ecosystem class over the 50°–40° N latitude belt provided by Moody et al. [52]. The assumption is also supported by the classification of the area as a cropland ecosystem in the IGBP Land Ecosystem Classification Map Image [55].

Table 4 summarizes the estimated land and sea spectral albedos for typical observation conditions (identified by the standard test case with ${\theta }_v=20°$, ${\varphi }_v=-75°$, and ${\varphi }_0=160°$). It is recalled that the sea albedo ${\rho }_{\mathrm{sea}}$, used to compute contributions from multiple reflections at the water surface [Eq. (12)], is obtained through an approximate relationship [Eq. (14)]. Acknowledging the qualitative relevance of ${\rho }_{\mathrm{sea}}$, tabulated data indicate very similar sea and land albedos up to about 510 nm, while they significantly differ in the NIR.

Table 4. Land ${\mathsf{\rho }}_{\mathsf{l}}$ and Sea ${\mathsf{\rho }}_{\mathsf{sea}}$ Spectral Albedos, and Related Standard Deviations $\mathsf{\sigma }$ for Typical Observation Conditions

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5. Evaluation of Uncertainties and Benchmark of Simulated Products

The quantification of the adjacency effects, the estimate of its horizontal range, as well the sensitivity analyses on selected geophysical parameters and the study of the impact of the commonly applied approximations, necessarily require an accurate evaluation of the involved uncertainties.

Uncertainties on simulated adjacency radiance ${\sigma }_{L_{\mathrm{adj}}}$ comprise independent contributions from the simulation procedure and from the definition of the input parameters. Assuming exact the definition of the atmospheric system, uncertainties on quantities simulated with the highly accurate FEM numerical code are considered negligible, and ${\sigma }_{L_{\mathrm{adj}}}$ can be simply expressed as

A. Uncertainties Connected to the NAUSICAA Simulation Procedure

The MC method is intrinsically associated with random noise: its computations are inherently affected by statistical uncertainties strictly depending on the number $N$ of initialized photons and on the selected threshold for photon survival. Computational products may additionally be affected by systematic uncertainties, so that ${\sigma }_{L_{\mathrm{adj}}}^{\mathrm{MC}}$ can be expressed as

In this study ${\sigma }_{L_{\mathrm{adj}}}^{\mathrm{rnd}}$ is required to not exceed the average radiometric resolution of selected ocean color sensors expressed in terms of the noise equivalent radiance difference, $\mathrm{NE}\mathrm{\Delta }\mathrm{L}$ [21]. At each wavelength, $\mathrm{NE}\mathrm{\Delta }\mathrm{L}$ is defined as the at-sensor incremental radiance that can still be discriminated from noise when observing a typical signal [21]. Making reference to the signal-to-noise-ratio (SNR) and the typical ocean color at-sensor radiance values provided by Hu et al. [56], $\mathrm{NE}\mathrm{\Delta }\mathrm{L}$ has been determined for each considered ocean color sensor and at each selected wavelength. In addition to these sensor specific values, an average ocean color sensor radiometric resolution has been defined as $\overline{\mathrm{NE}\mathrm{\Delta }\mathrm{L}}=2E_0\times {10}^{-5}{\mathrm{Wm}}^2{\mathrm{\mu m}}^{-1}{\mathrm{sr}}^{-1}$. It is noted that for SeaWiFS, especially in the visible channels, $\mathrm{NE}\mathrm{\Delta }\mathrm{L}$ is considerably higher than $\overline{\mathrm{NE}\mathrm{\Delta }\mathrm{L}}$ [56]. By initializing $N={10}^7$ photons the requirement ${\sigma }_{L_{\mathrm{adj}}}^{\mathrm{rnd}}<\overline{\mathrm{NE}\mathrm{\Delta }\mathrm{L}}$ is fully satisfied. Table 5 lists the 0.95 quantiles of ${\sigma }_{L_{\mathrm{adj}}}^{\mathrm{rnd}}$ for all test cases. The level of confidence on NAUSICAA computations is 99.7%, corresponding to the probability that the computed values fall within three standard deviations of the expected true value.

Table 5. Values of the 0.95 Quantiles at Representative Wavelengths of ${\mathsf{\sigma }}_{{\mathsf{L}}_{\mathsf{adj}}}^{\mathsf{rnd}}$ from Random Uncertainties of NAUSICAA MC Computations Performed by Initializing $\mathsf{N}={\mathsf{10}}^{\mathsf{7}}$ Photons and Assuming a Level of Confidence of 99.7%

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Contributions to ${\sigma }_{L_{\mathrm{adj}}}$ due to systematic uncertainties (biases) affecting the NAUSICAA computations, ${\delta }_{L_{\mathrm{adj}}}^{\mathrm{sys}}$, have been evaluated by comparing upwelling surface radiances at TOA, $L_{\mathrm{sfc}}^{\mathrm{TOA}}$, obtained with Eq. (12) for an infinite uniform reflecting surface, with the corresponding values from the totally converged solution of the plane-parallel highly accurate FEM numerical code [24], assumed as the reference. The analysis has been performed for all test cases by initializing $N={10}^7$ photons and assuming (i) an ideal (lossless) perfectly diffuse (Lambertian) surface and (ii) a Fresnel-reflecting surface. As illustrated by the distribution of ${\delta }_{\mathrm{MC}-\mathrm{FEM}}/E_0=(L_{{\mathrm{sfc}}_{\mathrm{MC}}}^{\mathrm{TOA}}-L_{{\mathrm{sfc}}_{\mathrm{FEM}}}^{\mathrm{TOA}})/E_0$ [${\mathrm{sr}}^{-1}$] in Fig. 4, results indicate the absence of biases between FEM $L_{{\mathrm{sfc}}_{\mathrm{FEM}}}^{\mathrm{TOA}}$, and NAUSICAA, $L_{{\mathrm{sfc}}_{\mathrm{MC}}}^{\mathrm{TOA}}$ results. Considering that the two codes rely on fundamentally different approaches to solve the RTE, the extremely good agreement between them provides a robust validation for the NAUSICAA code. Furthermore, ${\delta }_{\mathrm{MC}-\mathrm{FEM}}$ is always lower than the statistical uncertainty of NAUSICAA simulations with a 99.7% level of confidence. The above findings allow us to assume that ${\sigma }_{L_{\mathrm{adj}}}^{\mathrm{MC}}$ equals ${\sigma }_{L_{\mathrm{adj}}}^{\mathrm{rnd}}$ in the presence of an infinite homogenous underlying reflecting surface. Because of the high agreement obtained for both Lambertian and Fresnel reflecting surfaces, analogous considerations are expected to also apply for a nonuniform underlying reflecting surface.

 

Fig. 4. Normalized distributions of ${\delta }_{\mathrm{MC}-\mathrm{FEM}}/E_0=(L_{{\mathrm{sfc}}_{\mathrm{MC}}}^{\mathrm{TOA}}-L_{{\mathrm{sfc}}_{\mathrm{FEM}}}^{\mathrm{TOA}})/E_0$ [${\mathrm{sr}}^{-1}$] for all considered test cases ($N_{\text{cases}}$) obtained by initializing $N={10}^7$ photons and assuming (a) a uniform ideal Lambertian surface and (b) a uniform Fresnel-reflecting surface. The Gaussian fit of the distributions is displayed in black. Its mean $〈\delta /E_0〉$ and standard deviation ${\sigma }_{\delta }/E_0$ are also given. Note that the $x$ scale for the Lambertian case is one order of magnitude higher than that of the Fresnel case.

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B. Uncertainties Related to the Definition of the Surface Reflectance Properties

NAUSICAA simulations of adjacency effects require the assumption of a finite surface grid of nonuniform reflectance properties surrounded by an infinite uniform background surface (see Section 3).

This evaluation of ${\delta }_{\mathrm{MC}-\mathrm{FEM}}$ by performing NAUSICAA MC simulations at the center of a finite surface of the same size of the region illustrated in Fig. 1 and surrounded by a totally absorbing background surface showed that neglecting the background surface contribution can lead to ${\delta }_{\mathrm{MC}-\mathrm{FEM}}$ values significantly larger than the statistical uncertainties of NAUSICAA computations. This occurs in the case of both an ideal Lambertian and a Fresnel reflecting surface, and is particularly true towards the blue end of the spectrum where the atmosphere is more scattering. Consequently, results definitely indicate the need to evaluate the contribution from the background, i.e., to take into account the reflectance properties of the background surface and their related uncertainties.

It is worthwhile to mention that adjacency contributions from distant regions are likely due to photons that, once reflected by the surface itself, propagate until the upper atmospheric layers where they are eventually scattered by gas molecules in an almost horizontal direction. The very low optical thickness characterizing those atmospheric layers allows these photons to travel long distances before undergoing other scattering events and eventually propagating into the sensor field of view (FOV). This also suggests that volcanic aerosols, thin cirrus clouds, as well as any departure from the assumption of a plane-parallel atmosphere may affect the described phenomenon.

On the basis of the above considerations and assuming an exact knowledge of the sea surface BRDF, uncertainties on $L_{\mathrm{adj}}$ from input surface reflectances, ${\sigma }_{L_{\mathrm{adj}}}^{\mathrm{ref}}$ [see Eq. (18)], comprise contributions from the reflectance properties (i) of the land and water elements within the surface grid (${\sigma }_{L_{\mathrm{adj}}}^{{\rho }_l}$ and ${\sigma }_{L_{\mathrm{adj}}}^{R_{\mathrm{rs}}}$, respectively) and (ii) of the uniform background surface (${\sigma }_{L_{\mathrm{adj}}}^{\mathrm{bg}}$):

The modeling presented in Section 2 allows a simple evaluation of the contributions ${\sigma }_{L_{\mathrm{adj}}}^{{\rho }_l}$ and ${\sigma }_{L_{\mathrm{adj}}}^{R_{\mathrm{rs}}}$ from parameters ${\rho }_l$ and $R_{\mathrm{rs}}$ via error propagation of their uncertainties ${\sigma }_{{\rho }_l}$ and ${\sigma }_{R_{\mathrm{rs}}}$ (as given in Sections 4.D and 4.C, respectively). It is noted that uncertainties due to the nonuniformity of the land albedo are accounted for in the definition of ${\sigma }_{{\rho }_l}$. Conversely, uncertainties induced by the nonuniformity of the water albedo are considered negligible here. An evaluation of the uncertainties induced by the assumption of isotropic reflectance of land and water would require a dedicated study, which is out of the scope of this paper. The estimate of the adjacency radiance contribution originating from the background surface has been performed assuming that the infinite background surface is composed of an unknown combination of land and sea with the same reflectance properties of the land and sea elements within the surface grid of nonuniform reflectance properties. The background contribution has been thus computed along the whole transect for the standard case considering two extreme situations: a homogenous land background and a homogenous sea background. The average between these two computations has been adopted as the background contribution $L_{\mathrm{adj}}^{\mathrm{bg}}$ for all NAUSICAA simulations, and a spectral uncertainty ${\sigma }_{L_{\mathrm{adj}}}^{\mathrm{bg}}=\pm L_{\mathrm{adj}}^{\mathrm{bg}}$ has been assumed.

C. Uncertainties of Simulated Adjacency Radiance

For typical observation conditions and at sample wavelengths, Table 6 lists the overall uncertainties ${\sigma }_{L_{\mathrm{adj}}}$ affecting simulated $L_{\mathrm{adj}}$ at the AAOT, as well as individual contributions ${\sigma }_{L_{\mathrm{adj}}}^x$ to ${\sigma }_{L_{\mathrm{adj}}}$ from various sources. Notably, contributions from uncertainties in the simulation procedure, ${\sigma }_{L_{\mathrm{adj}}}^{\mathrm{MC}}$, are by far the lowest. Towards the blue end of the spectrum the highest uncertainty contributions are from $R_{\mathrm{rs}}$, while at larger wavelengths ${\sigma }_{L_{\mathrm{adj}}}^{{\rho }_l}$ and ${\sigma }_{L_{\mathrm{adj}}}^{\mathrm{bg}}$ predominate. Overall results indicate combined uncertainties lower than $\overline{\mathrm{NE}\mathrm{\Delta }\mathrm{L}}$.

Table 6. Combined and Contributing Uncertainties Affecting ${\mathsf{L}}_{\mathsf{adj}}$ Simulations ($·{\mathsf{E}}_{\mathsf{0}}\times {\mathsf{10}}^{-\mathsf{5}}$ [${\mathsf{sr}}^{-\mathsf{1}}$]) at the AAOT for Typical Observation Conditions

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D. Benchmark with Literature Data

To further assess the NAUSICAA code in the presence of a nonhomogeneous reflecting surface, simulated results have been benchmarked with data obtained with the algorithm developed by Sei [11]. Sei [11] has proposed a closed-form error analysis of the coastal adjacency problem for nadir view. The analysis has been performed adopting three common exponential approximations of the environment weighting function (which describes the contribution of the surface surrounding the target element) heuristically inferred from different MC simulations: specifically, those proposed by Reinersman and Carder [7], Vermote et al. [57], and Tanré et al. [58].

In order to perform the benchmark, equivalent observation conditions have been assumed, namely, two Lambertian half-planes in quasi-nadir observation. Additionally, the same surface albedos and the same total aerosol and Rayleigh optical thicknesses have been adopted. Values of $\mathrm{\Delta }{L}_{\mathrm{sfc}}^{\mathrm{TOA}}=(L_{\mathrm{sfc},\mathrm{NH}}^{\mathrm{TOA}}/L_{\mathrm{sfc},H}^{\mathrm{TOA}}-1)\times 100$ (where suffix NH indicates a nonhomogeneous underlying reflecting surface, and suffix $H$ an homogenous one) at sample wavelengths are displayed in Fig. 5 as a function of the distance from the coast. Results from Sei [11] refer to the approximation of the environment weighting function proposed by Reinersman and Carder [7]. Analogous results are obtained when adopting the approximations proposed by Vermote et al. [57] and by Tanré et al. [58]. Intercomparison results show optimal agreement. Still, a slight discrepancy with increasing distance from the coast indicates a slower convergence of NAUSICAA results. Differences are likely explained by an underestimate of adjacency effects with distance induced by the assumption of an exponential decay of the environment weighting functions. The applied exponential approximations were indeed acknowledged by the same authors “to account for the major part of the environment effect” [57,58], while Reinersman and Carder specifically stated that “the approximation was developed empirically [$\dots$] and no claim is made that this method is optimal” [7]. Results displayed in Fig. 5 are, moreover, in agreement with the slower convergence of MC simulations with respect to their exponential approximation as reported by Reinersman and Carder (see Fig. 27 of Ref. [7]).

 

Fig. 5. Plot of ${\mathrm{\Delta }L}_{\mathrm{sfc}}^{\mathrm{TOA}}=(L_{\mathrm{sfc},\mathrm{NH}}^{\mathrm{TOA}}/L_{\mathrm{sfc},H}^{\mathrm{TOA}}-1)\times 100$ as a function of the distance from the coast and at sample wavelengths, obtained using the parameterization proposed by Sei [11] with the approximation of the environment weighting function proposed by Reinersman and Carder [7] (black line) and the NAUSICAA code (empty circles) for equivalent observation conditions: two Lambertian half-planes in quasi-nadir observation, with the same surface albedos and the same total aerosol and Rayleigh optical thicknesses. Error bars indicate the NAUSICAA statistical uncertainty with a 99.7% level of confidence.

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6. Results and Discussion

Simulated results are presented and discussed through $L_{\mathrm{adj}}/E_0$ [${\mathrm{sr}}^{-1}$] and its percent contribution to the TOA signal, ${\xi }_{L_{\mathrm{tot}}}=(L_{\mathrm{adj}}/L_{\mathrm{tot}})\times 100$, along the study transect with specific focus on the AAOT site.

A. Adjacency Effects along the Transect

The quantitative evaluation of the adjacency effects in the selected region is herein illustrated through a statistical analysis of ${\xi }_{L_{\mathrm{tot}}}$ over all test cases and for each observed sea element along the considered transect. Results are illustrated in Fig. 6 at sample wavelengths as a function of the distance covered along the transect moving away from the coast. It is underlined that, due to the complex coastal pattern, the distance along the transect does not correspond to the minimum distance from the mainland. It is additionally noted that a different scale is applied to display data at 865 nm. It is finally pointed out that $L_w$ is assumed spatially invariant.

 

Fig. 6. Values of ${\overline{\xi }}_{L_{\mathrm{tot}}}$ at representative wavelengths as a function of the distance along the study transect. Error bars represent the standard deviation $\pm \sigma$ (black) and the sample variance (gray). $L_w$ is assumed constant all along the transect. The vertical dotted line identifies the position of the AAOT site.

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As expected, adjacency effects monotonically decrease when moving away from the coast. A minor non-monotonicity occurs in proximity of an isolated group of two land elements regarded as a small island in the applied surface grid [see Fig. 1(b)]. Values of mean adjacency contributions ${\overline{\xi }}_{L_{\mathrm{tot}}}$ lower than $\pm 1\%$ are observed throughout the transect up to 555 nm, where land and sea albedos are similar (see Table 4). At the other wavelengths, ${\overline{\xi }}_{L_{\mathrm{tot}}}$ always remains positive, exhibiting a steep decrease with distance. Values nearby the coast range from $\sim 2\%$ at 670 nm up to $\sim 11\%$ at 865 nm. As expected, the highest values occur at 865 nm, where the land is highly reflective and the water highly absorbing. All along the transect the sample variances are larger than the uncertainties on simulated results.

To further investigate the spectral and spatial trend of adjacency effects, Fig. 7 displays the average normalized adjacency radiance ${\overline{L}}_{\mathrm{adj}}/E_0$ for all test cases together with the separate average contributions originating from the nearby land, and from the masked water and sea surface [Eq. (15)]. It is noted that the two latter terms have negative sign. The mean land contribution ${\overline{L}}_{\text{land}}^{\mathrm{TOA}}$ exhibits a correlation with both the atmospheric scattering intensity and the land spectral albedo. This is deduced by observing higher values where the land albedo is higher (in the NIR) and/or where the atmosphere is more scattering (toward the blue end of the spectrum). The same holds, with respect to the water-leaving radiance, for the masked water contribution ${\overline{\tilde{L}}}_w^{\mathrm{TOA}}$. The masked sea surface contribution ${\overline{\tilde{L}}}_{\mathrm{ss}}^{\mathrm{TOA}}$ is strongly correlated to the scattering properties of the atmosphere: it decreases moving from the blue to the NIR. It is noted here that the land contribution is not always the largest: Fig. 7 clearly shows that towards the blue end of the spectrum mean contributions from the masked sea are larger than those from the land.

 

Fig. 7. Values of ${\overline{L}}_{\mathrm{adj}}/E_0$ (black circles) at representative wavelengths as a function of the distance along the study transect. Contributions from land, ${\overline{L}}_{\mathrm{land}}^{\mathrm{TOA}}/E_0$ (gray circles), masked sea surface, ${\overline{\tilde{L}}}_{\mathrm{ss}}^{\mathrm{TOA}}/E_0$ (empty triangles), and masked water ${\overline{\tilde{L}}}_w^{\mathrm{TOA}}/E_0$ (empty circles) are also displayed. Error bars represent the standard deviation $\pm \sigma$. $L_w$ is assumed constant all along the transect. The vertical dotted line identifies the position of the AAOT.

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Figure 7 furthermore eases the interpretation of the negative peaks in correspondence of the small island (peaks are more evident in Fig. 6), which occur even at wavelengths where the land albedo exceeds the sea one (e.g., 670 nm; see Table 4). Plots distinctly indicate that such peaks are a consequence of the significant sea surface contribution masked by the island itself. This contribution is particularly strong due to the extreme forward peak of the sea surface BRDF and the location of the island in the solar half-plane, identified by $\mathrm{\Delta }\varphi =|{\varphi }_v-{\varphi }_0|<90°$. To better illustrate the dependence of adjacency effects on the mutual position of land and Sun, $L_{\mathrm{adj}}/E_0$ is plotted along the study transect [Fig. 8(a)] and along a parallel transect positioned directly south of the small island [Fig. 8(b)]. The separate contributions originating from the nearby mainland, and the masked water and sea surface [Eq. (15)], are also indicated. Data are for typical observation conditions at 670 nm. The small island lays in the solar half-plane in the first case, and in the antisolar half-plane (for which $\mathrm{\Delta }\varphi >90°$) in the second case. Land and masked water contributions are clearly the same for sea elements located north or south of the island. Conversely, the masked sea surface contribution is much higher when the land is located in the solar half-plane, causing a global decrease in $L_{\mathrm{adj}}/E_0$.

 

Fig. 8. Values of $L_{\mathrm{adj}}/E_0$ (black circles) for typical observation conditions at 670 nm as a function of the distance (a) along the study transect and (b) along a parallel transect located just south of the small island. The land (gray circles), masked sea surface (empty triangles), and masked water (empty circles) contributions are also displayed. Error bars represent the standard deviation $\pm \sigma$. $L_w$ is assumed constant all along the transect.

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The masked sea surface contribution due to the mainland located in the solar half-plane is also the reason for negative adjacency contributions up to $-1\%$ at 490 nm (Fig. 6), where land and sea albedos are indeed very similar (Table 4).

Sensitivity of results to the mutual position of land and Sun is a direct consequence of the dependence of adjacency effects on the directional reflectance properties of the surface. The spectral and spatial trends of adjacency contributions can be fully appreciated only acknowledging that adjacency effects arise between neighboring areas of different reflectance in terms of both albedo (BHR) and BRDF, i.e., in terms both of the global reflected radiance and of its angular distribution. In this study an isotropic reflectance has been assumed for both land and water, while the BRDF of the rough sea surface is extremely forward peaked. Even at those wavelengths where the land albedo is larger than the sea albedo (see Table 4), the sea BRDF may still exceed the land BRDF at or near the forward scattering direction, implying masked sea contributions higher than those from the land. Further, at wavelengths where land and sea albedos are very similar and negligible adjacency perturbation would be expected, adjacency effects significantly originate from the different angular distribution of the radiance reflected by land and water (e.g., 412–510 nm).

While a constant $L_w$ is assumed along the whole transect, the increasing water turbidity in approaching the coast may actually lead to an increase in $L_w$, particularly in the red spectral region [20,50]. Yet, an analysis performed at 670 nm assuming an exponential increase of $L_w$ towards the coast up to values threefold those observed at the AAOT has not shown appreciable changes in $L_{\mathrm{adj}}$ and ${\xi }_{L_{\mathrm{tot}}}$.

To investigate the atmospheric, geometric, and seasonal dependencies of the adjacency effects in the study region, a sensitivity analysis has been performed accounting for the range of variation of the input parameters.

Adjacency effects show appreciable sensitivity to the aerosol load (see Fig. 9). As expected [11], the lower the concentration, the slower the decrease of adjacency effects with distance. On the contrary, data do not show any appreciable dependence on the aerosol type. Results are only displayed in the NIR, where contributions are the largest.

 

Fig. 9. Values of ${\overline{\xi }}_{L_{\mathrm{tot}}}$ at 865 nm as a function of the distance along the study transect. Plots on the left panel have been obtained for different values of the Ångstrom coefficient: $\alpha =0.02$ (empty circles), $\alpha =0.05$ (black circles), and $\alpha =0.08$ (gray circles). Plots on the right panel have been obtained for different values of the Ångstrom exponent: $\nu =1.4$ (empty circles), $\nu =1.7$ (black circles), and $\nu =1.9$ (gray circles). Error bars represent the standard deviation $\pm \sigma$. $L_w$ is assumed constant all along the transect.

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Adjacency contributions also display an anisotropic angular distribution [7]. Values of ${\overline{\xi }}_{L_{\mathrm{tot}}}$ at representative wavelengths are displayed in Fig. 10 for different zenith angles of observation ${\theta }_v$. The sole uncertainties ${\sigma }_{{\xi }_{L_{\mathrm{tot}}}}^{\mathrm{MC}}$ related to the simulation procedure and defined in accordance with the average radiometric resolution of the considered sensors (see Section 5) are displayed. Plots indicate an increase of adjacency effects with ${\theta }_v$, which is significant both from the point of view of simulation uncertainties and from the point of view of the average ocean color sensor radiometric resolution. Uncertainties related to the reflectance properties of the surface would mask the differences throughout the transect at 490 nm and at some distance from the coast at 865 nm. At both sample wavelengths, ${\overline{\xi }}_{L_{\mathrm{tot}}}$ for ${\theta }_v=50°$ is approximately 40% larger than for a quasi-nadir observation.

 

Fig. 10. Values of ${\overline{\xi }}_{L_{\mathrm{tot}}}$ at 490 nm (left panel) and at 865 nm (right panel) as a function of the distance along the study transect for different zenith angles of observation: ${\theta }_v=5°$ (empty circles), ${\theta }_v=20°$ (black circles), and ${\theta }_v=50°$ (gray circles). Bars solely represent the random uncertainties of simulated results $\pm {\sigma }_{{\xi }_{L_{\mathrm{tot}}}}^{\mathrm{rnd}}$.

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To better explain the origin of the angular dependence of adjacency effects, Fig. 11 displays results obtained for analogous conditions, but assuming an isotropic sea surface reflectance with albedo ${\rho }_{\mathrm{ss}}=0.04$. Increasing the ground-to-sensor path, the probability for the radiance reflected by the surface to be scattered into the sensor FOV increases. At those wavelengths where land and sea albedos only slightly differ (e.g., 490 nm), if the BRDFs are also equivalent, the increase is the same for both contributions, and a compensation occurs (Fig. 11 left panel). The zenith angular dependence of $L_{\mathrm{adj}}$ observed in Fig. 10 (left panel) is hence primarily a consequence of the strong reflectance anisotropy of the sea surface. Conversely, when the land albedo is consistently larger than the sea one (e.g., 865 nm) and no compensation can occur anymore, the dependence on the zenith angle of observation is simply a consequence of the longer ground-to-sensor path (see Figs. 10 and 11 right panels).

 

Fig. 11. Same as in Fig. 10 but assuming an isotropic sea surface reflectance with albedo ${\rho }_{\mathrm{ss}}=0.04$.

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Adjacency effects also show some dependence on the Sun-sensor relative azimuth (see Fig. 12): ${\overline{\xi }}_{L_{\mathrm{tot}}}$ is smaller for observations in the antisolar half-plane, where the masked sea surface contribution is larger due to the forward scattering properties of both the sea surface and atmospheric aerosol. At 865 nm (Fig. 12 right panel) this dependence is only appreciable in the proximity of the small island. At those wavelengths where the contribution of the masked sea surface is more important (e.g., 490 nm, Fig. 12 left panel), differences are also evident towards the coast. The dependence of adjacency contributions on $\mathrm{\Delta }\varphi$ is mostly related to the different directional reflectance properties of land and sea. By assuming an isotropic sea surface reflectance, the dependence on $\mathrm{\Delta }\varphi$ significantly decreases (Fig. 13). The angular anisotropy of adjacency contributions would be clearly enhanced by the presence of a more forward scattering atmosphere.

 

Fig. 12. Values of ${\overline{\xi }}_{L_{\mathrm{tot}}}$ at 490 nm (left panel) and at 865 nm (right panel) as a function of the distance along the study transect for different Sun-sensor relative azimuths: black circles are for $\mathrm{\Delta }\varphi =|{\varphi }_0-\varphi |=125°$; empty circles for $\mathrm{\Delta }\varphi =|{\varphi }_0-\varphi |=60°$. Bars solely represent the random uncertainties of simulated results $\pm {\sigma }_{{\xi }_{L_{\mathrm{tot}}}}^{\mathrm{rnd}}$.

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Fig. 13. Same as in Fig. 12 but assuming an isotropic sea surface reflectance with albedo ${\rho }_{\mathrm{ss}}=0.04$.

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Although not shown here, it is worthwhile to mention that adjacency effects are, as expected, slightly larger when the sensor is viewing the sea from over the land.

Finally, adjacency effects show a strong seasonal dependence (see Fig. 14). They increase in the summer, when the land is highly reflecting, the water is highly absorbing, and the Sun elevation is high. Conversely, they decrease in winter, when the land albedo is low, the water albedo is also at its maximum, and the Sun elevation is at its minimum. It is noted that ${\overline{\xi }}_{L_{\mathrm{tot}}}$ decreases with increasing solar zenith angle as a consequence of a corresponding decrease in the irradiance reaching the surface and an increase of the sea surface reflectance. In proximity of the coast, ${\overline{\xi }}_{L_{\mathrm{tot}}}$ at 865 nm seasonally varies from 4% to 17% from winter to summer, respectively.

 

Fig. 14. Seasonal values of ${\overline{\xi }}_{L_{\mathrm{tot}}}$ at 865 nm for standard atmospheric conditions, as a function of the distance along the study transect. Bars represent the standard deviation $\pm \sigma$. Black circles correspond to results obtained in November–February, empty circles in March–April (masked by empty diamonds), gray circles in May-August, and empty diamonds in September–October.

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The results illustrated in this section are summarized in Table 7 for representative wavelengths and at given distances from the coast. These latter distances are computed as the minimum distance between the center of the target element and the center of the closest land element. Statistical values (i.e., mean and standard deviation) are given for the whole set of test cases, as well as for subsamples of specific geophysical quantities.

Table 7. Values of ${\overline{\mathsf{\xi }}}_{{\mathsf{L}}_{\mathsf{tot}}}\pm {\mathsf{\sigma }}_{{\overline{\mathsf{\xi }}}_{{\mathsf{L}}_{\mathsf{tot}}}}$ in Percent at Representative Wavelengths $\mathsf{\lambda }$ [nm] and at Several Distances from the Coast $\mathsf{d}$ [km] for all Test Cases (Global), for Typical Observational Conditions, and for Test Cases Characterized by the Ångström Exponent $\mathsf{\alpha }=\mathsf{0.02}$ and 0.08, Viewing Zenith Angle ${\mathsf{\theta }}_{\mathsf{v}}=\mathsf{5}°$ and 50°, and November–February and May–August Observation Conditions

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B. Adjacency Effects at the AAOT

The AAOT, which is located at the 25th km along the selected transect and at 15 km from the closest mainland, is a recognized validation site for satellite ocean color data products. Consistent effort has been made over the years to investigate, quantify (and eventually correct) all possible sources of uncertainties affecting both in situ measurements and the satellite-retrieved products [18,59–61]. This sets the premises for an accurate theoretical quantification of adjacency perturbations in satellite ocean color data at the AAOT.

Spectral results from the statistical analysis of ${\xi }_{L_{\mathrm{tot}}}$ at the AAOT are presented in Fig. 15. Mean values, standard deviations, and sample variances are computed for all test cases. At the visible wavelengths, ${\overline{\xi }}_{L_{\mathrm{tot}}}$ never exceeds $\pm 0.5\%$, with slightly negative values up to 510 nm and positive afterwards. At the NIR wavelengths, ${\overline{\xi }}_{L_{\mathrm{tot}}}$ becomes definitely larger reaching 1.7% and 2.0% at 765 and 865 nm, respectively. Standard deviations indicate uncertainties ranging from $\pm 0.1\%$ at blue wavelengths up to $\pm 0.6\%$ and $\pm 0.5\%$ at 765 and 865 nm, respectively. Larger uncertainty values at the NIR wavelengths are mainly due to the larger uncertainties on the land albedo (see Fig. 3). Sample variances are of the same order of standard deviations at shorter wavelengths, while they reach $\pm 0.8\%$ in the NIR. These high values are mostly related to the pronounced seasonal variation of the land albedo in the NIR (Fig. 16), although the dependence on the atmospheric aerosol optical thickness [Fig. 17(a)] and the observation zenith angle [Fig. 18(a)] is also relevant. In agreement with Fig. 12, negligible sensitivity on the Sun-sensor relative azimuth is observed [Fig. 18(b)]. Sample variance in Figs. 16–18 indicates a larger spread of data at 865 nm in the May–August period, and for slanted observations from over the land. Note that the variance in Fig. 16 is low as a consequence of the assumption of mean land and water parameters.

 

Fig. 15. Spectral plot of ${\overline{\xi }}_{L_{\mathrm{tot}}}$ for all test cases ($N=108$) at the AAOT. Bars represent the standard deviation $\pm \sigma$ (black) and the sample variance (gray).

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Fig. 16. Seasonal values of ${\overline{\xi }}_{L_{\mathrm{tot}}}$ at representative wavelengths for standard atmospheric conditions. Bars represent the standard deviation $\pm \sigma$ (black) and the sample variance (gray). The dashed line indicates ${\overline{\xi }}_{L_{\mathrm{tot}}}$ determined from all test cases.

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Fig. 17. Values of ${\overline{\xi }}_{L_{\mathrm{tot}}}$ at 865 nm as a function (a) of the Ångstrom coefficient $\alpha$ and (b) of the Ångstrom exponent $\nu$ in selected ranges of variation (Table 2) for mean land and water parameters. Bars represent the standard deviation $\pm \sigma$ (black) and the sample variance (gray). The dashed line indicates ${\overline{\xi }}_{L_{\mathrm{tot}}}$ determined from all test cases.

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Fig. 18. Values of ${\overline{\xi }}_{L_{\mathrm{tot}}}$ at 865 nm as a function (a) of the zenith angle of observation ${\theta }_v$ and (b) of the Sun-sensor relative azimuth $\mathrm{\Delta }\varphi =|{\varphi }_0-\varphi |$ in degrees in their selected ranges of variation (see Table 1). Bars represent the standard deviation $\pm \sigma$ (black) and the sample variance (gray). The dashed line indicates ${\overline{\xi }}_{L_{\mathrm{tot}}}$ determined from all test cases.

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No significant differences are observed in the adjacency radiance contribution to the signal at the sensor when considering the geometries of observation of the selected sensors (Fig. 19). MERIS values, slightly below the average and with a smaller variance, are justified by the absence of very slanted observations (${\theta }_v<40°$). Consistently, the values slightly above the average and the larger variances observed for SeaWiFS are due to the absence of quasi-nadir observations (${\theta }_v\ge 20°$).

 

Fig. 19. Values of ${\overline{\xi }}_{L_{\mathrm{tot}}}$ at 865 nm for typical observation–illumination geometries at the AAOT for SeaWIFS (SWF), MODIS Aqua (MOD-A), MODIS Terra (MOD-T), and MERIS (MER). Bars represent the standard deviation $\pm \sigma$ (black) and the sample variance (gray). The dashed line indicates ${\overline{\xi }}_{L_{\mathrm{tot}}}$ determined from all test cases.

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C. Sensitivity Analysis on Commonly Applied Approximations

The need to provide easy-to-implement formulations of the adjacency effects in coastal waters has been commonly accomplished by applying one or more of the following assumptions: i) isotropic surface reflectance [7,11], ii) nadir observation [11], iii) straight coastline [11], and iv) single scattering approximation [9]. Nevertheless, the physical impact of most of these approximations has never been thoroughly assessed. Their effects are individually analyzed here for the considered study region.

The sensitivity analysis has been carried out with specific focus on the AAOT site and considering either all test cases or typical observation conditions. The study has been performed accounting for the estimated average radiometric resolution of the considered sensors ($\overline{\mathrm{NE}\mathrm{\Delta }\mathrm{L}}=2E_0\times {10}^{-5}$; see Section 5), which also represents the selected threshold for the uncertainties related to the NAUSICAA MC simulation procedure, ${\sigma }_{L_{\mathrm{adj}}}^{\mathrm{MC}}$. Misestimates are hence significant from the point of view of both the sensor radiometric resolution and the simulation accuracy, only when larger than such a threshold.

The need to take into account the Fresnel reflectance of the masked sea surface contribution was already underlined in [9] and as well identified in Section 6.A. Yet, most studies assume an isotropic reflectance for both land and sea [7,11]. Figure 20 presents at sample wavelengths the average normalized adjacency radiance over all test cases, ${\overline{L}}_{\mathrm{adj}}/E_0$, along the considered transect. Computations have been performed assuming an isotropic sea surface with albedo ${\rho }_{\mathrm{ss}}=0.04$ and accounting for the BRDF of a rough sea surface (defined by $W_s=3.3{\mathrm{ms}}^{-1}$). It is noted that latter results, significantly sensitive to the Sun azimuth, may appreciably vary for different land locations with respect to the Sun. For all considered cases, neglecting the anisotropy of the sea surface reflectance leads to misestimates of ${\overline{L}}_{\mathrm{adj}}$ particularly significant at the lower wavelengths (e.g., at 490 nm ${\overline{L}}_{\mathrm{adj}}$ is underestimated by approximately 40% throughout the transect). Moving towards the NIR, misestimates become appreciable only in the right vicinity of the coast. Actually, in correspondence of the AAOT (Fig. 21) differences are only appreciable up to 510 nm. Misestimates are expected to increase with the sensor and Sun zeniths. Indeed, the specular reflectance of the sea surface sharply increases for incident angles larger than 60° [27]. Overall results indicate the need to take into account the anisotropy of the sea surface reflectance, particularly towards the blue end of the spectrum.

 

Fig. 20. Values of ${\overline{L}}_{\mathrm{adj}}/E_0$ from all test cases ($N=108$) at representative wavelengths as a function of the distance along the study transect and obtained assuming i) the BRDF of a rough sea surface induced by $W_s=3.3{\mathrm{ms}}^{-1}$ (full markers) and ii) an isotropic sea surface reflectance with albedo ${\rho }_{\mathrm{ss}}=0.04$ (empty markers). Both error bars and the dashed lines indicate $\pm \overline{\mathrm{NE}\mathrm{\Delta }\mathrm{L}}/E_0$. $L_w$ is assumed constant all along the transect.

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Fig. 21. Spectral plot of ${\overline{L}}_{\mathrm{adj}}/E_0$ at the AAOT obtained assuming i) the BRDF of a rough sea surface induced by $W_s=3.3{\mathrm{ms}}^{-1}$ (full markers) and ii) an isotropic sea surface reflectance with albedo ${\rho }_{\mathrm{ss}}=0.04$ (empty markers). Both error bars and the dashed lines indicate $\pm \overline{\mathrm{NE}\mathrm{\Delta }\mathrm{L}}/E_0$.

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The results additionally suggest enquiring the need to consider the intrinsic directional reflectance properties of the land, here assumed isotropic (conversely, the water reflectance can be assumed with good approximation as isotropic). Even though a detailed investigation fully addressing the topic is out of the scope of this paper, some general considerations may be drawn from examples of land BRDF. The majority of terrestrial surfaces exhibit a bowl-shaped anisotropy, although surface heterogeneity may also lead to a bell-shaped pattern [62]. Realistic land BRDF can be obtained through the parametric semi-empirical model developed by Rahman et al. (the so-called RPV model [63]). The overall shape of the angular distribution is there defined through three parameters: ${\rho }_0$, representing the reflectance of the surface for illumination and viewing at the zenith; $k$, controlling the slope of the reflectance with respect to illumination and viewing angles [it is close to 1.0 for a quasi-Lambertian surface, lower (higher) than 1.0 when a bowl-shaped (bell-shaped) pattern dominates]; and $\mathrm{\Theta }$, which establishes the degree of forward (positive $\mathrm{\Theta }$) or backward (negative $\mathrm{\Theta }$) scattering. As an example, Fig. 22 reproduces the land BRDF obtained assuming $k=0.6$, 1.0, and 1.2 [62,63], $\mathrm{\Theta }=0$, accounting for the default relative increase in reflectance in the direction of illumination (the so-called hotspot), and selecting ${\rho }_0$ to obtain albedo values ${\rho }_l=0.04$ and 0.26 (corresponding to the values applied in the present work at 412 and 865 nm for typical observation conditions; see Table 4). The BRDFs of a sea surface roughed by wind speeds $W_s=2.3$, 3.3, and $4.3{\mathrm{ms}}^{-1}$ [34] are also shown. Data are plotted for an incident direction defined by ${\theta }_i=45°$ and ${\varphi }_i=0°$, and for observations performed in the principal solar plane (the vertical plane identified by the incoming direct solar radiation). In the plane normal to the principal solar plane, the sea surface BRDF is close to zero. The considered sample data suggest that accounting for the anisotropy in the land reflectance might affect simulations of adjacency effects towards the NIR, but not in the blue end of the spectrum. They additionally suggest the need to investigate the influence of wind speed on adjacency effects.

 

Fig. 22. BRDF values in the principal plane for (i) a sea surface roughed by wind speeds $W_s=2.3$ (dashed line), $3.3{\mathrm{ms}}^{-1}$ (solid line), and $4.3{\mathrm{ms}}^{-1}$ (dotted line) and (ii) a land surface defined by ${\rho }_l=0.04$ (diamonds) and ${\rho }_l=0.26$ (circles), with $\mathrm{\Theta }=0$ and $k=0.6$ (gray markers), 1.0 (black markers), and 1.2 (empty markers). The sea surface BRDFs are obtained from [34], while the land BRDFs through the so-called RPV model [63].

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Off-nadir views are characterized by increased adjacency effects [7,9]; nevertheless a quasi-nadir observation is often adopted in the simulation of adjacency contributions (e.g., [11]). The sensitivity of adjacency effects to the zenith angle of observation has already been presented and discussed in Section 6. Results indicated a significant angular anisotropy all along the transect, which, at shorter wavelengths, was strictly related to the directional properties of the sea surface reflectance. Results at the AAOT also show significant underestimates of adjacency effects for off-nadir observations in the blue and at NIR [Fig. 23(a)], which become appreciable only in the NIR when neglecting the anisotropy of the sea surface reflectance [Fig. 23(b)]. It is worthwhile to note that the estimate of average contributions is almost not affected by the assumption of a quasi-nadir view.

 

Fig. 23. Spectral values of ${\overline{L}}_{\mathrm{adj}}/E_0$ at the AAOT site for the test cases characterized by ${\theta }_v=50°$ (empty markers) and ${\theta }_v=5°$ (full markers) determined assuming (a) the BRDF of a rough sea surface induced by $W_s=3.3{\mathrm{ms}}^{-1}$ and (b) an isotropic sea surface reflectance with albedo ${\rho }_{\mathrm{ss}}=0.04$. Both error bars and dashed lines indicate $\pm \overline{\mathrm{NE}\mathrm{\Delta }\mathrm{L}}/E_0$.

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The influence of assuming a straight coastline clearly depends on the actual coastal morphology. Additionally, due to the strong anisotropy of the sea surface reflectance in the forward direction, it also depends on the orientation of the coast with respect to Sun and sensor (as already evidenced in Section 6.A). The sensitivity analysis has been performed separately for each sea element along the transect by assuming a straight coastline located at the minimum distance from the land and oriented along the south–north direction. By assuming typical observation conditions, the results summarized in Fig. 24 indicate significant differences only in the Venice Lagoon, where the coastal morphology highly differs from a straight coastline. Differences at the AAOT are negligible throughout the spectral region. Misestimates might be more appreciable when the land contribution is larger, as in summer and for slanted observations.

 

Fig. 24. Values of $L_{\mathrm{adj}}/E_0$ for typical observation conditions at representative wavelengths as a function of the distance along the study transect. Full markers represent values obtained accounting for the actual coastal pattern. Empty markers indicate values obtained separately assuming for each observed sea element along the transect a straight coastline located at the minimum distance from the land and oriented along the south–north direction. Both error bars and the dashed lines indicate $\pm \overline{\mathrm{NE}\mathrm{\Delta }\mathrm{L}}/E_0$. $L_w$ is assumed constant all along the transect.

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Finally, Fig. 25 displays results obtained by solving the RTE accounting or not for atmospheric multiple scattering. Values are given at representative wavelengths and for typical observation conditions. All separate contributions to the adjacency radiance [$L_{\text{land}}^{\mathrm{TOA}}$ originating from the land and ${\tilde{L}}_w^{\mathrm{TOA}}$ and ${\tilde{L}}_{\mathrm{ss}}^{\mathrm{TOA}}$ originating from the masked sea; see Eq. (15)] are largely underestimated in single scattering approximation. Their underestimation consistently increases towards the blue end of the spectrum (where the atmosphere is more scattering) and moving away from the coast (where the multiple scattering contribution is larger due to the longer path the radiance reflected by the land has to cover before entering the sensor IFOV). However, the underestimation of the sea mask contribution is more pronounced as a consequence of the strong anisotropy of the sea surface reflectance: Fig. 26 clearly illustrates how the misestimates induced at 490 nm by the single scattering approximation drastically decrease under the assumption of an isotropic sea surface reflectance (${\rho }_{\mathrm{ss}}=0.04$). This is explained by acknowledging that in single scattering approximation ${\tilde{L}}_{\mathrm{ss}}^{\mathrm{TOA}}$ originates only from reflection of the direct solar irradiance. This generates an even stronger anisotropic angular distribution of ${\tilde{L}}_{\mathrm{ss}}$, which becomes non-null only in a narrow cone centered on the direction specular to that of the direct Sun beam. As a consequence, the masked sea surface contribution practically disappears throughout the spectrum, except in the right vicinity of the land. Misestimates induced on the adjacency radiance clearly depend on the relative importance of the different contributions. Where the masked sea contribution is more important (e.g., at 490 nm), the misestimate of the adjacency radiance is particularly large. Compensations may eventually occur at some wavelengths (e.g., at 555 and 670 nm for typical observation conditions). In the NIR the underestimate of the adjacency radiance reflects the underestimate of $L_{\text{land}}^{\mathrm{TOA}}$, which is the only relevant contribution. In conclusion, the single scattering approximation would not allow appreciating any adjacency contribution up to 510 nm, except in the right vicinity of the land, and it would induce an appreciable underestimate of adjacency effects even in the NIR. Some compensation may arise at certain wavelengths, depending on the specific observation conditions. Results for the AAOT are summarized in Fig. 27.

 

Fig. 25. Values of $L_{\mathrm{adj}}/E_0$ for typical observation conditions at representative wavelengths as a function of the distance along the study transect obtained accounting for multiple scattering (full markers) and adopting the single scattering approximation (empty markers). Both error bars and the dashed lines indicate $\pm \overline{\mathrm{NE}\mathrm{\Delta }\mathrm{L}}/E_0$. $L_w$ is assumed constant all along the transect.

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Fig. 26. Same as in Fig. 25 at 490 nm but assuming an isotropic sea surface reflectance with albedo ${\rho }_{\mathrm{ss}}=0.04$.

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Fig. 27. Spectral plot of $L_{\mathrm{adj}}/E_0$ at the AAOT for typical observation conditions obtained in single scattering approximation (empty markers) and accounting for multiple scattering (full markers). Both error bars and dashed lines indicate $\pm \overline{\mathrm{NE}\mathrm{\Delta }\mathrm{L}}/E_0$.

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Overall results indicate that, out of the Venice Lagoon, simulations could be performed for a straight coastline, opportunely oriented, without a significant loss in accuracy. Conversely, they highlight the need to properly take into account the anisotropy of the sea surface reflectance, particularly at the visible wavelengths, the actual zenith angle of observation, and the atmospheric multiple scattering.

The above considerations have been drawn accounting for an average radiometric resolution of the sensor. Clearly, for a less sensitive sensor such as SeaWiFS, the misestimates induced by the approximations may become less significant.

Additionally, it must not be forgotten that other sources of uncertainties (e.g., those related to the definition of the input parameters) may be so high to dominate and mask the observed differences.

D. On the Horizontal Range of Adjacency Effects

While the horizontal scale of the adjacency effects is defined as the distance at which the adjacency radiance is decreased by a factor $e$, the horizontal range of the adjacency effects describes the distance at which the adjacency radiance becomes negligible (i.e., lower than $\mathrm{NE}\mathrm{\Delta }\mathrm{L}$). In order to evaluate the spatial extent of the impact of adjacency perturbations on ocean color satellite data, the second is the most appropriate, since it is strictly related to the actual sensor radiometric resolution.

For typical observation conditions and a straight coastline along the south–north direction, adjacency effects have been analyzed as a function of the distance from the coast by accounting for the radiometric resolutions of different ocean color sensors [56]. Figure 28 illustrates $L_{\mathrm{adj}}/E_0$ at representative wavelengths along a transect perpendicular to the coast. The radiometric noise thresholds for SeaWiFS and MODIS sensors are also indicated [56]. The horizontal scale of $L_{\mathrm{adj}}$ is $\sim 10\mathrm{km}$. Considering that the selected observation conditions correspond to a clear atmosphere, results are consistent with those from Tanré et al. [58] and Vermote et al. [57], indicating a horizontal scale for the environment weighting function of $\sim 1\mathrm{km}$ for the aerosol scattering and $\sim 10\mathrm{km}$ for the Rayleigh scattering.

 

Fig. 28. Values of $L_{\mathrm{adj}}/E_0$ at representative wavelengths and for typical observation conditions as a function of the distance from a straight-line coast. Error bars represent the standard deviation. $L_w$ is assumed constant all along the transect. The dashed and dotted lines correspond to $\pm \mathrm{NE}\mathrm{\Delta }\mathrm{L}/E_0$ for the SeaWiFS and the MODIS sensors, respectively.

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However, results clearly indicate that for a sensor such as SeaWiFS the adjacency radiance becomes lower than the radiometric noise threshold only at $\sim 25\mathrm{km}$ off the coast in the NIR. Differently, for a more sensitive sensor such as MODIS, the adjacency radiance at 555 and 670 nm remains above the estimated sensor radiometric noise up to $\sim 20\mathrm{km}$, while in the NIR it may still be appreciable at distances larger than 30 km. At blue wavelengths the uncertainties on simulated results are much larger than at other wavelengths (particularly towards the coast), and most importantly, results at blue wavelengths may be particularly sensitive to the orientation of the coast with respect to the Sun and to a correct characterization of the sea surface roughness (see Section 6.C). Results from the analysis presented in this section are summarized in Table 8.

Table 8. Values of ${\mathsf{\xi }}_{{\mathsf{L}}_{\mathsf{tot}}}\pm {\mathsf{\sigma }}_{{\mathsf{\xi }}_{{\mathsf{L}}_{\mathsf{tot}}}}$ in Percent at Representative Wavelengths $\mathsf{\lambda }$ [nm] and at Several Distances from the Coast $\mathsf{d}$ [km] Under Typical Observation Conditions for a Straight Coastline Oriented in the South–North Direction

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The estimates presented for the horizontal range of adjacency effects have been determined for typical observation conditions. Recalling considerations drawn in Section 6.A, the spatial extent of adjacency effects is expected to be larger than the average in summer, for off-nadir views, for observations from over the land, and for a lower aerosol optical thickness. Conversely, it is predicted to be smaller in winter, for nadir viewing, for observations from over the sea, and for a higher aerosol optical thickness.

It is finally underlined that the above results were obtained with the assumption of a plane-parallel atmosphere. Any actual departure from this assumption (i.e., any horizontal inhomogeneity of the atmospheric properties) may consistently influence the horizontal range of adjacency effects.

7. Summary and Conclusions

A methodology to accurately quantify and analyze the adjacency effects in coastal waters accounting for the Fresnel reflectance of a wind-roughed sea surface, off-nadir views, the actual coastal morphology, and atmospheric multiple scattering has been illustrated. A modeling of the adjacency radiance has been proposed assuming isotropic and homogeneous land and water reflectances, allowing the decoupling of land and water optical properties from atmospheric scattering. The proposed model additionally allow distinguishing between the different contributions to the adjacency radiance, namely the land contribution (i.e., the radiance at the sensor originating from the area covered by the land) and the masked water and sea surface contributions (i.e., the water-leaving radiance and the reflected sea surface radiance that would reach the sensor from the same area if still covered by the sea).

Simulations requiring a full 3D description of the propagating system have been performed with the newly developed NAUSICAA backward MC code. Other simulations have been carried out with the highly accurate FEM plane-parallel numerical code.

Adjacency effects have been simulated for realistic and representative observation conditions along a transect crossing the Venice Lagoon and intercepting the AAOT site in the northern Adriatic Sea (utilized to support satellite ocean color validation activities). The description of the propagating system includes observation geometries representative for SeaWiFS, MODIS, and MERIS ocean color sensors; seasonal land and water optical properties and typical illumination conditions; the atmospheric variability observed for clear conditions in the northern Adriatic Sea; and a detailed picture of the complex coastal morphology.

An accurate evaluation of the uncertainties affecting simulated data has been performed accounting for an average ocean color sensor radiometric resolution. This included an assessment procedure of NAUSICAA simulated results through comparison with results from the FEM numerical plane-parallel code for all considered test cases, and a benchmark with respect to data from the literature.

The analysis of the average contribution ${\overline{\xi }}_{L_{\mathrm{tot}}}$ of adjacency radiance to the total signal at TOA indicates values lower than $\pm 1\%$ throughout the transect in the 412–555 nm spectral interval, where land and water spectral albedos are similar. At the other wavelengths, ${\overline{\xi }}_{L_{\mathrm{tot}}}$ always remains positive and shows a steep spatial gradient with values at the coast ranging from $\sim 2\%$ at 670 nm to $\sim 11\%$ at 865 nm. At the AAOT (about 15 km offshore) ${\overline{\xi }}_{L_{\mathrm{tot}}}$ never exceeds $\pm 0.5\%$ at the visible wavelengths, but becomes appreciably larger in the NIR where it reaches $\sim 2\%$. The highest values occur in the NIR, where the land is highly reflective and the water highly absorbing.

Adjacency effects exhibit sensitivity to the aerosol load and no appreciable dependence on the aerosol type. Furthermore, adjacency effects show a strong seasonal dependence, with increased values in summer when the land is highly reflecting, the water is highly absorbing, and the Sun elevation is high. Conversely, a decrease is observed in winter. At 865 nm and in proximity of the coast, ${\overline{\xi }}_{L_{\mathrm{tot}}}$ seasonally spans from 5% in winter to 18% in summer.

Adjacency effects are slightly larger for a sensor looking at the sea from over the land. Values display a significant increase with the zenith angle of observation, a slight variation with the Sun-sensor relative azimuth, and an appreciable dependence on the mutual position of land and Sun. The latter two are a consequence of the strong anisotropy of the sea surface reflectance, which induces a significant dependence of adjacency effects on the Sun azimuth. The sea surface reflectance anisotropy is also mostly responsible for adjacency effects at those wavelengths where land and sea albedos are similar (412–510 nm).

No significant differences have been found when distinguishing among the observation geometries of the considered satellite sensors.

The need to provide easy-to-implement formulations of the adjacency effects in coastal waters has been commonly accomplished by applying approximations, such as an exponential decay of the environment weighting function [7,9,11], Lambertian reflecting surfaces [7,11], a nadir geometry of observation [11], a straight coastline [11], and the single scattering approximation [9]. The separate impact of such approximations on the simulation of adjacency effects in the selected study region has been analyzed considering significant any misestimate larger than the average radiometric resolution of the selected sensor, which in turn has been assumed as the upper limit for random uncertainties on simulated data.

The analysis shows that the assumption of an exponential decay of the environment weighting functions leads to a slight underestimate of adjacency effects with distance. The impact of neglecting the anisotropy of the sea surface reflectance is particularly significant towards shorter wavelengths: the dependence of the adjacency effects on the angle of observation and on the mutual position of land and Sun disappears, while the average adjacency contributions are significantly reduced (e.g., by $\sim 40\%$ throughout the transect at 490 nm). Moving towards the NIR, misestimates become appreciable only in the right vicinity of the coast. A quasi-nadir geometry of observation considerably underestimates off-nadir adjacency contributions throughout the transect. At 865 nm average adjacency contributions for ${\theta }_v=50°$ are $\sim 40\%$ larger than those for ${\theta }_v=5°$. The assumption of a straight coastline located for each sea element at the minimum distance from the land and oriented along the south–north direction indicates significant differences only for observations in the Venice Lagoon, where the coastal morphology decidedly differs from a straight coastline. Finally, solving the RTE in single scattering approximation induces misestimates that are particularly important at shorter wavelengths, but still appreciable in the NIR.

An evaluation of the horizontal range of adjacency effects, describing the distance at which the adjacency radiance becomes negligible (i.e., lower than $\mathrm{NE}\mathrm{\Delta }\mathrm{L}$), has been performed to estimate the spatial extent of adjacency perturbations on ocean color satellite data. It is noted that the horizontal scale of adjacency effects, describing the distance at which adjacency radiance is decreased by a factor $e$, is not the best indicator for this purpose, since it is not related to the sensor radiometric resolution. Estimates performed for typical observation conditions and assuming a coastline oriented in the south–north direction indicate values of $\sim 25\mathrm{km}$ off the coast in the NIR for SeaWiFS. For a more sensitive sensor such as MODIS the adjacency radiance at 555 and 670 nm remains above the sensor radiometric resolution up to $\sim 20\mathrm{km}$, while in the NIR it might still be appreciable at distances larger than 30 km. Results at the blue wavelengths may be particularly sensitive to the orientation of the coast with respect to the Sun and to a correct characterization of the sea surface roughness. The horizontal range of adjacency effects is additionally expected to be larger than the average in summer, for off-nadir views, for observations from over the land, and for a lower aerosol optical thickness. Conversely, it is predicted to be smaller than the average in winter, for nadir viewing, for observations from over the sea, and for a higher aerosol optical thickness.

In conclusion, overall results point out that, within the accuracy limits defined by the radiometric resolution of the ocean color sensors, adjacency effects in coastal waters might be significant at both visible and NIR wavelengths up to several kilometers off the coast. The theoretical quantification of adjacency effects should rely on a proper description of the atmospheric aerosol concentration, account for the seasonal dependence of geophysical parameters, and include an appropriate description of multiple scattering events. It should definitely consider the actual angle of observation, allowing for off-nadir views. Additionally, and particularly at shorter wavelengths, it should take into account the radiance contributions from the masked sea region, include a proper description of the intrinsic directional properties of the sea surface, and take into consideration the land location with respect to the Sun and sensor position. Conversely, a straight coastline, opportunely oriented with respect to the Sun, can be assumed without a significant loss in accuracy, besides cases of complex coastal patterns such as the semi-enclosed basin of the Venice Lagoon.

The above considerations are relevant indications for the development of approximate algorithms for the actual quantification of adjacency effects in coastal waters. It is, however, remarked that the influence on retrieved products of the approximations applied in an operational algorithm for the correction of adjacency effects strictly depends on the overall correction procedure. Because of this, targeted analysis should be conducted for each specific data reduction scheme.

The conclusive remarks are valid for mid-latitude coastal regions characterized by a deciduous vegetation type in the absence of snow. In the presence of nondeciduous vegetation results would likely show a limited seasonal variation and higher average values. The presence of snow would exhibit radically different results, leading to consistent adjacency effects in the blue end part of the spectrum [12].

Once the geometric and atmospheric inputs are defined, the proposed modeling of the adjacency radiance allows an easy evaluation of the adjacency effects for a wide variety of land and water spectral signatures. Thus, the possibility to assume a straight coastline without a significant loss in accuracy suggests that, widening the range of the selected geometric and atmospheric features, the proposed methodology could be easily applied to evaluate adjacency effects in the presence of a large variety of realistic conditions. Look-up tables or parameterizations of the modeling functions $C$ and $W$ could be the basis for the development of an operational scheme to correct adjacency effects in coastal waters.

The results from this study furthermore constitute an important reference frame for the assessment of any algorithm for the quantification and correction of adjacency effects. This is particularly relevant when considering the difficulties encountered in the experimental quantification of adjacency effects.

The dependence of adjacency effects on the intrinsic directional reflectance properties of the surface suggests the need to further investigate the sensitivity of results on land reflectance anisotropy and wind velocity. Finally, the influence of different aerosol profiles (including the presence of volcanic aerosols), thin cirrus clouds, as well as any departure from the adopted assumption of a plane-parallel atmosphere should also be investigated.