influence of surface waves on measured and modeled irradiance profiles

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Influence of surface waves on measured andmodeled irradiance profiles

J. Ronald V. Zaneveld, Emmanuel Boss, and Andrew Barnard

Classical radiative transfer programs are based on the plane-parallel assumption. We show that theGershun equation is valid if the irradiance is averaged over a sufficiently large area. We show that theequation is invalid for horizontal areas of the order of tens of meters in which horizontal gradients ofirradiance in the presence of waves are much larger than vertical gradients. We calculate the distri-bution of irradiance beneath modeled two-dimensional surface waves. We show that many of thefeatures typically observed in irradiance profiles can be explained by use of such models. We derive amethod for determination of the diffuse attenuation coefficient that is based on the upward integrationof the irradiance field beneath waves, starting at a depth at which the irradiance profile is affected onlyweakly by waves. © 2001 Optical Society of America

OCIS codes: 010.4450, 030.5620, 000.4430, 070.4560.

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1. Introduction

Optical remote sensing by use of satellites averagesthe upwelling radiance over the footprint of the sat-ellite. For the Sea-Viewing Wide Field-of-View Sen-sor ~SeaWiFS! this is approximately 1 km 3 1 km.

hen the experimental optical oceanographer goes toea to undertake observations to calibrate the satel-ite data or to validate inversion algorithms, he or sheypically takes measurements from a ship using pro-ling radiometers and profiling devices for the mea-urement of absorption, scattering, and attenuationharacteristics ~inherent optical properties, IOP’s!.bservations from ships or moored profilers have a

ootprint of less than 1 m 3 1 m. It is then usuallyssumed @for example, in NASA’s Sensor Intercom-arison and Merger for Biological and Interdiscipli-ary Oceanic Studies ~SIMBIOS! program# that thehipboard observation is a valid ground truth mea-urement for the one million times larger satelliteootprint. It is clear that the much larger satelliteootprint averages over many more surface waves

J. R. V. Zaneveld [email protected]!, E. Boss [email protected]!, and A. Barnard [email protected]! are with theCollege of Oceanic and Atmospheric Sciences, Oregon State Uni-versity, 104 Ocean Administration Building, Corvallis, Oregon97331-0000.

Received 7 July 2000; revised manuscript received 6 November2000.

0003-6935y01y091442-08$15.00y0© 2001 Optical Society of America

1442 APPLIED OPTICS y Vol. 40, No. 9 y 20 March 2001

han an in situ profile. Because of this, shipboardrradiance profiles nearly always show large varia-ions in the absolute value of the downwelling irra-iance near the surface. Figures 1 and 2 showypical measured profiles. Variability in the irradi-nce is largest near the surface, yet for remote sens-ng purposes near-surface measurements are by farhe most important.

Much has been written about the statistics of theight field variations caused by surface waves. Theook by Walker1 gives a thorough review on the sub-

ject, including extensive literature from Poland andthe former Soviet Union. Near-surface fluctuationsin measured irradiance profiles are usually treated asnoise and are filtered out.2 In this paper we showthat the horizontal gradient in irradiance profiles isoften an order-of-magnitude larger than the verticalgradient. An individual irradiance profile clearlyshows the superposition of wave-induced irradiancefluctuations on the vertical profile. Here we addressthe vertical structure of irradiance profiles in thepresence of waves and examine the variability ofthese profiles for constant light absorption and scat-tering conditions ~IOP’s! and its influence on radia-tive transfer. We examine characteristic verticalstructures of these profiles and relate them to seasurface properties.

2. Plane-Parallel Assumption

When radiometric data are used to infer IOP’s, orwhen radiative transfer calculations are performed,it is often assumed that the ocean is horizontally

w

homogeneous. This is the so-called plane-parallelassumption. This assumption has major implica-tions for the understanding and applicability of ship-board data to radiative transfer modeling.

If the equation of radiative transfer is integratedover 4p sr, we derive Gershun’s3 equation:

¹ z E~x, y, z! 5 2a~x, y, z!E0~x, y, z!, (1)

where E is the net plane irradiance, a is the absorp-tion coefficient, and E0 is the scalar irradiance. Thisequation is almost always cited in the oceanographicliterature as

dE~z!

dz5 2a~z!E0~z!, (2a)

or

K~z!E~z! 5 a~z!E0~z!, (2b)

or

a~z!

K~z!5

E~z!

E0~z!5 m# ~z!, (2c)

where K~z! is the attenuation coefficient for the planeirradiance and m# ~z! is the average cosine of the lightfield. These equations ~which we refer to as theplane-parallel Gershun or PPG equations! imply thatthe horizontal gradients in E are small comparedwith the vertical ones. Thus the plane-parallel as-sumption implies that

]

]xE~x, y, z! ,,

]

]zE~x, y, z!,

]

]yE~x, y, z! ,,

]

]zE~x, y, z!. (3)

We examine whether the large-scale horizontally av-eraged irradiance profile obeys the PPG equations.

3. Near-Surface Horizontal and Vertical Gradients ofIrradiance

The sea surface nearly always contains at least somesmall waves. Typical downwelling plane irradianceprofiles for off the Oregon coast and the Gulf of Cal-ifornia are shown in Figs. 1 and 2. These profileswere taken under calm conditions ~winds were lessthan 2 mys for Fig. 1 and less than 0.5 mys for Fig. 2!.Near the surface the profiles fluctuate by a factor of2–3 several times at a depth interval of 1 m. Thesefluctuations are not due to small-scale vertical vari-ations in the IOP’s because the IOP measurementsthat were made almost simultaneously show uniformvertical profiles. A tilt sensor in the instrumentshowed values of approximately 1 deg for these pro-files. The irradiance fluctuations are thus almostentirely due to variations in the underwater lightfield related to the wavy surface.

We now calculate approximately the horizontaland vertical gradients of the irradiance for the profilein Fig. 2, taken during quite calm conditions in the

Gulf of California. The profiler with which the datawere obtained drops at approximately 80 cmys, andthe vertical peaks of irradiance in Fig. 2 ~in the Gulfof California! were measured approximately 0.4 sapart. If we assume that these peaks are due to awave with a period T of approximately 0.4 s, we findthat for gravity waves in deep water4 their wave-length is approximately 0.7 m ~using T2 5 1ygk,where g is the gravitational constant and k is the

ave number!. If we assume that E is proportionalto sin~kx!, we can see that ~]y]x!E~x, z! is propor-tional to k cos~kx! and the average value of the hor-izontal gradient of E, , ~]y]x!E~x, z! . ' ~ky2!E '

Fig. 1. Irradiance profile taken off the Oregon coast, September1997, with a Satlantic irradiance profiler. The profiler drops atapproximately 0.8 mys and samples at 8 Hz. Conditions werecalm.

Fig. 2. Irradiance profile taken in the Gulf of California, October1998, with the same profiler as used in Fig. 1. Conditions werecalm.

20 March 2001 y Vol. 40, No. 9 y APPLIED OPTICS 1443

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1

4.5 E. On the other hand, the total absorption coef-ficient was measured to be 0.1 m21. With m# ' 0.8and using Eqs. 2~a!–2~c!, we see that ~]y]z!E~x, z! '0.1y0.8 5 0.125 E. The horizontal gradient near thesurface is thus more than an order-of-magnitudelarger than the vertical gradient. The conditions ininequalities ~3! therefore do not apply at all, and theocean is anything but horizontally homogeneous asfar as irradiance is concerned. It should be notedthat the example just provided is for extremely calmoceanic conditions. For conditions with largerwaves, the horizontal gradients would be larger.For the PPG equations to apply, the horizontal gra-dient of irradiance would have to be approximately 2orders-of-magnitude smaller than that observed inthe ocean under calm conditions.

4. Large-Area Approximation of the Gershun Equation

Given that the plane-parallel assumption does notapply to typical ocean surfaces over a horizontal scaleof meters, even if the IOP’s are constant, does it applyfor kilometer scales? We show here that it does in-deed apply when irradiances are averaged over suit-ably large areas, and the PPG equation @Eq. ~2!# canbe used for irradiances averaged horizontally overlarge scales.

For simplicity we use the two-dimensional form ofEq. ~1!:

]

]xE~x, z! 1

]

]zE~x, z! 5 2a~z!E0~x, z!, (4)

where the absorption coefficient is assumed to be afunction of z only.

Integrating Eq. ~4! from x1 to x2, we obtain

E~x1, z! 2 E~x2, z! 1 *x1

x2 ]

]zE~x, z!dx

5 2a~z! *x1

x2

E0~x, z!dx. (5)

We then divide by Dx and denote the average over xby an overbar:

E~x1, z! 2 E~x2, z!

Dx1

]

]zE~z! 5 2a~z!E0~z!. (6)

In the limit of Dx 3 `, we obtain

]

]zE~z! 5 2a~z!E0~z!. (7)

We have thus shown that large horizontal averages ofplane and vector irradiance obey the plane-parallelapproximation and the PPG equation. How large ahorizontal area needs to be averaged in practice? Inthe above example we had variations approximatelyof the magnitude of E in one wavelength of approxi-mately 0.7 m which gave a gradient of 4.5 E. Thevertical gradient was estimated to be an order-of-magnitude smaller. To invert this, with the hori-


zontal component being an order-of-magnitudesmaller than the vertical gradient, we would have toaverage over 100 wavelengths, or approximately70 m in this case for calm conditions. In the pres-ence of a large swell with wavelengths of tens ofmeters, 100 wavelengths could well exceed the foot-print of the satellite radiance sensor. Note that theabove does not imply that the large-scale averageirradiance is the same in the presence of waves as intheir absence.

5. Model for Radiometer Profiles under Waves

When the experimentalist measures the profile of aradiometric property in a wavy environment, a singlerealization of the profile is obtained. Such profilesdepend on the specific location at which the profile istaken. Averages of several different profiles will dif-fer, yet it is usually assumed that such averages rep-resent the large-scale average. For that to be true,the ergodic principle must hold, and a sufficientlylarge number of profiles must be averaged to satisfythe conditions in inequalities ~3!. In Section 6 westudy the relationship of a single realization of theirradiance profile to the large-scale average. So farwe have dealt with the net plane irradiance E~z!.The parameter that is usually measured is the down-welling plane irradiance Ed~z!. In practice thesetwo quantities differ by only a few percent. Forpractical purposes, the scale arguments made for thenet plane irradiance thus also apply to the down-welling irradiance. We are particularly interestedin determining how such single profiles can be depthor space averaged to obtain an approximation of thelarge-scale average Ed~z!. It is obvious from theabove discussion that averaging a large number ofprofiles of Ed~z! will approximate the large-area av-rage Ed~z!.Many radiative transfer models take waves into

account. In such models1,5 the radiance just be-neath the sea surface is nevertheless assumed to behorizontally homogeneous. We can accomplish thisby using the statistics of the wave slopes and byobtaining an ensemble-average radiance distribu-tion. This radiance distribution would be the sameas the large-area average alluded to above providedthat all the waves are taken into account ~we showelow that even small waves have a significant im-act on the underwater light field!. Such radiativeransfer models thus apply to remote sensing prob-ems when the conditions in inequalities ~3! are met.hey do not apply when we want to understand inore detail the experimental vertical irradiance pro-les such as those in Figs. 1 and 2.To study the effects of waves on vertical irradiance

rofiles, we developed a two-dimensional wave modelhat allows us to model measured radiometer profilesnder a variety of wave conditions. Our model takesarallel incident rays, refracts them at the sea sur-ace according to Snell’s law, and reduces their inten-ity according to the Fresnel equations. Theocations of intersection with the z plane and angle ofncidence are calculated. By using a large number

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of rays and by sorting their irradiances according tothe intersection with the z plane, we can obtain theirradiance as a function of distance along the z plane.We add absorption and scattering effects by turningeach ray intersection with z into a beam spread func-tion pattern and recalculating the distribution of Ealong z. To do this we follow the model of McLeanand Freeman,6 which in turn is based on the theoret-ical development of Wells.7 The program that wedeveloped allows the superposition of any number ofsinusoids and use of any spatial distribution of theabsorption coefficient and volume-scattering func-tion. Solar zenith angle and sky conditions will af-fect the results. In this paper we chose to limit thealready extensive number of parameters that affectthe underwater irradiance distribution. We there-fore used a vertical Sun in a black sky and homoge-neous IOP’s. Figure 3 is an example of the output ofour radiative transfer program. In this example wecalculated the downwelling irradiance pattern rela-tive to the downwelling irradiance just above the sur-face for homogeneous IOP’s and a random wavesurface with a 1.1-m dominant wave. Figure 3 alsoshows the sampling points of a typical irradianceprofiler, which is discussed further below.

The ray pattern beneath a sinusoidal wave hasbeen studied before1; we present additional detailshere. We calculated the irradiance fields beneathwave surfaces of increasing complexity to study theirinfluence on the spatial distribution of downwellingirradiance. Figure 4 shows the plane irradiance be-neath a one-dimensional sinusoidal wave. No ab-sorption or scattering of light was introduced. Anobvious feature is the extreme horizontal variation ofEd~z!, with a focal area beneath the wave crest.Note that the focal point is spread out because we donot have a simple spherical surface. Also note that

Fig. 3. Irradiance pattern beneath a random wave surface with a1.1-m dominant wave: a 5 0.06 m21 and b 5 0.15 m21. APetzold volume-scattering function was used ~see text!. Crossesare sampling points of an irradiance sensor dropping at 0.8 myswith a sampling rate of 8 Hz.

beams from successive waves intersect multipletimes below the primary focal area and form a dia-mond pattern. These beams are due to the sinusoi-dal wave form, which has two nearly flat areas at z 50. These areas generate nearly parallel rays unlikethe converging or diverging rays from the rest of thesinusoidal surface. These beams persist to greatdepths if not diffused by scattering. When there isweak absorption and scattering, the maximum irra-diance occurs in the primary focal point and the min-imum just above the first intersection of the beamsfrom successive waves. These minima decrease inintensity with depth because, at each successivecrossover, the light from more waves is averaged.Simple geometry shows that the vertical size of thediamond pattern is given by

zc 5L2

tan(p

21 a sinH1

nsinFa tanS2pA

L DGJ2 a tanS2pA

L D) , (8)

where n is the index of refraction of the water, A is themplitude, and L is the wavelength of the sinusoidalave. AyL is the wave steepness.Various authors1 have approximated the depth of

the focal area. Our own calculation is based on theassumption that the crest of the wave can be approx-imated by a spherical segment with the sameamplitude-to-wavelength ratio ~AyL! as the wave.This approximation gives the depth of the focal pointas

zf

L5 2

AL

118

LA

, (9)

where zf is the depth of the focal point below theaverage surface elevation. The ratio of the focal

Fig. 4. Irradiance pattern beneath a generic sine wave. Thevertical extent of the basic diamond pattern is a function of thewavelength and wave steepness only ~see text!.


mlbwpbra0asinwallnct

aiato

d

1

depth to the wavelength, zfyL, depends only on thewave steepness.

The irradiance pattern beneath a sinusoidal wavecan thus be scaled in the x direction to wavelength Land in the z direction to the fundamental size of thediamond pattern described above. With this scalingin mind, Fig. 4 thus applies to the light pattern be-neath any sinusoidal wave. Note that absorptionand scattering were neglected here to show a clearpicture of the irradiance pattern.

Figure 5 shows a wave with L 5 2.25 m and A 5 0.1with the addition of a capillary wave with a wave-

ength of 0.05 m and an amplitude of 0.002 m. It cane immediately seen that such a very small capillaryave completely changes the underwater irradianceattern from that of a simple sinusoid. The focaleams of the 1-m wave alluded to above are greatlyeduced in intensity. Figure 6 shows the addition of

smaller wave with a wavelength of 0.2 m and a.01-m amplitude to the wave in Fig. 5. The changelso emphasizes that very small ripples in the seaurface have a significant effect on the underwaterrradiance distribution. It can be seen that theumber of focal areas are increased by the number ofave crests added and that their divergence belownd above the focal area is governed by the wave-ength and the wave steepness. When diving orooking over the side of a ship, one can see that thesearrow, well-defined rays converge. The apparentonvergence is due to the perspective of parallel rayshat meet at infinity.

The strong dependence of the near-surface irradi-nce distribution on the surface waves has a majornfluence on the measured vertical profile of irradi-nce. To model the dependence of the measured ver-ical irradiance profile on the surface waves, we carryut the following analysis. The radiometer profiler


rops through the water at a speed wd. The largestwave travels horizontally at a wave speed nw. Thuswe can visualize the track of the profiler through theirradiance field beneath the waves by considering thewaves to be stationary and the instrument to be de-scending at a speed of wd while traveling horizontallyat a speed nw. This approach is not exactly correctinasmuch as the different waves that constitute thesurface travel at different speeds ~wave dispersion!.The method works well enough, however, to under-stand observed Ed~z! profiles. Figure 3 shows thetrack of a profiler through a random wave surfacegenerated by the superposition of a number of sinu-soids. This wave surface corresponds to a windspeed of 10 mys.

The ensemble of possible profiles has some inter-esting characteristics. Figure 7 shows the superpo-sition of 20 irradiance profiles with random startingpoints in the surface wave pattern. Irradiance pro-files were calculated for a profiler with a drop speed of0.8 mys and a sampling rate of 8 Hz. The greatestvariance occurs between 2 and 3 m, the focal area ofthe largest wave. The variance decreases below thefocal area and decreases sharply at 6 m. This is thedepth of the intersection of the focal rays discussedabove. Beneath this point the light from morewaves is averaged, reducing the variance. It canalso be seen that the minimum irradiance at a givendepth actually increases below 6 m for the same rea-son.

6. Analysis and Reconstruction of Irradiance Profiles

Using the above model results we are now in a posi-tion to understand features of specific profiles muchbetter. Figure 2, for example, shows a region ofmaximum variance between 2 and 5 m, which is pre-sumably the focal area of the largest wave. The

Fig. 5. Downwelling irradiance field beneath a sea surface com-posed of a gravity and a capillary wave. The gravity wave has awavelength of 2.25 m and an amplitude of 0.1 m. The capillarywave has a wavelength of 0.05 m and an amplitude of 0.002 m.

Fig. 6. Irradiance pattern beneath a superposition of sinusoidalwaves with wavelengths of 2.25, 0.2, and 0.05 m and with ampli-tudes of 0.1, 0.01, and 0.002 m. Note that the addition of verysmall amplitude waves significantly alters the irradiance pattern.

0

a

minimum irradiance in the 10-m profile occurs justabove a 6-m depth, not at the greatest depth. Below6 m the profile is much more damped, indicating that6 m is probably the depth of the crossover point of thefocal rays. Finally, the profile is increasinglysmooth, probably because of light scattering, al-though the presence of capillary waves would alsocontribute to damping of irradiance fluctuations.

We have already shown that the irradiance patternbeneath a wave is highly dependent on even thesmallest surface elevations and the specific locationin the wave train where the profile is taken. Thereconstruction of a specific measured wave profilethus is not possible. One can, however, generateprofiles that display similar features to the measuredone by varying the components of the surface wave,the starting location in the wave train, and the dropspeed. For the profile in Fig. 2, for example, wemeasured the IOP’s to be absorption coefficient a 5.07 m21 and scattering coefficient b 5 0.39 m21.

We can estimate the period of the dominant wavefrom the measured profile by determining the timeinterval of vertical irradiance maxima in the profile.The wavelength is then calculated from the period.Figure 8 shows an irradiance profile generated by useof the measured IOP’s and an average Petzold phasefunction.5 The modeled profiles are extremely sen-sitive to the details of the surface, the point in thewave pattern at which the probe enters the water,and the drop speed of the probe.

7. Determination of an Average Diffuse AttenuationCoefficient

The attenuation coefficient of the downwelling planeirradiance Kd~z!~[21yEd]Edy]z! is a frequently usedparameter in remote sensing algorithms. Its deter-mination from a single profile of irradiance beneath a

wavy surface is not straightforward. Depending onwhere in the wave train the profiler penetrated thesurface, one obtains a different profile. These pro-files will provide larger or smaller Kd values near thesurface depending on whether they profiled through afocal region. It is not unusual to see an irradianceprofile with the maximum at a 2–3-m depth. Thisdoes not imply that the maximum large-scale hori-zontally averaged irradiance occurs at that depth, butonly that the local irradiance measured during theprofile had a maximum. Such local profiles in turncan lead to a negative Kd value in the upper 2–3 m ifany of the usual averaging schemes are used. Sim-ilarly, analysis of single profiles can lead to Kd valuesthat are less than the absorption coefficient. Onemethod is to take many profiles and to average them.In the limit, this approaches the horizontal averageirradiance at each depth.

It was shown that the PPG equation holds for thehorizontally averaged light field. Let the horizon-tally averaged downwelling irradiance be given byEd~z!. We measure a single realization of Ed~z! be-neath a wavy surface. Each profile is different eventhough Ed~z! remains constant. Is there an opti-mum way to derive Kd for Ed~z! when we have only asingle profile of Ed~z!?

We assume that Ed~z! has an exponential profileEd~z! 5 Ed~0!exp~2Kdz!. If we wish to obtain anestimate of the depth-integrated irradiance presentbetween a depth z0 at which fluctuations in irradi-ance that are due to the sea surface are small and thesea surface, we could write

I 5 *z0

0

Ed~z!dz 5Ed~0!

Kd@exp~Kd z0! 2 1#. (10)

Fig. 7. Superposition of 20 modeled irradiance profiles with ran-dom offsets. a 5 0.1 m21 and b 5 0 m21. The surface consists oftwo waves with wavelengths of 2.25 and 0.2 m and amplitudes of0.29 and 0.01 cm, respectively.

Fig. 8. Modeled irradiance profile with waves of wavelengths 3,1.2, 0.6, and 0.05 m and amplitudes of 0.1, 0.05, 0.01, and 0.0006m, respectively. a 5 0.07 m21, b 5 0.39 m21, offset is 0.355 m,nd drop speed is 0.6 mys.


W

o1

p

w

c

cciwne

t

1

We obtained the integrated irradiance I, in numericalpractice, by subdividing space into small volumesDxDz and allowing Ed~x, z! to represent the value ofthe irradiance in the immediate neighborhood. In-tegration by use of the sampling points of the irradi-ance profiler uses the same principle, but thesampling points ~x, z! are different. We thus can setintegration over the two sets of sampling points ap-proximately equal; in the limit of infinitely close spac-ing of Dx and Dz, they would be

I9 5 *z0

0

Ed~z!dz <Ed~0!

Kd@exp~Kdz0! 2 1#. (11)

This would be sufficient to determine Kd if we knewEd~0!, but, given single irradiance profiles, this isdifficult to determine. We can avoid having to knowEd~0! by integrating from z0 to successively shallowerz:

I9~z! 5 *z0

z

Ed~z!dz < I~ z!

5Ed~0!

Kd@exp~Kdz0! 2 exp~Kdz!#. (12)

e then see that the slope of ln@dI9~z!ydz# is Kd.For profiles such as those shown in Figs. 1 and 2,

ne can choose z0 5 10 m, integrate the profile from0 to 9 m, from 10 to 8 m, etc. to obtain I9~9!, I9~8!, etc.

Kd is calculated when we least-squares fit an expo-nential to DI9~z!yDz.

We ran an example of a superposition of three si-nusoidal waves. The absorption coefficient was setat 0.05 m21. The scattering coefficient was set at 0m21 so that Kd was highly predictable. In the ab-sence of scattering, only the increased path lengththat was due to the refraction at the sea surfacecauses Kd to deviate from a. This deviation is, how-ever, small. We thus know that the large-scale Kd isnearly equal to 0.05 m21. How do Kd’s for individual

rofiles deviate from the large-scale Kd? The statis-tics for 2000 calculated profiles through the wave-induced irradiance field were as follows: mean Kd 50.0503 m21, median Kd 5 0.0503 m21, standard de-viation of 0.0037 m21, Kmin 5 0.0413 m21, andKmax 5 0.0597 m21. The distribution of Kd values

as nearly Gaussian. Thus the mean Kd was veryclose to a as expected. Kd’s for individual profilesshowed deviations as large as 20%. Thus it is per-fectly reasonable for the Kd determined from an in-dividual profile to be less than the absorptioncoefficient because not all the divergence in Ed~x, z! isontained in the vertical direction.

8. Discussion and Conclusions

We have shown that the variability of irradiance pro-files as typically determined by experimentalists isnot due to instrument noise, but probably representsthe true irradiance field as present beneath the wavysea surface. Even waves with small amplitudes


have a significant effect on the redistribution of irra-diance @Figs. 4 and 5; Eqs. ~8! and ~9!#. Currentcommercially available irradiance sensors typicallydrop at 0.8 mys and sample at 8 Hz. Because of thestatistical nature of the sea surface and the discretesampling of the irradiance field, it is to be expectedthat no two irradiance profiles will be the same, eventhough the wind speed and IOP’s are constant. Thisobservation is supported by Fig. 7 in which 20 irra-diance profiles are superimposed. For these same20 profiles, the IOP profiles are identical. Whereasone measured IOP profile when used with appropri-ate wave statistics can lead to the calculation of anaccurate, spatially averaged irradiance profile ~by useof the plane-parallel assumption and classical radia-tive transfer!, one measured irradiance profile ~or aombination of other local radiometric measurables!annot be inverted to obtain the IOP accurately, evenf the wave statistics are known. Simply stated,aves directly affect apparent optical properties, butot IOP’s. Indirectly, of course, breaking waves gen-rate bubbles that in turn affect the IOP’s.Zaneveld8 showed that IOP’s can be uniquely de-

termined if the radiance and its depth derivative areknown. We have shown here that this proof appliesonly if the plane-parallel assumption is valid, i.e., ifthe horizontal gradient of the radiance is negligible.

We have shown that, for a sufficiently large hori-zontal scale, the horizontally averaged irradianceobeys the PPG equations @Eqs. ~2a!–~2c!#. On thatscale classical radiative transfer calculations basedon the plane-parallel assumption can be used. Forincreasingly smaller scales, calculations based onthese assumptions will show increasingly larger de-viations between the model and the observations.We will direct our future research to comparing ra-diative transfer results using the plane-parallel as-sumption and wave optics models. This is ofparticular interest when we look at the remote sens-ing reflectance for different scales. Horizontal vari-ability in the remote sensing reflectance is ofparticular interest as sensors are being developedwith footprints of the order of tens of meters.

An important problem is the connection between asingle irradiance profile and the large-scale averageirradiance profile. It is obvious that within the limitof an infinitely high sampling rate and infinitely slowdrop rates the two will converge. Irradiance pro-files, however, are designed to drop quite fast tomaintain vertical stability. We demonstrated thatfor such devices a value for the diffuse attenuationcoefficient Kd can be derived @Eqs. ~10!–~12!# that inhe mean converges to the true large-scale Kd. Any

individual calculation can still be in error, however.The standard deviation of the error depends on thewave surface as well as the IOP, so that no general-ization can be provided. In the example provided,the standard deviation of the error was approxi-mately 7%. This is fairly typical for wavelengths ofthe order of a meter.

The measurement of Kd by hanging irradiance sen-sors at a given depth below a mooring may have some

1

i

2

interesting consequences inasmuch as the sensorswould follow the surface elevations and not bepresent at a constant depth relative to the averagesurface. This would result in an overestimate of theirradiance because of the averaging of an exponentialfunction. The error would thus depend on the valueof Kd.

In the calculations for Figs. 4, 5, and 6, we set theIOP equal to zero. We did this to show the results ofwave focusing only. With increasing absorption, theirradiance values decrease with depth, whereas in-creased scattering smears the pattern. Stramskaand Dickey9 showed that the power spectrum of ir-radiance fluctuations in the green had a peak at ahigher frequency than the wave power spectrum.The observation depths in their paper were 15 and35 m. The focal depth for large L is approximately

y8 L2yA @Eq. ~9!#. The large waves thus will havefocal depths beneath the observation point. Thepeak frequency of the irradiance fluctuation spec-trum at a given depth will be due to the wave with asimilar focal depth. In addition, IOP’s will decreasethe variance of irradiance with depth.

There are many expressions for the remote sensingreflectance as a function of the IOP. These also as-sume a plane-parallel situation because the up-welling radiance is less variable horizontally than theirradiance. Upwelling radiance taken at the sametime as the irradiance, as shown in Figs. 1 and 2, hasapproximately 10% variability near the surface com-pared with the downwelling irradiance. The remotesensing reflectance ~Rrs 5 LuyEd! thus varies approx-mately inversely proportional to Ed~z! with depth.

Clearly this is not allowed under the plane-parallelassumption if the IOP’s are constant with depth.The usual model,

Rrs 5fQ

bb

a, (13)

does not apply in a wavy environment on the scale ofthe wavelength of the surface wave. In the aboveexample, where E varies by a factor of 2 at a wave-length of 70 cm, Rrs would also vary by approximatelya factor of 2 horizontally at 70 cm, even though bb anda are constants. The extension of this argument towavelengths of tens of meters has implications for theinterpretation of upwelling radiance signals takenwith satellite sensors that have footprints of thatorder. Variability in the measured reflectance canbe reduced when the downwelling irradiance is mea-sured above the sea surface.

This research was supported by the OceanographyProgram of the National Aeronautics and Space Ad-ministration.

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influence of surface waves on measured and modeled irradiance profiles

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