Abstract

The direct and diffuse components of downwelling irradiance have in general different path lengths in water, and hence they decrease differently with sensor depth. Furthermore, the ever-changing geometry of a wind-roughened and wave-modulated water surface induces uncorrelated intensity changes to these components. To cope with both effects, an analytic model of the downwelling irradiance in water was developed that calculates the direct and diffuse components separately. By assigning weights fdd and fds to the intensities of the two components, measurements performed at arbitrary surface conditions can be analyzed by treating fdd and fds as fit parameters. The model was validated against HydroLight and implemented into the public-domain software WASI. It was applied to data from three German lakes to determine the statistics of fdd and fds, to derive the sensor depth of each measurement and to estimate the concentrations of water constituents.

©2012 Optical Society of America

1. Introduction

The irradiance illuminating the water surface is composed of two spectrally and geometrically distinct components: the direct irradiance from the Sun, Edd, which is an almost parallel beam of light from the direction of the Sun; and the diffuse irradiance of the sky, Eds, formed by rays with incidence angles covering the upper hemisphere. A number of measurement techniques and simulation tools exist to determine both components experimentally or by modeling.

In water, the two components cannot be determined reliably, not by measurement or by simulation, because the water surface is almost never perfectly flat. Waves, ripples, and foam incline the water surface with spatially highly variable slopes, and this locally variable geometry of the water surface is changing quickly in time because of wind and currents. Because the surface geometry determines the refraction angle, the angular distribution of downwelling radiation is changing for an observer in water locally and temporally in an unpredictable way. The induced changes depend on wind speed, solar elevation, and depth [1,2]. Observations show that variations are typically of the order of 20% to 40% in the upper few meters concerning intensity [3,4] and 5% concerning spectral shape across the visible [5], but flashes can increase intensity up to a factor of five [6]. Because of this huge variability, in-water measurements of irradiance, and consequently of reflectance, are quite challenging [7,8]. For this reason, some measurement concepts, e.g., the widely used Ocean optics protocols [9], recommend to make all incident irradiance measurements above the surface.

Radiative transfer models usually account for the influence of the water surface on the underwater light field by an empirical relationship between wind speed and the inclination of the wave facets [1012]. Because of the unpredictable behavior of individual facets, these models can describe the influence on irradiance only as statistical averages but not for individual measurements. The analytic model developed in this paper uses a different approach that is not based on parameters describing wind and waves, but instead it is based on the induced changes of Edd and Eds intensity. Edd and Eds are treated as two spectrally well-defined light sources with unknown intensities. Their spectral shapes at depth z are calculated individually using an analytic model, and their intensities are fit parameters during data analysis. In this way, inverse modeling allows to analyze irradiance measurements even for geometrically undefined radiance distributions. Results of data analysis are a number of model parameters like sensor depth and concentrations of water constituents as well as separation of Edd and Eds.

The depth dependency of irradiance is commonly parameterized by the diffuse attenuation coefficient, Kd, which is an apparent optical property. The new approach replaces Kd by an inherent optical property, a+bb, where a is the average absorption coefficient and bb is the average backscattering coefficient of the water column between surface and sensor. The model was validated using the well-established radiative transfer program HydroLight [10] and implemented into the public-domain software WASI [1315] for forward calculation and inverse modeling.

2. Parameterization of Irradiance

The downwelling irradiance, Ed, is the sum of a direct (Edd) and a diffuse (Eds) component. In this study, these components are defined according to their spectral shapes at the location of the observer in air or in water: Edd(λ,z) represents the spectral irradiance of the Sun disk for an observer at depth z, and Eds(λ,z) the average irradiance of the sky excluding the Sun. Upwelling radiation that is reflected at the water surface or scattered in the water in downward direction is neglected. Hence,

Ed(λ,z)=fddEdd(λ,z)+fdsEds(λ,z).
λ denotes wavelength. The parameters fdd and fds describe the intensities of Edd and Eds relative to conditions with undisturbed illumination geometry. For an observer in air, these reference conditions are defined by a cloudless atmosphere and unobscured sky view, for an observer in water additionally by a plane water surface. 0fdd<1 corresponds to measurements when obstacles or waves decrease the magnitude of the direct component compared with a plane surface (shadowing effect), fdd>1 when Edd intensity is increased (wave focusing effect). Similarly, a decrease of the diffuse component compared with a plane surface is described by 0fds<1, and an increase by fds>1. Note that wavelength-independent errors of Ed(λ,z), introduced, e.g., by erroneous sensor calibration and expressed by a multiplicative factor (1+εd), correspond to (1+εd)fdd and (1+εd)fds; hence, all model parameters except fdd and fds are insensitive to such errors.

The diffuse component at depth z is related to that below the surface as follows:

Edd(λ,z)=Eds(λ,0)exp{Kds(λ)zlds}.
The symbol 0 indicates that the sensor is in water and just beneath the water surface. Kds is the attenuation coefficient of the water column for the diffuse irradiance between the depths 0 and z. A factor lds is introduced as the average path length of diffuse radiation relative to sensor depth.

The direct component is attenuated along a path with length z/cosθsun:

Edd(λ,z)=Edd(λ,0)exp{Kdd(λ)zlddcosθsun}·
Kdd is the average attenuation coefficient of the water column for the direct irradiance component between the depths 0 and z. A path length factor for direct radiation, ldd, is introduced as the ratio of the mean path length of all radiation with spectral shape Edd(λ,z) to the geometric path length of directly transmitted sunrays. This definition treats all that radiation as direct irradiance that comes, above water, from the Sun direction, even if in water the angle is different from the Sun zenith angle. Such angular deviations arise as a consequence of refraction at water surface elements that are inclined due to waves or foam, or by scattering processes in the water. The Sun zenith angle in water, θsun, is related to that in air, θsun, by Snell’s law nWsinθsun=sinθsun, with nW denoting the refractive index of water.

The irradiance components beneath the surface (0) are related to those in air (0+) through Edd(λ,0)=Edd(λ,0+)(1ρdd) and Eds(λ,0)=Eds(λ,0+)(1ρds), where the reflectance factors ρdd and ρds describe the losses of the direct and diffuse components of the downwelling irradiance at the air-water interface, respectively. Typical values are ρdd0.02 and ρds0.06 for small Sun zenith angles. The actual values are calculated as function of the Sun zenith angle as follows:

ρdd=12|sin2(θsunθsun)sin2(θsun+θsun)+tan2(θsunθsun)tan2(θsun+θsun)|,
ρds=0.06087+0.03751(1cosθsun)+0.1143(1cosθsun)2.

Equation (4) is the Fresnel equation for unpolarized radiation [16]. Equation (5) was derived from radiative transfer simulations as described below. Both equations are valid for a plane water surface. Waves and foam alter locally the inclination of the water surface, and the resulting effective values of θsun lead to changes of ρdd and ρds, which can be calculated only for well-known wind speed and wind direction as statistical averages [11,12]. However, as the wavelength dependencies of ρdd and ρds are small, the impact of these changes on Edd and Eds can be described by wavelength-independent correction factors εdd and εds, which change fdd and fds toward (1+εdd)fdd and (1+εds)fds. Consequently, the parameterization of Ed through Eq. (1) accounts for changes of surface reflections induced by a wind-roughened water surface.

A suitable model to calculate Edd(λ,0) and Eds(λ,0) for undisturbed illumination geometry has been developed by Gregg and Carder [17]. This widely used model is adopted here. The equations used in the software implementation of WASI 4 are recalled; for more details, see Gregg and Carder [17]. The direct component of downwelling irradiance is calculated as follows:

Edd(λ,0)=F0(λ)cosθsunTr(λ)Taa(λ)Tas(λ)Toz(λ)To(λ)Twv(λ)(1ρdd).
F0(λ) is the extraterrestrial solar irradiance corrected for Earth–Sun distance and orbital eccentricity ϵ=0.0167:F0(λ)=H0(λ){1+ϵcos[2π(d3)/365]}2, where H0(λ) is the mean extraterrestrial irradiance and d is day of the year (measured from 1 January). Ti is the transmittance of the atmosphere after scattering or absorption of component i (Tr: Rayleigh scattering, Taa: aerosol absorption, Tas: aerosol scattering, Toz: ozone absorption, To: oxygen absorption, and Twv: water vapor absorption).

The diffuse component of downwelling irradiance is given by

Eds(λ,0)=[Edr(λ)+Eda(λ)](1ρds).
Edr denotes the diffuse component caused by Rayleigh scattering, and Eda the diffuse component due to aerosol scattering. These are parameterized as follows:
Edr(λ)=12F0(λ)cosθsun(1Tr(λ)0.95)Taa(λ)Toz(λ)To(λ)Twv(λ),
Eda(λ)=F0(λ)cosθsunTr(λ)1.5Taa(λ)Toz(λ)To(λ)Twv(λ)(1Tas(λ))Fa.

The aerosol forward scattering probability, Fa, is calculated using the empirical equation Fa=10.5exp{(B1+B2cosθsun)cosθsun} with B1=B3[1.459+B3(0.1595+0.4129B3)], B2=B3[0.0783+B3(0.38240.5874B3)], B3=ln(1cosθsun). The asymmetry factor of the aerosol scattering phase function is calculated as cosθsun=0.1417α+0.82 when α is in the range 0 to 1.2, and set equal 0.82 for α<0 and 0.65 for α>1.2.

The atmospheric transmittance spectra are calculated as follows:

Taa(λ)=exp[(1ωa)τa(λ)M],
Tas(λ)=exp[ωaτa(λ)M],
Tr(λ)=exp[M/(115.6406λ41.335λ2)],
Toz(λ)=exp[aoz(λ)HozMoz],
To(λ)=exp1.41ao(λ)·M[1+118.3ao(λ)·M]0.45,
Twv(λ)=exp0.2385awv(λ)·WV·M[1+20.07awv(λ)·WV·M]0.45.

Aerosol is parameterized in terms of aerosol optical thickness, τa(λ)=β(λ/λa)α, and aerosol single scattering albedo, ωa=(0.0032AM+0.972)exp{3.06×104RH}. The Angström exponent α determines the wavelength dependency, and the turbidity coefficient β is a measure of aerosol concentration. The reference wavelength λa is set to 550 nm. α typically ranges from 0.2 to 2, and β ranges from 0.16 to 0.50. β is related to horizontal visibility V and aerosol scale height Ha: β=τa(550)=3.91Ha/V. Typical values are 8 to 24 km for V, and 1 km for Ha. The parameters of ωa are air mass type, AM, which ranges from 1 (typical of open-ocean aerosols) to 10 (typical of continental aerosols), and relative humidity, RH, with typical values from 46% to 91%. WV is the total precipitable water vapor content (in units of cm) in a vertical path from the top of the atmosphere to the surface.

The atmospheric path length is M=1/cosθsun+a(90°+bθsun)c]. The numerical values used by Gregg and Carder (a=0.15, b=3.885°, c=1.253) were replaced by updated values a=0.50572, b=6.07995°, c=1.6364 from Kasten and Young [18]. M=MP/(1013.25mbar) is the path length corrected for nonstandard atmospheric pressure P, and Moz=1.0035/(cos2θsun+0.007)1/2 is the path length for ozone.

Aerosol optical thickness (τa) and the absorption coefficients of ozone (aoz), oxygen (ao), and water vapor (awv) are wavelength dependent; the other parameters (ωa, τa, M, M, Hoz, Moz, WV) are independent of λ. Gregg and Carder [17] list the values of aoz(λ), ao(λ), awv(λ) for the range 400–700 nm in 1 nm intervals. I extended these spectra to the range 300–1000 nm, as described in Section 3.

Because the ratio of direct to diffuse irradiance, rd, is the key parameter of Ed variance in water [5], analytic equations are derived that express rd as function of the parameters of the irradiance model. Just below the water surface, the ratio is given by

rd(λ,0)=fddfdsEdd(λ,0)Eds(λ,0)=fddfds2Tr(λ)Tas(λ)(1ρdd)[1Tr(λ)0.95+2Tr(λ)1.5(1Tas(λ))Fa](1ρds).

Equation (16) shows that the wavelength dependency of rd(0) is determined by the scattering components of the atmosphere but not by its absorbing components. Consequently, the distinctive spectral gradients of Ed, which are caused by the extraterrestrial solar irradiance and the absorbing components of the atmosphere, are not present in rd(0). Rather rd(0) has a smooth spectral shape, which is increasing almost linearly from the shortwave to the longwave {see Fig. 3(a) in Gege and Pinnel [5]}. For depth z the following relationship is obtained:

rd(λ,z)=fddfdsEdd(λ,z)Eds(λ,z)=fddfdsEdd(λ,0)exp{Kdd(λ)zldd/cosθsun}Eds(λ,0)exp{Kds(λ)zlds}
or
rd(λ,z)=rd(λ,0)exp{(Kds(λ)ldsKdd(λ)lddcosθsun)z}.

 

Fig. 1. Atmospheric transmission after absorption by ozone, oxygen, and water vapor.

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Fig. 2. Effective absorption coefficients of the atmospheric components ozone, oxygen, and water vapor.

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Fig. 3. Comparison of analytically estimated sensor depths with results from inverse modeling.

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Hence, the wavelength dependency of rd(0) is altered at depth z>0 by a factor whose spectral shape is determined by Kds(λ) and Kdd(λ). For the concentrations and depths studied here (z<5m), these changes are relevant only above 700 nm {see Fig. 3(b) in Gege and Pinnel [5]}.

3. Optical Properties of the Atmosphere

The adopted model of downwelling irradiance is a semi-empirical parameterization of the radiative transfer within the atmosphere. In particular, absorption of radiation is simplified by ignoring the temperature and pressure dependencies of the absorption coefficients of the atmospheric gases. These coefficients are replaced by “effective” absorption spectra representing averages over the vertical profile. Gregg and Carder [17] provide their values for the wavelength interval 400–700 nm. Because the measurements of this study cover a wider range, the effective absorption values were derived for an extended spectral range (300–1000 nm) by simulation.

The calculations were performed using version 1.5 of the radiative transfer model MODTRAN-3 [19], which includes highly resolved spectral absorption coefficients from the HITRAN database [20]. Program settings were as follows. Altitude of surface relative to sea level: 0, observer height: 0, Sun zenith angle: 30°, horizontal visibility: 23 km, rain rate: 0, multiple scattering: yes, model: midlatitude summer, aerosol model: rural, clouds: none. The data interval was set to 40cm1 for the range 300–500 nm, 20cm1 for 500–725 nm, and 10cm1 for 725–1000 nm. Figure 1 shows the calculated transmission spectra for ozone, oxygen, and water vapor (labeled WASI4).

The figure shows for comparison transmission spectra calculated using Eqs. (11)–(13) and the absorption spectra aoz(λ), ao(λ), awv(λ) of Gregg and Carder [17]. Ozone scale height and water vapor concentration were adjusted to match the MODTRAN spectra.

A new set of absorption spectra aoz(λ), ao(λ), awv(λ) was calculated for the spectral range 300–1000 nm by inverting the MODTRAN transmission spectra using Eqs. (1113) and parameter values obtained from matching the transmission spectra: Hoz=0.38cm, WV=2.5cm, and Moz=1.07479, M=M=1.07834. A comparison of the new absorption spectra (labeled WASI4) with those of Gregg and Carder [17] is shown in Fig. 2.

The comparison indicates that the new spectra not only cover a wider wavelength range but also are spectrally finer resolved. Gregg and Carder provide no information concerning spectral resolution of their data. To illustrate the effect of spectral sampling, a third data set is included in Fig. 2 (labeled HE5), which is used in the software HydroLight [10]. The three data sets are similar but not identical. The differences concern mainly spectral fine structures, in particular those of awv(λ). The mismatches are caused by the gases’ numerous absorption lines with bandwidths far below 1 nm. For spectral regions with strong gradients, the irradiance measured by a sensor depends very much on the spectral response of each band, i.e., on center wavelength, width, and spectral shape. The HE5 and WASI4 spectra were calculated using slightly different spectral weighting; hence, their noticeable differences indicate that spectral fine structures of Ed measurements using real sensors should be interpreted with care. Neither the WASI4 nor the HE5 spectra H0(λ), aoz(λ), ao(λ), and awv(λ) are suited to model properly spectral fine structures of Ed measurements of sensors with a resolution of the order of 1 nm or higher.

4. Optical Properties of the Water Body

A beam of light passing a water layer is affected by absorption and scattering processes along its path, resulting in spectrally dependent intensity changes. For irradiances, a diffuse attenuation coefficient K parameterizes the average changes along the various paths. The bulk coefficient for Ed, denoted Kd, has been studied extensively (see Bukata [21] for an overview), but I am not aware of analytic models for the coefficients Kds and Kdd as defined by Eqs. (2) and (3), respectively.

Because an irradiance sensor detects besides the direct also radiation from angles covering a hemisphere, only a part of the photons that are scattered out of the incident direction is lost for detection. For a beam incident perpendicular on an irradiance sensor, these are the backscattered photons. They are parameterized by the backscattering coefficient bb(λ), which measures the resulting decrease of photon flux per length (in units of m1). Hence, the following approximation is made:

Kds(λ)=Kdd(λ)=a(λ)+bb(λ),
with a(λ) denoting the absorption coefficient of the water layer. Equation (18) corresponds to a widely used approximation of the wavelength dependency of Kd(λ) [22]. The bb term is exactly valid, however, only for the idealized condition of perpendicular incidence of all radiation, which is never the case during in situ measurements. For beams with non-nadir incidence, a portion of backscattered photons is detected, and a fraction of the forward scattered radiation becomes undetectable. To validate the approach, radiative transfer simulations using the well-established model HydroLight [10] are performed below for different depths and Sun zenith angles. These confirm that Eq. (18) describes the wavelength dependency of Kds and Kdd with high accuracy.

The optical properties of water are calculated as follows:

a(λ)=aW(λ)+aph(λ)+aY(λ)+ad(λ),
bb(λ)=bb,W(λ)+bb,X(λ),
where aW(λ) and bb,W(λ) are the absorption and backscattering coefficients of pure water, respectively. The spectrum aW(λ) used in this study is a combination from different sources: 350–390 nm, interpolation between Quickenden and Irvin [23] and Buiteveld et al. [24]; 391–787 nm, Buiteveld et al. [24]; 788–874 nm, author’s unpublished measurements on UV-treated pure water; 875–1000 nm, Palmer and Williams [25]. For bb,W(λ) the relation of Morel [26] is used: bb,W(λ/500)4.32 (λ in nm) with b1=0.00111m1 for fresh water and b1=0.00144m1 for oceanic water with a salinity of 35–38%.

Four types of water constituents are considered: phytoplankton, gelbstoff, detritus, and suspended particles. The first three are parameterized by their spectral absorption coefficients aph(λ), aY(λ), and ad(λ), respectively; suspended particles by their spectral backscattering coefficient, bb,X(λ). Backscattering by phytoplankton cells is included in bb,X(λ).

Phytoplankton concentration is expressed as mass of the pigments chlorophyll-a plus phaeophytin-a per water volume (mgm3), its specific absorption coefficient is species dependent. Frequently, a mixture of several phytoplankton species is present in the water. The resulting phytoplankton absorption coefficient is the sum of the individual contributions:

aph(λ)=Ciai*(λ).
Ci is the concentration of phytoplankton class number i, ai*(λ) is the specific absorption coefficient of that class. The database of the software WASI, which is used in this study, provides six spectra ai*(λ) representing the phytoplankton in Lake Constance [27,28]. Spectrum 0 represents a typical phytoplankton mixture in that lake, Spectrum 1 cryptophyta with low concentration of the pigment phycoerythrin, Spectrum 2 cryptophyta with high phycoerythrin concentration, Spectrum 3 diatoms, Spectrum 4 dinoflagellates, and Spectrum 5 green algae. For the calculations below, Ci is set zero for i1.

Gelbstoff (yellow substance) is the colored dissolved organic matter (CDOM) in the water and is composed of a huge variety of organic molecules. Its absorption coefficient is calculated as aY(λ)=YaY*(λ), where aY*(λ) is the specific absorption spectrum, normalized at λ0, and Y=aY(λ0) is the absorption coefficient at λ0. The spectrum aY*(λ) can either be imported from file, or it can be modeled by the frequently used approximation exp[S(λλ0)]. For the calculations below, the exponential approximation is used with λ0=440nm and S=0.014nm1, which is representative of a great variety of water types [29,30].

Detritus (also known as tripton [31] or bleached particles [32]) is the collective name for all absorbing nonalgal particles in the water. Its absorption spectrum is parameterized as ad(λ)=Dad*(λ), with ad*(λ) denoting specific absorption, normalized at the same wavelength λ0 as Gelbstoff, and D=ad(λ0) describing the absorption coefficient at λ0. The spectrum ad*(λ) is imported from file. In this study, detritus is neglected.

Suspended particle concentration is expressed as mass per water volume (gm3). Its backscattering coefficient is calculated as

bb,X(λ)=X·bb,X*·bX(λ)+CMie·bb,Mie*·(λ/λS)n.
This equation allows to model mixtures of two spectrally different types of suspended matter. The first type is defined by a scattering coefficient with arbitrary wavelength dependency, bX(λ), which is imported from file. X is the concentration and bb,X* is the specific backscattering coefficient. The second type is defined by the normalized scattering coefficient (λ/λS)n, where the Angström exponent n is related to the particle size distribution. CMie is the concentration and bb,Mie* is the specific backscattering coefficient. The parameters of Eq. (22) are set by default to bb,X*=0.0086m2g1, bX(λ)=1, bb,Mie*=0.0042m2g1, λS=500nm, n=1, which are representative for Lake Constance [28]. For the simulations below, bX(λ)=1 and CMie=0 is set.

The relative path lengths of diffuse and direct radiation were estimated for z in the range from 0.5 to 5 m and C0=2mgm3, X=0.6gm3, Y=0.3m1 as a function of the Sun zenith angle in water using HydroLight simulations. As described in Section 6, the following relationships were obtained:

lds=1.1156+0.5504(1cosθsun),
ldd=1.

5. Software Implementation

The described model was implemented into version 4 of the Water Color Simulator WASI [1315]. The public-domain software WASI can be used to simulate and analyze the most common types of spectral measurements of shipborne instruments.

Simulation of a measurement (forward modeling) is done by attributing a value to each model parameter and then by calculating the corresponding model curve using a set of equations that represents the model. For simulation of an Ed(λ) measurement at depth z, the spectra Taa(λ), Tas(λ), Tr(λ), Toz(λ), To(λ), Twv(λ), Edd(λ,z), Eds(λ,z), Edr(λ,z), Eda(λ,z), rd(λ,z), a(λ), Kdd(λ), and Kds(λ) are calculated as intermediate results. WASI allows visualization and export to file of each of these spectra. Forward modeling is used in Subsection 6.F to compare the developed irradiance model with a reference model.

Data analysis (inverse modeling) aims to determine the values of unknown model parameters, called fit parameters. The fit parameters of the irradiance model are listed in Table 1. Their values are determined iteratively as follows. In the first iteration, a model spectrum Ed(λ) is calculated using Eq. (1) for user-defined initial values of the fit parameters. This spectrum is compared with the measured one, Edmeas(λ), by calculating the residuum |Edmeas(λi)Ed(λi)|2 as a measure of correspondence. Then, in the further iterations, the fit parameter values are altered using the Downhill Simplex algorithm [33,34], resulting in altered model curves and altered residuals. The procedure is stopped when the calculated and the measured spectrum agree as good as possible, which corresponds to the minimum residuum. The values of the fit parameters of the final step are the fit results. WASI provides different methods to tune this algorithm, in particular, wavelength-dependent weighting and changing the residuum definition; for details, see Gege and Albert [14]. Inverse modeling is used in Section 7 to analyze field measurements.

Tables Icon

Table 1. Parameters of the Irradiance Model That Can Be Used in WASI as Fit Parameters

If the initial value of a fit parameter is very different from its correct value, the inversion algorithm may not be able to find the parameter combination for which the model curve has the best correspondence with the measured spectrum. This problem, which increases with the number of fit parameters, can be reduced if a reasonable “first guess” of the fit parameters can be made, in particular, of those parameters that can cause large deviations. Sensor depth is such a parameter. An analytic algorithm has been implemented into WASI, which estimates a first guess, z0, from the ratio r=Ed(λ1)/Ed(λ2) of an irradiance measurement at two wavelengths, λ1 and λ2, using the following equation:

z0=gcosθsunlnrlnr(0)Kdd(λ2)Kdd(λ1).
Equation (25) is obtained (for g=1/ldd) from Eq. (3) by solving the ratio r(z)=Edd(λ1,z)/Edd(λ2,z) for z. The ratio r(0) at depth z=0 is estimated using Eq. (6). Note that the measured ratio r is different from r because an irradiance sensor detects the diffuse component. The empirical parameter g accounts for this additional component. An experimental test of Eq. (25) is shown in Fig. 3 for λ1=800nm, λ2=680nm, g=1.2. It can be seen that z0 is highly correlated to z up to a depth of approximately 3 m. The mean absolute difference is 0.06 m, and the mean relative difference is 9% for z in the range from 0.1 to 3 m. Sensor noise was large in the range of 735 to 900 nm at depths above approximately 2 m. This may explain the deviations for depths >2m, which are, however, irrelevant for the purpose to get a first guess of z.

6. Comparison with a Reference Model

The commercial software HydroLight [10] is taken as reference to determine some model parameters (ldd, lds, ρds) and to estimate the accuracy of the analytical model. HydroLight is probably the most widely used numerical model for calculating the radiative transfer in water bodies. The current version is called HE5 (HydroLight-EcoLight version 5.1). It solves the radiative transfer equation in scattering media using the invariant embedding method [35].

A. Consistency of Input Spectra

A comparison of the extraterrestrial solar irradiance spectrum, H0(λ), is shown in Fig. 4. The HE5 and WASI4 spectra of H0(λ) correspond well. Typical differences are of the order of a few percent, except concerning spectral fine structures. Similar results were obtained in Section 3 (see Fig. 2) for the absorption spectra of the atmospheric gases ozone, oxygen, and water vapor. Consequently, all input spectra of WASI for calculating irradiance above the surface can be considered consistent with those of HE5. Remember that spectral fine structures of the order of 1 nm cannot be modeled accurately. To avoid discrepancies resulting from slightly different input spectra, the RADTRAN-X database file of HE5 was exchanged for all further runs by the spectra H0(λ), aoz(λ), ao(λ), and awv(λ) used in WASI. Also, the optical properties of the water body were made consistent by using the spectra aW(λ) and api*(λ) of the WASI database in HE5.

 

Fig. 4. Extraterrestrial solar irradiance.

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B. Consistency of Above-Water Irradiance Model

Irradiance spectra Ed(λ), Edd(λ), Eds(λ) were compared above the water surface for the spectral range from 350 to 900 nm. HE5 calculates these spectra using RADTRAN-X, which is an implementation of the Gregg and Carder model [17] using an extended database covering the range 300 to 1000 nm. The bands are specified in HE5 by their border wavelengths, i.e., irradiance E of band k is calculated as the average E(λi), where λi runs from the center wavelength of band k1 to the center wavelength of band k+1 in steps of 1 nm. For the calculations described below, such spectral averaging was also implemented in WASI4, and a sampling interval of 2 nm was chosen; i.e., irradiance at wavelength λ was calculated as [E(λ1nm)+E(λ)+E(λ+1nm)]/3.

The common parameters of WASI4 and HE5 were set to d=94 (April 4), AM=1, RH=60%, P=1013.25mbar, WV=2.5cm, and Hoz=0.3cm. Horizontal visibility was chosen as V=15km in HE5. The aerosol parameters were set to α=1.317, β=0.2606, and Ha=1km in WASI4. Because these cannot be selected directly in HE5, their values were obtained as follows. The sourcecode of HE5 was checked to obtain Ha=1km and to confirm β=3.91Ha/V. α could not be derived from the sourcecode because it is calculated in a complex way using a marine aerosol model consisting of three components. Hence, α and β were tuned in WASI4 until the Eds spectra for θsun=30° corresponded to the HE5 spectra. An example for the correspondence of WASI4 and HE5 irradiance spectra is shown in Fig. 5.

 

Fig. 5. Comparison of above water irradiance spectra for θsun=30°.

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The correspondence is excellent: differences between WASI4 and HE5 irradiance spectra are less than line thickness for the majority of spectral bands. For this reason only the HE5 spectra are displayed. The ratio plot shows that the differences are always below 0.1%. Correspondence is of similar quality to Fig. 5 also for other Sun zenith angles ranging from 0° to 80° (data not shown), i.e., HE5 and WASI4 provide consistent irradiance spectra Ed(λ), Edd(λ), and Eds(λ)above water, indicating that the Gregg and Carder model [17] is implemented identically in both software.

C. Parameterization of ρds

The reflectance of the diffuse component of irradiance at the water surface, ρds, depends on the Sun zenith angle. To obtain a parameterization of ρds(θsun), pairs of irradiance spectra Eds(λ,0+), Eds(λ,0) were calculated for Sun zenith angles ranging from 0° to 80°. Because Eds(λ,0) is no output of HE5, Edd(λ,0+)=0 was set in the sourcecode of EcoLight. The calculated irradiance is then just the diffuse component. Furthermore, that part of Eds(λ,0) that is caused by reflection of upwelling radiation at the water surface has been made zero by setting the scattering coefficients of water and all water constituents to zero. ρds was then calculated as the average of ρds(λ)=1Eds(λ,0)/Eds(λ,0+) for the wavelength range from 400 to 700 nm. The result is shown in Fig. 6.

 

Fig. 6. Dependency of the reflectance factor for diffuse irradiance on the Sun zenith angle.

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The obtained ρds values can be approximated by Eq. (5), which was derived by fitting the values shown in Fig. 6 with a second-order polynomial function.

D. Parameterization of lds

A factor lds was introduced in Eq. (2) as the average path length of diffuse radiation relative to sensor depth z. To obtain numerical values for lds, a set of 130 spectra Eds(λ,z,θsun) was calculated using HE5 for z ranging from 0.5 to 5 m and θsun values between 0° and 80°. The concentrations of water constituents were set to C0=2mgm3, X=0.6gm3, and Y=0.3m1. As before, Edd(λ,0+)=0 was set in the sourcecode of EcoLight to get just the diffuse component. lds was then calculated as

lds=1[a(λ)+bb(λ)]zlnEds(λ,z,θsun)Eds(λ,0,θsun)·
This equation is obtained by solving Eq. (2) for lds and using Eq. (18) for Kds(λ). The obtained values of lds were averaged from 400 to 800 nm. As shown in Fig. 7, lds is proportional to the cosine of the Sun zenith angle in water up to θsun=65°. Hence, a regression line was fitted to the depth-averaged values for θsun between 0° and 65°. The obtained function is given by Eq. (23) and shown as the solid line in Fig. 7. The equation ignores the slight depth dependency of lds, causing some overestimation for depths above approximately 2 m and minor underestimation below. The dependency on C0, X, and Y was not investigated. A value of lds=1.18 was used in Gege and Pinnel [5] to analyze the sources of Ed variance; it is valid for θsun=40°.

 

Fig. 7. Dependency of the relative path length of diffuse radiation on depth and on the Sun zenith angle.

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E. Parameterization of ldd

Similarly, as before, relative path length factors for direct radiation, ldd, were obtained by solving Eq. (3) for ldd and generating a number of spectra Edd(λ,z,θsun) using HE5. Eds(λ,0+)=0 was set in the sourcecode to force HE5 to calculate Edd instead of Ed. The simulations were performed using a modified version of EcoLight, which has an angular resolution of 2° instead of the standard 10°. In this way, a set of spectra Edd was calculated for the same wavelengths, Sun zenith angles, and depths as above for Eds. The derived values of ldd have an average of 1.000 and a standard deviation of 0.004. Hence, the dependency on λ, z, and θsun can be neglected for the studied conditions, and ldd=1 is set.

F. Validation

The developed model was validated by comparing irradiance spectra obtained from WASI4 using forward modeling with the corresponding spectra from HE5 for 99 different combinations of z and θsun. The ranges were 0 to 5 m for z and 0° to 80° for θsun. The atmosphere parameters were chosen as in Section 6.B. The water parameters were as follows: C0=2mgm3, X=0.6gm3, Y=0.3m1, and nW=1.33. For calculating Edd(λ) using HE5, Eds(λ,0+)=0 was set in the HE5 sourcecode, and Edd(λ,0+)=0 was set to obtain Eds(λ). The computation time for a single spectrum with 276 spectral bands was on average 120 s for the 2° version of EcoLight and 103s for WASI4 on a 2 GHz Intel Core 2 Duo processor, i.e., the analytical model is approximately 105 times faster. Recently a fast version of EcoLight has been developed, called EcoLight-S, with typical computation times of less than 1 s [36].

A typical matchup is shown in Fig. 8 for z=4m and θsun=40°. The HE5 and WASI4 curves are hardly distinguishable by eye. They differ less than ±1.5% in the range from 400 to 700 nm; the average ratio for that range is 0.9991 for Edd, 1.0097 for Eds, and 1.0057 for Ed.

 

Fig. 8. Comparison of spectral shape for WASI4 and HE5 simulations of irradiance.

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Figure 9 shows the spectrally averaged ratios for all matchups of Ed, Edd, and Eds. Averaging was performed from 400 to 700 nm. In most cases, the deviations are below ±1%. Deviations of more than 1.5% occur only for Sun zenith angles above 65° and depths above 3 m. The average ratio is 1.002±0.003 for Ed, 0.998±0.005 for Edd, and 1.006±0.003 for Eds for z ranging from 0.5 to 5 m and θsun from 0° to 70°. Hence, WASI4 and HE5 provide consistent spectra Ed(λ), Edd(λ), and Eds(λ) for a plane water surface under the studied conditions.

 

Fig. 9. Comparison of intensity for WASI4 and HE5 simulations of irradiance.

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7. Application to Field Measurements

Measurements were performed in 2003 and 2004 in the German lakes Bodensee (Lake Constance), Starnberger See, and Waginger See mostly in shallow waters using a small boat [37]. A data set of 421 Ed measurements was collected using a RAMSES-ACC-VIS irradiance sensor (TriOS, Oldenburg, Germany). Each measurement consisted of a sequence of 4 to 50 irradiance spectra, which cover the range from 350 to 900 nm at a spectral sampling interval of 5 nm. 98% of the data have integration times between 16 and 64 ms, and the average is 34 ms. Measurement time of a sequence varied from 21 to 700 s with an average of 105 s. The Ed sensor was lowered into the water at the sunlit side of the boat at a distance of 2 to 3 m to avoid shadowing. More details can be found in Gege and Pinnel [5].

A representative example of a single spectrum and the corresponding fit curve obtained by inverse modeling using WASI is shown in Fig. 10(a). The measurement was performed on a cloudless day at Bodensee (July 28, 2004, 15:12 h local time). For illustration purposes, the model curve was calculated at higher spectral resolution (1 nm) than the measurement (5 nm). The plot demonstrates that the downwelling irradiance reveals many spectral fine structures that are not resolved by the instrument; thus, the measured spectrum is smooth compared with the model curve. Except this difference of spectral resolution, the calculated spectrum fits well to the measurement. The obtained fit parameters are listed in the figure’s legend. A further result of inverse modeling using WASI is the separation of the direct and diffuse components of irradiance, Edd(λ) and Eds(λ); see Fig. 10(b). Their spectral shapes differ significantly. Consequently, changes of their relative intensities alter the spectral shape of Ed.

 

Fig. 10. Example for inverse modeling of an irradiance measurement. Fit parameters: X=0.85gm3, Y=0.21m1, fdd=1.02, fds=1.02, z=1.06m. (a) Measured spectrum and model curve. (b) Direct and diffuse component obtained from inverse modeling.

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Inverse modeling was performed for all 4375 individual Ed spectra of the 421 measurements. The model curves were calculated for the spectral range from 400 to 800 nm in 5 nm steps using X, Y, fdd, fds, and z as fit parameters. For θsun the actual angle at the time of the measurement was taken, and C0=2mgm3 was set, which is between the average concentration of Starnberger See (1.4±0.3mgm3) and Bodensee (2.5±0.9mgm3) as derived from simultaneous in situ measurements [38].

The results for fdd and fds are shown in Fig. 11. These parameters describe the variability of Ed induced by the changing geometry of a wind-roughened and wave-modulated water surface. The histograms of both parameters peak near the expected value for undisturbed geometry, fdd=1 and fds=1, suggesting that Eq. (1) is a reasonable model to separate the direct and diffuse components of Ed. The asymmetry of the fdd histogram, which favors values <1(fdd=0.88±0.38), is caused by the numerous measurements that were performed under cloud cover. The fds histogram is symmetrical around fds=1.04 with a standard deviation of 0.46. The fdd and fds values shown here were analyzed in more detail in Gege and Pinnel [5] to determine quantitatively the depth dependent influence of wave focusing on Ed during a measurement. It was found that fdd and fds are only weakly correlated (r2 is between 0.02 and 0.16), changes of fdd explain up to 82% of intensity variability and alter the spectral shape of Ed between 400 and 700 nm on average by 5.4%, and changes of fds have only minor impact on Ed.

 

Fig. 11. Frequency distributions of the parameters fdd and fds.

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Because the spectral shape of Ed depends on the ratio rd of direct to diffuse irradiance [5], the variability of rd during a measurement is of interest to estimate wavelength-dependent changes of Ed. The rd values of all 4375 spectra are shown in Fig. 12 as a function of the Sun zenith angle. They were calculated as spectral average of fddEdd(λ)/fdsEds(λ) in the range from 400 to 700 nm. The solid red line shows for comparison the function rd(θSun) as expected for fdd=fds=1; it was calculated using Eq. (17) for the average depth of 0.7 m of all measurements. Obviously, the large and uncorrelated variability of fdd and fds induces very large variability to rd. The standard deviation of rd is 4.5 for the actual data set, which is far above the average of 2.6.

 

Fig. 12. Variability of the ratio of direct to diffuse irradiance.

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Sensor depth z changes during a measurement because of waves that alter the thickness of the water column above the sensor, and because of roll movements of the boat. Figure 13 shows a statistics of these changes. The z values obtained by inverse modeling of the individual spectra forming a measurement were averaged, and then the differences Δz between the individual z values and the mean were calculated. The standard deviation of Δz is 0.043 m. Because a portion of this variability is caused by wave action and boat movement, it can be concluded that the uncertainty of z determination by inverse modeling is below 4 cm.

 

Fig. 13. Sensor depth variability during a measurement.

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The accuracy of water constituents determination increases with the thickness of the water column above the irradiance sensor. An analysis of the Bodensee and Starnberger See data set showed that the uncertainty of phytoplankton concentration decreases exponentially with depth and reaches class-specific lower limits at depths between 1 and 1.5 m [38]. For this reason, only measurements from depths z>1.5m were used to calculate lake-specific averages of the parameters X and Y. These are summarized in Table 2. The standard deviations reflect the natural concentration variability together with the uncertainties of inverse modeling. The averages for all three lakes, X=0.6gm3 and Y=0.3m1, were used in Gege and Pinnel [5] as typical concentrations to study the impact of environmental and experimental conditions on the variance of downwelling irradiance in water.

Tables Icon

Table 2. Mean and Standard Deviation of Suspended Matter Concentration (X, σX) and Gelbstoff Concentration (Y, σY) Derived by Inverse Modeling of Measurements at Depths>1.5m

Because the in situ measurements indicated only little variability of phytoplankton concentration, the parameter C0 was set to 2  mgm3 and kept constant during inverse modeling in this study. It was shown in Gege [38] that the described model also can be used to determine the concentration of phytoplankton. By applying an adapted fit strategy to the same data set from Bodensee and Starnberger See, it was possible to determine the concentrations of three phytoplankton classes (diatoms, dinoflagellates, green algae) above class-specific thresholds between 0.4 and 0.9mgm3. The uncertainty of total pigment concentration (sum of chlorophyll-a and phaeophytin-a in all classes) was 0.7mgm3.

8. Summary

An analytic model of the downwelling irradiance in water (Ed) was developed that calculates the direct (Edd) and diffuse (Eds) components separately. This separation allows to account for the different path lengths of the two components and to handle the large variability of Ed at typical field conditions. Because the computation time is of the order of 103s, the model can be used for computationally extensive simulations like inverse modeling.

The intensities and wavelength dependencies of Edd and Eds just beneath the water surface are parameterized using the model of Gregg and Carder for cloudless maritime atmospheres. Their database, which is restricted to the spectral range of 400–700 nm, was extended to a range of 300–1000 nm through radiative transfer simulations using MODTRAN. Changes of Edd and Eds with depth (z) are calculated individually to account for the different path lengths of direct (z/cosθSun) and diffuse (zlds) radiation. The radiative transfer program HydroLight was used to show that using these path lengths, the Lambert—Beer law describes accurately the depth dependency of both irradiance components with a common attenuation coefficient, a+bb, where a is the average absorption coefficient and bb is the average backscattering coefficient of the water column between surface and depth z. The relative path length of diffuse irradiance (lds) was parameterized as a function of the Sun zenith angle in water (θsun) for concentrations of water constituents that are typical for the German lakes Bodensee, Starnberger See, and Waginger See.

The wind-roughened and wave-modulated water surface induces large and uncorrelated intensity changes to the direct and diffuse irradiance components. To account for this variability, the downwelling irradiance in water is calculated as a weighted sum of both components, Ed=fddEdd+fdsEds, where the weights fdd, fds are the actual intensities of Edd and Eds relative to the corresponding intensities for a plane water surface. By treating fdd and fds as fit parameters, underwater measurements performed at arbitrary surface conditions can be analyzed.

The described model was implemented into the public-domain software WASI for the simulation and analysis of spectral Ed measurements. It uses inverse modeling to determine unknown values of model parameters and to separate the direct and diffuse components of Ed. The model was applied to data from the three above-mentioned lakes to study the magnitude of short-term intensity changes and accompanying spectral changes for the depth range 0–5 m. The large observed variability could be attributed to changes of fdd and fds. Despite the high variability of Ed, the model was able to determine sensor depth and analyze its variability during a measurement and to estimate the concentrations of water constituents.

This study was initiated and motivated by many stimulating discussions with Nicole Pinnel about the reasons for the large variability of her irradiance measurements and how to model these. She measured all data used in this study; I am deeply grateful to her for providing this valuable data set. Curtis Mobley is acknowledged for helpful discussions on HydroLight and for providing a modified EcoLight version with a zenith angle discretization of 2°. Philipp Grötsch modified the HydroLight sourcecode such that the diffuse and direct components of downwelling irradiance could be calculated separately. Two anonymous reviewers provided helpful suggestions.

References

1. R. E. Walker, Marine Light Field Statistics (Wiley, 1994).

2. J. R. V. Zanefeld, E. Boss, and A. Barnard, “Influence of surface waves on measured and modeled irradiance profiles,” Appl. Opt. 40, 1442–1449 (2001). [CrossRef]  

3. J. Dera and D. Stramski, “Focusing of sunlight by sea surface waves: new results from the Black Sea,” Oceanologia 34, 13–25 (1993).

4. H. Hofmann, A. Lorke, and F. Peeters, “Wave-induced variability of the underwater light climate in the littoral zone,” Verh. Internat. Verein. Limnol. 30, 627–632 (2008).

5. P. Gege and N. Pinnel, “Sources of variance of downwelling irradiance in water,” Appl. Opt. 50, 2192–2203 (2011). [CrossRef]  

6. J. Dera and D. Stramski, “Maximum effects of sunlight focusing under a wind-disturbed sea surface,” Oceanologia 23, 15–42 (1986).

7. D. A. Toole, D. A. Siegel, D. W. Menzies, M. J. Neumann, and R. C. Smith, “Remote-sensing reflectance determinations in the coastal ocean environment: impact of instrumental characteristics and environmental variability,” Appl. Opt. 39, 456–469 (2000). [CrossRef]  

8. S. B. Hooker and S. Maritorena, “An evaluation of oceanographic radiometers and deployment methodologies,” J. Atmos. Oceanic Technol. 17, 811–830 (2000).

9. J. L. Mueller, “In-water radiometric profile measurements and data analysis protocols,” in Ocean Optics Protocols for Satellite Ocean Color Sensor Validation, Rev. 4, Vol. III, J. L. Mueller, G. S. Fargion, and C. R. McClain, eds. (NASA, 2003), pp. 7–20.

10. C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, and R. Stavn, “Comparison of numerical models for the computation of underwater light fields,” Appl. Opt. 32, 7484–7504 (1993). [CrossRef]  

11. C. Cox and W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–227 (1954).

12. C. Cox and W. Munk, “The measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. 44, 838–850 (1954). [CrossRef]  

13. P. Gege, “The water color simulator WASI: an integrating software tool for analysis and simulation of optical in situ spectra,” Comput. Geosci. 30, 523–532 (2004). [CrossRef]  

14. P. Gege and A. Albert, “A tool for inverse modeling of spectral measurements in deep and shallow waters,” in Remote Sensing of Aquatic Coastal Ecosystem Processes: Science and Management Applications, L. L. Richardson and E. F. LeDrew, eds. (Springer, 2006), pp. 81–109.

15. Executable code of WASI and manual, ftp://ftp.dfd.dlr.de/pub/WASI.

16. N. G. Jerlov, Marine Optics (Elsevier, 1976).

17. W. W. Gregg and K. L. Carder, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnol. Oceanogr. 35, 1657–1675 (1990). [CrossRef]  

18. F. Kasten and A. T. Young, “Revised optical air mass tables and approximation formula,” Appl. Opt. 28, 4735–4738 (1989). [CrossRef]  

19. F. X. Kneizys, L. W. Abreu, and G. P. Anderson, “The MODTRAN 2/3 Report and LOWTRAN 7 MODEL,” Technical report, Phillips Laboratory, Geophysics Directorate, Hanscom, Massachusetts (1996).

20. L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. Malathy Devi, J.-M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, and R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992). [CrossRef]  

21. R. Bukata, J. H. Jerome, K. Y. Kondratyev, and D. V. Pozdnyakov, Optical Properties and Remote Sensing of Inland and Coastal Waters (CRC, 1995).

22. H. R. Gordon, “Can the Lambert–Beer law be applied to the diffuse attenuation coefficient of ocean water?” Limnol. Oceanogr. 34, 1389–1409 (1989). [CrossRef]  

23. T. I. Quickenden and J. A. Irvin, “The ultraviolet absorption spectrum of liquid water,” J. Chem. Phys. 72, 4416–4428 (1980). [CrossRef]  

24. H. Buiteveld, J. H. M. Hakvoort, and M. Donze, “The optical properties of pure water,” Proc. SPIE 2258, 174–183(1994). [CrossRef]  

25. K. F. Palmer and D. Williams, “Optical properties of water in the near infrared,” J. Opt. Soc. Am. A 64, 1107–1110 (1974). [CrossRef]  

26. A. Morel, “Optical properties of pure water and pure sea water,” in Optical Aspects of Oceanography, N. G. Jerlov and E. Steemann Nielsen, eds. (Academic, 1997), pp. 1–24.

27. P. Gege, “Characterization of the phytoplankton in Lake Constance for classification by remote sensing,” in Lake Constance—Characterisation of an Ecosystem in Transition, E. Bäuerle and U. Gaedke, eds. (Archiv für Hydrobiologie53, 1998), pp. 179–193.

28. T. Heege, “Flugzeuggestützte Fernerkundung von Wasserinhaltsstoffen am Bodensee,” Ph.D. thesis (DLR-Forschungsbericht, 2000).

29. A. Bricaud, A. Morel, and L. Prieur, “Absorption by dissolved organic matter of the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43–53 (1981). [CrossRef]  

30. K. L. Carder, G. R. Harvey, and P. B. Ortner, “Marine humic and fulvic acids: their effects on remote sensing of ocean chlorophyll,” Limnol. Oceanogr. 34, 68–81 (1989). [CrossRef]  

31. A. A. Gitelson, G. Dall’Olmo, W. Moses, D. C. Rundquist, T. Barrow, T. R. Fisher, D. Gurlin, and J. Holz, “A simple semi-analytical model for remote estimation of chlorophyll-a in turbid waters: validation,” Remote Sens. Environ. 112, 3582–3593 (2008). [CrossRef]  

32. R. Doerffer and H. Schiller, “The MERIS Case 2 water algorithm,” Int. J. Remote Sens. 28, 517–535 (2007). [CrossRef]  

33. J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965). [CrossRef]  

34. M. S. Caceci and W. P. Cacheris, “Fitting curves to data,” Byte 9, 340–362 (1984).

35. C. D. Mobley, Light and Water (Academic, 1994).

36. C. D. Mobley, “Fast light calculations for ocean ecosystem and inverse models,” Opt. Express 19, 18927–18944 (2011). [CrossRef]  

37. N. Pinnel, “A method for mapping submerged macrophytes in lakes using hyperspectral remote sensing,” Ph.D. thesis (Technical University Munich, 2007).

38. P. Gege, “Estimation of phytoplankton concentration from downwelling irradiance measurements in water,” Israel J. Plant Sci. (to be published).

References

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  1. R. E. Walker, Marine Light Field Statistics (Wiley, 1994).
  2. J. R. V. Zanefeld, E. Boss, and A. Barnard, “Influence of surface waves on measured and modeled irradiance profiles,” Appl. Opt. 40, 1442–1449 (2001).
    [Crossref]
  3. J. Dera and D. Stramski, “Focusing of sunlight by sea surface waves: new results from the Black Sea,” Oceanologia 34, 13–25 (1993).
  4. H. Hofmann, A. Lorke, and F. Peeters, “Wave-induced variability of the underwater light climate in the littoral zone,” Verh. Internat. Verein. Limnol. 30, 627–632 (2008).
  5. P. Gege and N. Pinnel, “Sources of variance of downwelling irradiance in water,” Appl. Opt. 50, 2192–2203 (2011).
    [Crossref]
  6. J. Dera and D. Stramski, “Maximum effects of sunlight focusing under a wind-disturbed sea surface,” Oceanologia 23, 15–42 (1986).
  7. D. A. Toole, D. A. Siegel, D. W. Menzies, M. J. Neumann, and R. C. Smith, “Remote-sensing reflectance determinations in the coastal ocean environment: impact of instrumental characteristics and environmental variability,” Appl. Opt. 39, 456–469 (2000).
    [Crossref]
  8. S. B. Hooker and S. Maritorena, “An evaluation of oceanographic radiometers and deployment methodologies,” J. Atmos. Oceanic Technol. 17, 811–830 (2000).
  9. J. L. Mueller, “In-water radiometric profile measurements and data analysis protocols,” in Ocean Optics Protocols for Satellite Ocean Color Sensor Validation, Rev. 4, Vol. III, J. L. Mueller, G. S. Fargion, and C. R. McClain, eds. (NASA, 2003), pp. 7–20.
  10. C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, and R. Stavn, “Comparison of numerical models for the computation of underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
    [Crossref]
  11. C. Cox and W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–227 (1954).
  12. C. Cox and W. Munk, “The measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. 44, 838–850 (1954).
    [Crossref]
  13. P. Gege, “The water color simulator WASI: an integrating software tool for analysis and simulation of optical in situ spectra,” Comput. Geosci. 30, 523–532 (2004).
    [Crossref]
  14. P. Gege and A. Albert, “A tool for inverse modeling of spectral measurements in deep and shallow waters,” in Remote Sensing of Aquatic Coastal Ecosystem Processes: Science and Management Applications, L. L. Richardson and E. F. LeDrew, eds. (Springer, 2006), pp. 81–109.
  15. Executable code of WASI and manual, ftp://ftp.dfd.dlr.de/pub/WASI .
  16. N. G. Jerlov, Marine Optics (Elsevier, 1976).
  17. W. W. Gregg and K. L. Carder, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnol. Oceanogr. 35, 1657–1675 (1990).
    [Crossref]
  18. F. Kasten and A. T. Young, “Revised optical air mass tables and approximation formula,” Appl. Opt. 28, 4735–4738 (1989).
    [Crossref]
  19. F. X. Kneizys, L. W. Abreu, and G. P. Anderson, “The MODTRAN 2/3 Report and LOWTRAN 7 MODEL,” Technical report, Phillips Laboratory, Geophysics Directorate, Hanscom, Massachusetts (1996).
  20. L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. Malathy Devi, J.-M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, and R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
    [Crossref]
  21. R. Bukata, J. H. Jerome, K. Y. Kondratyev, and D. V. Pozdnyakov, Optical Properties and Remote Sensing of Inland and Coastal Waters (CRC, 1995).
  22. H. R. Gordon, “Can the Lambert–Beer law be applied to the diffuse attenuation coefficient of ocean water?” Limnol. Oceanogr. 34, 1389–1409 (1989).
    [Crossref]
  23. T. I. Quickenden and J. A. Irvin, “The ultraviolet absorption spectrum of liquid water,” J. Chem. Phys. 72, 4416–4428 (1980).
    [Crossref]
  24. H. Buiteveld, J. H. M. Hakvoort, and M. Donze, “The optical properties of pure water,” Proc. SPIE 2258, 174–183(1994).
    [Crossref]
  25. K. F. Palmer and D. Williams, “Optical properties of water in the near infrared,” J. Opt. Soc. Am. A 64, 1107–1110 (1974).
    [Crossref]
  26. A. Morel, “Optical properties of pure water and pure sea water,” in Optical Aspects of Oceanography, N. G. Jerlov and E. Steemann Nielsen, eds. (Academic, 1997), pp. 1–24.
  27. P. Gege, “Characterization of the phytoplankton in Lake Constance for classification by remote sensing,” in Lake Constance—Characterisation of an Ecosystem in Transition, E. Bäuerle and U. Gaedke, eds. (Archiv für Hydrobiologie53, 1998), pp. 179–193.
  28. T. Heege, “Flugzeuggestützte Fernerkundung von Wasserinhaltsstoffen am Bodensee,” Ph.D. thesis (DLR-Forschungsbericht, 2000).
  29. A. Bricaud, A. Morel, and L. Prieur, “Absorption by dissolved organic matter of the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43–53 (1981).
    [Crossref]
  30. K. L. Carder, G. R. Harvey, and P. B. Ortner, “Marine humic and fulvic acids: their effects on remote sensing of ocean chlorophyll,” Limnol. Oceanogr. 34, 68–81 (1989).
    [Crossref]
  31. A. A. Gitelson, G. Dall’Olmo, W. Moses, D. C. Rundquist, T. Barrow, T. R. Fisher, D. Gurlin, and J. Holz, “A simple semi-analytical model for remote estimation of chlorophyll-a in turbid waters: validation,” Remote Sens. Environ. 112, 3582–3593 (2008).
    [Crossref]
  32. R. Doerffer and H. Schiller, “The MERIS Case 2 water algorithm,” Int. J. Remote Sens. 28, 517–535 (2007).
    [Crossref]
  33. J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
    [Crossref]
  34. M. S. Caceci and W. P. Cacheris, “Fitting curves to data,” Byte 9, 340–362 (1984).
  35. C. D. Mobley, Light and Water (Academic, 1994).
  36. C. D. Mobley, “Fast light calculations for ocean ecosystem and inverse models,” Opt. Express 19, 18927–18944 (2011).
    [Crossref]
  37. N. Pinnel, “A method for mapping submerged macrophytes in lakes using hyperspectral remote sensing,” Ph.D. thesis (Technical University Munich, 2007).
  38. P. Gege, “Estimation of phytoplankton concentration from downwelling irradiance measurements in water,” Israel J. Plant Sci. (to be published).

2011 (2)

2008 (2)

A. A. Gitelson, G. Dall’Olmo, W. Moses, D. C. Rundquist, T. Barrow, T. R. Fisher, D. Gurlin, and J. Holz, “A simple semi-analytical model for remote estimation of chlorophyll-a in turbid waters: validation,” Remote Sens. Environ. 112, 3582–3593 (2008).
[Crossref]

H. Hofmann, A. Lorke, and F. Peeters, “Wave-induced variability of the underwater light climate in the littoral zone,” Verh. Internat. Verein. Limnol. 30, 627–632 (2008).

2007 (1)

R. Doerffer and H. Schiller, “The MERIS Case 2 water algorithm,” Int. J. Remote Sens. 28, 517–535 (2007).
[Crossref]

2004 (1)

P. Gege, “The water color simulator WASI: an integrating software tool for analysis and simulation of optical in situ spectra,” Comput. Geosci. 30, 523–532 (2004).
[Crossref]

2001 (1)

2000 (2)

1994 (1)

H. Buiteveld, J. H. M. Hakvoort, and M. Donze, “The optical properties of pure water,” Proc. SPIE 2258, 174–183(1994).
[Crossref]

1993 (2)

1992 (1)

L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. Malathy Devi, J.-M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, and R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
[Crossref]

1990 (1)

W. W. Gregg and K. L. Carder, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnol. Oceanogr. 35, 1657–1675 (1990).
[Crossref]

1989 (3)

F. Kasten and A. T. Young, “Revised optical air mass tables and approximation formula,” Appl. Opt. 28, 4735–4738 (1989).
[Crossref]

H. R. Gordon, “Can the Lambert–Beer law be applied to the diffuse attenuation coefficient of ocean water?” Limnol. Oceanogr. 34, 1389–1409 (1989).
[Crossref]

K. L. Carder, G. R. Harvey, and P. B. Ortner, “Marine humic and fulvic acids: their effects on remote sensing of ocean chlorophyll,” Limnol. Oceanogr. 34, 68–81 (1989).
[Crossref]

1986 (1)

J. Dera and D. Stramski, “Maximum effects of sunlight focusing under a wind-disturbed sea surface,” Oceanologia 23, 15–42 (1986).

1984 (1)

M. S. Caceci and W. P. Cacheris, “Fitting curves to data,” Byte 9, 340–362 (1984).

1981 (1)

A. Bricaud, A. Morel, and L. Prieur, “Absorption by dissolved organic matter of the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43–53 (1981).
[Crossref]

1980 (1)

T. I. Quickenden and J. A. Irvin, “The ultraviolet absorption spectrum of liquid water,” J. Chem. Phys. 72, 4416–4428 (1980).
[Crossref]

1974 (1)

K. F. Palmer and D. Williams, “Optical properties of water in the near infrared,” J. Opt. Soc. Am. A 64, 1107–1110 (1974).
[Crossref]

1965 (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[Crossref]

1954 (2)

C. Cox and W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–227 (1954).

C. Cox and W. Munk, “The measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. 44, 838–850 (1954).
[Crossref]

Abreu, L. W.

F. X. Kneizys, L. W. Abreu, and G. P. Anderson, “The MODTRAN 2/3 Report and LOWTRAN 7 MODEL,” Technical report, Phillips Laboratory, Geophysics Directorate, Hanscom, Massachusetts (1996).

Albert, A.

P. Gege and A. Albert, “A tool for inverse modeling of spectral measurements in deep and shallow waters,” in Remote Sensing of Aquatic Coastal Ecosystem Processes: Science and Management Applications, L. L. Richardson and E. F. LeDrew, eds. (Springer, 2006), pp. 81–109.

Anderson, G. P.

F. X. Kneizys, L. W. Abreu, and G. P. Anderson, “The MODTRAN 2/3 Report and LOWTRAN 7 MODEL,” Technical report, Phillips Laboratory, Geophysics Directorate, Hanscom, Massachusetts (1996).

Barnard, A.

Barrow, T.

A. A. Gitelson, G. Dall’Olmo, W. Moses, D. C. Rundquist, T. Barrow, T. R. Fisher, D. Gurlin, and J. Holz, “A simple semi-analytical model for remote estimation of chlorophyll-a in turbid waters: validation,” Remote Sens. Environ. 112, 3582–3593 (2008).
[Crossref]

Benner, D. C.

L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. Malathy Devi, J.-M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, and R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
[Crossref]

Boss, E.

Bricaud, A.

A. Bricaud, A. Morel, and L. Prieur, “Absorption by dissolved organic matter of the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43–53 (1981).
[Crossref]

Brown, L. R.

L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. Malathy Devi, J.-M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, and R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
[Crossref]

Buiteveld, H.

H. Buiteveld, J. H. M. Hakvoort, and M. Donze, “The optical properties of pure water,” Proc. SPIE 2258, 174–183(1994).
[Crossref]

Bukata, R.

R. Bukata, J. H. Jerome, K. Y. Kondratyev, and D. V. Pozdnyakov, Optical Properties and Remote Sensing of Inland and Coastal Waters (CRC, 1995).

Caceci, M. S.

M. S. Caceci and W. P. Cacheris, “Fitting curves to data,” Byte 9, 340–362 (1984).

Cacheris, W. P.

M. S. Caceci and W. P. Cacheris, “Fitting curves to data,” Byte 9, 340–362 (1984).

Camy-Peyret, C.

L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. Malathy Devi, J.-M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, and R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
[Crossref]

Carder, K. L.

W. W. Gregg and K. L. Carder, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnol. Oceanogr. 35, 1657–1675 (1990).
[Crossref]

K. L. Carder, G. R. Harvey, and P. B. Ortner, “Marine humic and fulvic acids: their effects on remote sensing of ocean chlorophyll,” Limnol. Oceanogr. 34, 68–81 (1989).
[Crossref]

Cox, C.

C. Cox and W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–227 (1954).

C. Cox and W. Munk, “The measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. 44, 838–850 (1954).
[Crossref]

Dall’Olmo, G.

A. A. Gitelson, G. Dall’Olmo, W. Moses, D. C. Rundquist, T. Barrow, T. R. Fisher, D. Gurlin, and J. Holz, “A simple semi-analytical model for remote estimation of chlorophyll-a in turbid waters: validation,” Remote Sens. Environ. 112, 3582–3593 (2008).
[Crossref]

Dera, J.

J. Dera and D. Stramski, “Focusing of sunlight by sea surface waves: new results from the Black Sea,” Oceanologia 34, 13–25 (1993).

J. Dera and D. Stramski, “Maximum effects of sunlight focusing under a wind-disturbed sea surface,” Oceanologia 23, 15–42 (1986).

Doerffer, R.

R. Doerffer and H. Schiller, “The MERIS Case 2 water algorithm,” Int. J. Remote Sens. 28, 517–535 (2007).
[Crossref]

Donze, M.

H. Buiteveld, J. H. M. Hakvoort, and M. Donze, “The optical properties of pure water,” Proc. SPIE 2258, 174–183(1994).
[Crossref]

Fisher, T. R.

A. A. Gitelson, G. Dall’Olmo, W. Moses, D. C. Rundquist, T. Barrow, T. R. Fisher, D. Gurlin, and J. Holz, “A simple semi-analytical model for remote estimation of chlorophyll-a in turbid waters: validation,” Remote Sens. Environ. 112, 3582–3593 (2008).
[Crossref]

Flaud, J.-M.

L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. Malathy Devi, J.-M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, and R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
[Crossref]

Gamache, R. R.

L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. Malathy Devi, J.-M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, and R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
[Crossref]

Gege, P.

P. Gege and N. Pinnel, “Sources of variance of downwelling irradiance in water,” Appl. Opt. 50, 2192–2203 (2011).
[Crossref]

P. Gege, “The water color simulator WASI: an integrating software tool for analysis and simulation of optical in situ spectra,” Comput. Geosci. 30, 523–532 (2004).
[Crossref]

P. Gege and A. Albert, “A tool for inverse modeling of spectral measurements in deep and shallow waters,” in Remote Sensing of Aquatic Coastal Ecosystem Processes: Science and Management Applications, L. L. Richardson and E. F. LeDrew, eds. (Springer, 2006), pp. 81–109.

P. Gege, “Characterization of the phytoplankton in Lake Constance for classification by remote sensing,” in Lake Constance—Characterisation of an Ecosystem in Transition, E. Bäuerle and U. Gaedke, eds. (Archiv für Hydrobiologie53, 1998), pp. 179–193.

P. Gege, “Estimation of phytoplankton concentration from downwelling irradiance measurements in water,” Israel J. Plant Sci. (to be published).

Gentili, B.

Gitelson, A. A.

A. A. Gitelson, G. Dall’Olmo, W. Moses, D. C. Rundquist, T. Barrow, T. R. Fisher, D. Gurlin, and J. Holz, “A simple semi-analytical model for remote estimation of chlorophyll-a in turbid waters: validation,” Remote Sens. Environ. 112, 3582–3593 (2008).
[Crossref]

Goldman, A.

L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. Malathy Devi, J.-M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, and R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
[Crossref]

Gordon, H. R.

Gregg, W. W.

W. W. Gregg and K. L. Carder, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnol. Oceanogr. 35, 1657–1675 (1990).
[Crossref]

Gurlin, D.

A. A. Gitelson, G. Dall’Olmo, W. Moses, D. C. Rundquist, T. Barrow, T. R. Fisher, D. Gurlin, and J. Holz, “A simple semi-analytical model for remote estimation of chlorophyll-a in turbid waters: validation,” Remote Sens. Environ. 112, 3582–3593 (2008).
[Crossref]

Hakvoort, J. H. M.

H. Buiteveld, J. H. M. Hakvoort, and M. Donze, “The optical properties of pure water,” Proc. SPIE 2258, 174–183(1994).
[Crossref]

Harvey, G. R.

K. L. Carder, G. R. Harvey, and P. B. Ortner, “Marine humic and fulvic acids: their effects on remote sensing of ocean chlorophyll,” Limnol. Oceanogr. 34, 68–81 (1989).
[Crossref]

Heege, T.

T. Heege, “Flugzeuggestützte Fernerkundung von Wasserinhaltsstoffen am Bodensee,” Ph.D. thesis (DLR-Forschungsbericht, 2000).

Hofmann, H.

H. Hofmann, A. Lorke, and F. Peeters, “Wave-induced variability of the underwater light climate in the littoral zone,” Verh. Internat. Verein. Limnol. 30, 627–632 (2008).

Holz, J.

A. A. Gitelson, G. Dall’Olmo, W. Moses, D. C. Rundquist, T. Barrow, T. R. Fisher, D. Gurlin, and J. Holz, “A simple semi-analytical model for remote estimation of chlorophyll-a in turbid waters: validation,” Remote Sens. Environ. 112, 3582–3593 (2008).
[Crossref]

Hooker, S. B.

S. B. Hooker and S. Maritorena, “An evaluation of oceanographic radiometers and deployment methodologies,” J. Atmos. Oceanic Technol. 17, 811–830 (2000).

Irvin, J. A.

T. I. Quickenden and J. A. Irvin, “The ultraviolet absorption spectrum of liquid water,” J. Chem. Phys. 72, 4416–4428 (1980).
[Crossref]

Jerlov, N. G.

N. G. Jerlov, Marine Optics (Elsevier, 1976).

Jerome, J. H.

R. Bukata, J. H. Jerome, K. Y. Kondratyev, and D. V. Pozdnyakov, Optical Properties and Remote Sensing of Inland and Coastal Waters (CRC, 1995).

Jin, Z.

Kasten, F.

Kattawar, G. W.

Kneizys, F. X.

F. X. Kneizys, L. W. Abreu, and G. P. Anderson, “The MODTRAN 2/3 Report and LOWTRAN 7 MODEL,” Technical report, Phillips Laboratory, Geophysics Directorate, Hanscom, Massachusetts (1996).

Kondratyev, K. Y.

R. Bukata, J. H. Jerome, K. Y. Kondratyev, and D. V. Pozdnyakov, Optical Properties and Remote Sensing of Inland and Coastal Waters (CRC, 1995).

Lorke, A.

H. Hofmann, A. Lorke, and F. Peeters, “Wave-induced variability of the underwater light climate in the littoral zone,” Verh. Internat. Verein. Limnol. 30, 627–632 (2008).

Malathy Devi, V.

L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. Malathy Devi, J.-M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, and R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
[Crossref]

Maritorena, S.

S. B. Hooker and S. Maritorena, “An evaluation of oceanographic radiometers and deployment methodologies,” J. Atmos. Oceanic Technol. 17, 811–830 (2000).

Massie, S. T.

L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. Malathy Devi, J.-M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, and R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
[Crossref]

Mead, R.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[Crossref]

Menzies, D. W.

Mobley, C. D.

Morel, A.

C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, and R. Stavn, “Comparison of numerical models for the computation of underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
[Crossref]

A. Bricaud, A. Morel, and L. Prieur, “Absorption by dissolved organic matter of the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43–53 (1981).
[Crossref]

A. Morel, “Optical properties of pure water and pure sea water,” in Optical Aspects of Oceanography, N. G. Jerlov and E. Steemann Nielsen, eds. (Academic, 1997), pp. 1–24.

Moses, W.

A. A. Gitelson, G. Dall’Olmo, W. Moses, D. C. Rundquist, T. Barrow, T. R. Fisher, D. Gurlin, and J. Holz, “A simple semi-analytical model for remote estimation of chlorophyll-a in turbid waters: validation,” Remote Sens. Environ. 112, 3582–3593 (2008).
[Crossref]

Mueller, J. L.

J. L. Mueller, “In-water radiometric profile measurements and data analysis protocols,” in Ocean Optics Protocols for Satellite Ocean Color Sensor Validation, Rev. 4, Vol. III, J. L. Mueller, G. S. Fargion, and C. R. McClain, eds. (NASA, 2003), pp. 7–20.

Munk, W.

C. Cox and W. Munk, “The measurement of the roughness of the sea surface from photographs of the sun’s glitter,” J. Opt. Soc. Am. 44, 838–850 (1954).
[Crossref]

C. Cox and W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–227 (1954).

Nelder, J. A.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[Crossref]

Neumann, M. J.

Ortner, P. B.

K. L. Carder, G. R. Harvey, and P. B. Ortner, “Marine humic and fulvic acids: their effects on remote sensing of ocean chlorophyll,” Limnol. Oceanogr. 34, 68–81 (1989).
[Crossref]

Palmer, K. F.

K. F. Palmer and D. Williams, “Optical properties of water in the near infrared,” J. Opt. Soc. Am. A 64, 1107–1110 (1974).
[Crossref]

Peeters, F.

H. Hofmann, A. Lorke, and F. Peeters, “Wave-induced variability of the underwater light climate in the littoral zone,” Verh. Internat. Verein. Limnol. 30, 627–632 (2008).

Perrin, A.

L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. Malathy Devi, J.-M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, and R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
[Crossref]

Pinnel, N.

P. Gege and N. Pinnel, “Sources of variance of downwelling irradiance in water,” Appl. Opt. 50, 2192–2203 (2011).
[Crossref]

N. Pinnel, “A method for mapping submerged macrophytes in lakes using hyperspectral remote sensing,” Ph.D. thesis (Technical University Munich, 2007).

Pozdnyakov, D. V.

R. Bukata, J. H. Jerome, K. Y. Kondratyev, and D. V. Pozdnyakov, Optical Properties and Remote Sensing of Inland and Coastal Waters (CRC, 1995).

Prieur, L.

A. Bricaud, A. Morel, and L. Prieur, “Absorption by dissolved organic matter of the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43–53 (1981).
[Crossref]

Quickenden, T. I.

T. I. Quickenden and J. A. Irvin, “The ultraviolet absorption spectrum of liquid water,” J. Chem. Phys. 72, 4416–4428 (1980).
[Crossref]

Reinersman, P.

Rinsland, C. P.

L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. Malathy Devi, J.-M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, and R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
[Crossref]

Rothman, L. S.

L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. Malathy Devi, J.-M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, and R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
[Crossref]

Rundquist, D. C.

A. A. Gitelson, G. Dall’Olmo, W. Moses, D. C. Rundquist, T. Barrow, T. R. Fisher, D. Gurlin, and J. Holz, “A simple semi-analytical model for remote estimation of chlorophyll-a in turbid waters: validation,” Remote Sens. Environ. 112, 3582–3593 (2008).
[Crossref]

Schiller, H.

R. Doerffer and H. Schiller, “The MERIS Case 2 water algorithm,” Int. J. Remote Sens. 28, 517–535 (2007).
[Crossref]

Siegel, D. A.

Smith, M. A. H.

L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. Malathy Devi, J.-M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, and R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
[Crossref]

Smith, R. C.

Stamnes, K.

Stavn, R.

Stramski, D.

J. Dera and D. Stramski, “Focusing of sunlight by sea surface waves: new results from the Black Sea,” Oceanologia 34, 13–25 (1993).

J. Dera and D. Stramski, “Maximum effects of sunlight focusing under a wind-disturbed sea surface,” Oceanologia 23, 15–42 (1986).

Tipping, R. H.

L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. Malathy Devi, J.-M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, and R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
[Crossref]

Toole, D. A.

Toth, R. A.

L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. Malathy Devi, J.-M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, and R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
[Crossref]

Walker, R. E.

R. E. Walker, Marine Light Field Statistics (Wiley, 1994).

Williams, D.

K. F. Palmer and D. Williams, “Optical properties of water in the near infrared,” J. Opt. Soc. Am. A 64, 1107–1110 (1974).
[Crossref]

Young, A. T.

Zanefeld, J. R. V.

Appl. Opt. (5)

Byte (1)

M. S. Caceci and W. P. Cacheris, “Fitting curves to data,” Byte 9, 340–362 (1984).

Comput. Geosci. (1)

P. Gege, “The water color simulator WASI: an integrating software tool for analysis and simulation of optical in situ spectra,” Comput. Geosci. 30, 523–532 (2004).
[Crossref]

Comput. J. (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[Crossref]

Int. J. Remote Sens. (1)

R. Doerffer and H. Schiller, “The MERIS Case 2 water algorithm,” Int. J. Remote Sens. 28, 517–535 (2007).
[Crossref]

Israel J. Plant Sci. (1)

P. Gege, “Estimation of phytoplankton concentration from downwelling irradiance measurements in water,” Israel J. Plant Sci. (to be published).

J. Atmos. Oceanic Technol. (1)

S. B. Hooker and S. Maritorena, “An evaluation of oceanographic radiometers and deployment methodologies,” J. Atmos. Oceanic Technol. 17, 811–830 (2000).

J. Chem. Phys. (1)

T. I. Quickenden and J. A. Irvin, “The ultraviolet absorption spectrum of liquid water,” J. Chem. Phys. 72, 4416–4428 (1980).
[Crossref]

J. Mar. Res. (1)

C. Cox and W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–227 (1954).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

K. F. Palmer and D. Williams, “Optical properties of water in the near infrared,” J. Opt. Soc. Am. A 64, 1107–1110 (1974).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (1)

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Figures (13)

Fig. 1.
Fig. 1. Atmospheric transmission after absorption by ozone, oxygen, and water vapor.
Fig. 2.
Fig. 2. Effective absorption coefficients of the atmospheric components ozone, oxygen, and water vapor.
Fig. 3.
Fig. 3. Comparison of analytically estimated sensor depths with results from inverse modeling.
Fig. 4.
Fig. 4. Extraterrestrial solar irradiance.
Fig. 5.
Fig. 5. Comparison of above water irradiance spectra for θ sun = 30 ° .
Fig. 6.
Fig. 6. Dependency of the reflectance factor for diffuse irradiance on the Sun zenith angle.
Fig. 7.
Fig. 7. Dependency of the relative path length of diffuse radiation on depth and on the Sun zenith angle.
Fig. 8.
Fig. 8. Comparison of spectral shape for WASI4 and HE5 simulations of irradiance.
Fig. 9.
Fig. 9. Comparison of intensity for WASI4 and HE5 simulations of irradiance.
Fig. 10.
Fig. 10. Example for inverse modeling of an irradiance measurement. Fit parameters: X = 0.85 g m 3 , Y = 0.21 m 1 , f d d = 1.02 , f d s = 1.02 , z = 1.06 m . (a) Measured spectrum and model curve. (b) Direct and diffuse component obtained from inverse modeling.
Fig. 11.
Fig. 11. Frequency distributions of the parameters f d d and f d s .
Fig. 12.
Fig. 12. Variability of the ratio of direct to diffuse irradiance.
Fig. 13.
Fig. 13. Sensor depth variability during a measurement.

Tables (2)

Tables Icon

Table 1. Parameters of the Irradiance Model That Can Be Used in WASI as Fit Parameters

Tables Icon

Table 2. Mean and Standard Deviation of Suspended Matter Concentration ( X , σ X ) and Gelbstoff Concentration ( Y , σ Y ) Derived by Inverse Modeling of Measurements at Depths > 1.5 m

Equations (27)

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E d ( λ , z ) = f d d E d d ( λ , z ) + f d s E d s ( λ , z ) .
E d d ( λ , z ) = E d s ( λ , 0 ) exp { K d s ( λ ) z l d s } .
E d d ( λ , z ) = E d d ( λ , 0 ) exp { K d d ( λ ) z l d d cos θ sun } ·
ρ d d = 1 2 | sin 2 ( θ sun θ sun ) sin 2 ( θ sun + θ sun ) + tan 2 ( θ sun θ sun ) tan 2 ( θ sun + θ sun ) | ,
ρ d s = 0.06087 + 0.03751 ( 1 cos θ sun ) + 0.1143 ( 1 cos θ sun ) 2 .
E d d ( λ , 0 ) = F 0 ( λ ) cos θ sun T r ( λ ) T a a ( λ ) T a s ( λ ) T oz ( λ ) T o ( λ ) T wv ( λ ) ( 1 ρ d d ) .
E d s ( λ , 0 ) = [ E d r ( λ ) + E d a ( λ ) ] ( 1 ρ d s ) .
E d r ( λ ) = 1 2 F 0 ( λ ) cos θ sun ( 1 T r ( λ ) 0.95 ) T a a ( λ ) T oz ( λ ) T o ( λ ) T wv ( λ ) ,
E d a ( λ ) = F 0 ( λ ) cos θ sun T r ( λ ) 1.5 T a a ( λ ) T oz ( λ ) T o ( λ ) T wv ( λ ) ( 1 T a s ( λ ) ) F a .
T a a ( λ ) = exp [ ( 1 ω a ) τ a ( λ ) M ] ,
T a s ( λ ) = exp [ ω a τ a ( λ ) M ] ,
T r ( λ ) = exp [ M / ( 115.6406 λ 4 1.335 λ 2 ) ] ,
T oz ( λ ) = exp [ a oz ( λ ) H oz M oz ] ,
T o ( λ ) = exp 1.41 a o ( λ ) · M [ 1 + 118.3 a o ( λ ) · M ] 0.45 ,
T wv ( λ ) = exp 0.2385 a wv ( λ ) · W V · M [ 1 + 20.07 a wv ( λ ) · W V · M ] 0.45 .
r d ( λ , 0 ) = f d d f d s E d d ( λ , 0 ) E d s ( λ , 0 ) = f d d f d s 2 T r ( λ ) T as ( λ ) ( 1 ρ d d ) [ 1 T r ( λ ) 0.95 + 2 T r ( λ ) 1.5 ( 1 T as ( λ ) ) F a ] ( 1 ρ d s ) .
r d ( λ , z ) = f d d f d s E d d ( λ , z ) E d s ( λ , z ) = f d d f d s E d d ( λ , 0 ) exp { K d d ( λ ) z l d d / cos θ sun } E d s ( λ , 0 ) exp { K d s ( λ ) z l d s }
r d ( λ , z ) = r d ( λ , 0 ) exp { ( K d s ( λ ) l d s K d d ( λ ) l d d cos θ sun ) z } .
K d s ( λ ) = K d d ( λ ) = a ( λ ) + b b ( λ ) ,
a ( λ ) = a W ( λ ) + a ph ( λ ) + a Y ( λ ) + a d ( λ ) ,
b b ( λ ) = b b , W ( λ ) + b b , X ( λ ) ,
a ph ( λ ) = C i a i * ( λ ) .
b b , X ( λ ) = X · b b , X * · b X ( λ ) + C Mie · b b , Mie * · ( λ / λ S ) n .
l d s = 1.1156 + 0.5504 ( 1 cos θ sun ) ,
l d d = 1.
z 0 = g cos θ sun ln r ln r ( 0 ) K d d ( λ 2 ) K d d ( λ 1 ) .
l d s = 1 [ a ( λ ) + b b ( λ ) ] z ln E d s ( λ , z , θ sun ) E d s ( λ , 0 , θ sun ) ·

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