1. Introduction

The strength of the returned signal for passive and active optical sensors depends on a large number of parameters, one of which is the shape of the sea surface. In this paper we address the dependence of the water-leaving radiance (L_w) on the wavelength and amplitude of a sinusoidal wave. The magnitude of L_w furthermore depends on the Inherent Optical Properties of the water column, the bottom depth and the bottom reflectance.

Optical remote sensing from aircraft and satellites [1] is widely used to infer properties of the dissolved and suspended materials in water. Satellite data finds widespread application in studies of global climate change, the carbon cycle, global productivity, air-sea interaction, etc.

In the presence of waves, L_w can be larger than that for a flat surface. This effect, which is in addition to Fresnel effects, has been termed intensified backscattering [2–4]. Intensified backscattering cannot be modeled using plane parallel radiative transfer models, as it depends on the shape (not only slope spectrum) of the sea surface. The papers referenced above show that for a random sea surface generated using an observed power spectrum, the effects of intensified backscattering are limited to angles less than approximately 12° between the detector and the source. The maximum intensification is 6.5% in the blue and green and 12% in the red for a windspeed of 7.5 m/s. This enhanced backscattering effect is thus extremely important for active remote sensing (Lidar), but not so important for passive remote sensing in which an effort is made to avoid sun glint, and hence small source–detector separation angles. The sea surface is however, not completely represented by random slope statistics. If there are organized wave structures, subsurface irradiance patterns will also be organized and these can lead to intensified backscattering with different spatial patterns compared to random surfaces.

There are significant gaps between application models and our state of knowledge on the surface roughness properties. Recent advances in observations or theoretical development in hydrodynamics have been largely ignored by the optical, radar, and acoustic communities, who all need realistic surface realizations for their work. In the remote sensing related publications, typically Cox and Munk [5] mean square slopes and Pierson and Moskowitz spectra [6] are used. Small capillary and capillary-gravity waves are important contributors to the sea surface slope due to their relatively large steepness [4]. The larger the slope, the larger the refraction of the incident light. Hwang et al. [7] have shown that the curvature wavenumber spectra of ocean waves have a peak at k_p=900 rad m^-1. This peak corresponds to a wavelength of 0.007 m. Longuet-Higgins [8,9] has shown that such capillary waves are generated on the forward face of small gravity waves by high curvature and vortices. These small capillary waves thus occur in regular patches. Within these patches they are coherent. This coherence is lost when, for modeling purposes, a surface is generated randomly from a power spectrum.

The assumption of ocean with a surface made of random waves is valid for wind generated gravity waves of relatively long wavelengths. Short waves that are influenced by surface tension are very different. Although direct wind forcing may be still in effect, steep curvature induced surface tension force plays an important role in their formation [8, 11]. The steep curvature results in a localized surface tension force at the crests of short gravity waves and the surface tension force acts as local wave generator to produce trains of capillary waves. Therefore the capillary waves in a natural wave field are almost always phase-locked with the short gravity waves that served as the wave generator. Example of such coherent structure of wind-generated short waves has been recorded using an optical slope measurement technique [12] in a wind wave tank in the late 50s. Hwang et al. [7, 13, 14] use a scanning slope sensing technique to obtain the space-time evolution of short wind-generated waves in-situ and in the lab.

Intensified backscattering is the result of wave focusing, because there are multiple single scattering return paths. This can be demonstrated by means of a ray diagram. Fig.1 shows the ray pattern beneath a generic sinusoidal wave with a wavelength of 1 m and amplitude of 10 cm. The focal depth to wavelength ratio (Z_f/L) can be shown [10] to be a function of the wave steepness (amplitude to wavelength ratio, A/L):

The location of the scattering event is at the intersection of the rays. In the case of collimated incident light and a flat sea surface, rays do not intersect, and singly scattered light must exit along the same ray it entered the ocean, having undergone one 180° scattering event. If the surface is not flat, light can enter the ocean along one ray and be scattered in another direction in the backward hemisphere along another ray. The number and location of intersections depends on the shape of the sea surface. Focal areas beneath the crests of waves contain many intersections and so are brighter than the troughs. The more ray intersections there are, the more possible return paths there are, and the larger the water leaving radiance. For random sea surfaces there can only be one maximum (towards the zenith) in the intensified backscattering as crests occur randomly. If crests do not occur randomly, but are equally spaced as in Fig.1, secondary focal areas occur and it would be possible to have multiple maxima. Note that here we use the term “backscattering” to denote photons scattered in the backward hemisphere into the receiver. The exact scattering angle depends on the source and receiver geometry.

 

Fig.1. Ray pattern beneath a sinusoidal wave with a wavelength of 1m and amplitude of 10 cm.

Download Full Size | PDF

2. Reflectance Calculations

We have used the ray-tracing model described in Zaneveld et al. [10] to calculate the intensified backscattering in the presence of coherent waves. We use the reciprocity principle, i.e. the fact that the detector sensor response pattern, termed “sensor irradiance”, is the same as that of a light source with the same optics as the detector [15]. The intersection of source and detector rays is where a photon that is backscattered at just the right angle can make the trip from source to detector, and hence contribute to the perceived reflectance. The more ray intersections there are, the more possible return paths there are, and the larger L_w. Reflectance is proportional to the product of the source irradiance and the sensor “irradiance” integrated over the ocean. Using a modulation transfer function in our ray-tracing model we include both scattering and absorption effects.

We have modeled a coherent capillary wave and investigated its effect on remote sensing. We have chosen a wavelength (0.01m) near the peak in the curvature spectrum (see above) and with variable amplitude. The wave amplitude was decreased from 0.3 mm to nearly 0 in powers of 2.

3. Results

Fig.2 shows a typical calculation of irradiance relative to the irradiance just above the sea surface. The absorption coefficient is 0.07 m^-1 and the scattering coefficient is 0.39 m^-1; the volume scattering function used is one of Petzold’s (1972, NUC 2040). The modeled wavy sea surface consists of the superposition of three sinusoids with a wavelengths L=[1.1 0.55 0.05]m and an amplitudes A=[0.05 0.01 0.0008]m. The beam is 0.8 m wide and enters vertically. The left panel shows the beam entering near the trough of a wave, generating a divergent beam. On the right the light enters near the crest of a wave, generating a convergent beam with a focal area.

 

Fig. 2. An example of calculated irradiance relative to the irradiance just above the sea surface. The absorption coefficient is 0.07 m^-1 and the scattering coefficient is 0.39 m^-1; the wave surface is wavelength L=[1.1 0.55 0.05] m and amplitude A=[0.05 0.01 0.0008]m. The beam is 0.8 m wide and enters vertically. Fig.3a shows the beam entering near the trough of a wave, generating a divergent beam. On Fig.3b the light enters near the crest of a wave, generating a convergent beam with a focal area.

Download Full Size | PDF

Active systems, such as Lidar, are likely to see intensified backscattering if waves are present and they have a large footprint. Lidar source and detector are collocated, so that source and sensor irradiance patterns overlap. The resultant signal from a given small region in the ocean is thus the square of the irradiance at that location (the product of the source and the receiver ‘irradiances’). The large area average of the product of redistributed (due to wave refraction), overlapping source and sensor irradiances are always larger than the product of the flat surface irradiances, ignoring scattering and absorption effects. Redistribution of non-zero Δ irradiance from one area to another would result in a lidar signal (1+Δ)²+(1-Δ)²>2. If the footprint is small, the product of source and sensor irradiances is different from the large-scale average. Thus, in Fig. 2, the trough reflectance would be much less than the crest reflectance.

Passive systems do not experience intensified backscattering due to random waves if the source–sensor separation angle is greater than approximately 12° [2–3]. Coherent structure of waves can change this conclusion, as shown below.

Fig. 3 shows the dependence of the remote sensing reflectance on wave amplitude for a wave of wavelength of 0.01 m and a variable angle of separation between the light source and the detector. The reflectance is normalized to the reflectance for a flat sea surface just below the sea surface. It is seen that for large capillary wave amplitudes, there is only a single maximum and that the intensified back scattering was larger than for the random sea surface. For intermediate capillary wave amplitudes we found an oscillatory behavior with source zenith angle. For small wave amplitudes the signal approaches that of a flat surface.

 

Fig. 3. The dependence of the remote sensing reflectance on wave amplitude for a wave of wavelength of 0.01 m and on the angle of separation between the light source and the light detector, normalized to the reflectance for a flat sea surface just below the sea surface. The amplitude of the wave is given in the legend box in cm.

Download Full Size | PDF

 

Fig. 4. Source rays in red and detector rays in blue for a sine wave with wavelength of 0.01m and amplitude of 0.003m. The source detector separation is 15 degrees. Horizontal and vertical axes are not to the same scale. When the wave is steep, the primary focal point is “trapped” in the wave. The source and detector focal points are then further apart the larger the separation angle. This results in a steadily decreasing reflectance as a function of separation angle as can be seen in Fig. 3 for the steeper waves (curves 1and 2 in Fig. 3).

Download Full Size | PDF

From the reciprocity principle [15] it can be inferred that the reflectance is a function of the product of the source irradiance and the “sensor irradiance”. This simple mechanism can explain the observations from Fig.4. If the wave slope is high, the focal point is close to the surface, and changing the separation angle moves the focal points of the light source and detector only a small distance apart. Hence there is only one possible overlap of the primary source and detector focal points. Fig. 4 shows an example of this situation. When the wave is steep, as in Fig. 4, the primary focal point is “trapped” in the wave. The source and detector focal points are then further apart the larger the separation angle. This results in a steadily decreasing reflectance as a function of separation angle as can be seen in Fig. 3 for the steeper waves (curves 1 and 2, Fig.3).

 

Fig. 5. Same as for Fig.5, but for wave amplitude of 0.0001 m. Note different vertical scale. When the wave is less steep there can be multiple locations where the source and receiver focal points overlap. This sets up a periodic response function (curves 5-7 in Fig.3).

Download Full Size | PDF

In Fig.5 we have plotted a capillary wave with much smaller amplitude of 0.0001 m. Note the different vertical scale compared to Fig. 5. In the case of a less steep wave, the focal length is much larger and the “lever arm” of the focal point as the separation angle increases is much larger than in Fig.4. This results in the source and detector primary focal areas alternately overlapping and missing as the separation angle increases. We then get the oscillatory behavior as seen in cases 5, 6, and 7 in Fig. 3.

4. Conclusions

We have shown, using a simple illustrative example that coherent capillary waves can result in a far different back scattering pattern than random waves. This return can be much larger than for random wave surfaces, and will result in estimated backscattering from remote sensing being biased compared to in-situ measurements. The angle of the observation plane relative to the wind is important as capillary waves are not isotropically distributed. Fresnel effects at the sea surface are in addition to the effects studied here. The effects discussed here must be adjusted for the proportion of the sea surface covered by coherent waves. Swell is also coherent and will have similar effects, except that the IOP are more important, since photon will travel a larger distance within the ocean before emerging to the sensor.

Enhanced backscattering has been found to be negligible at source-receiver angles separation bigger than 12° when the ocean surface was modeled as a random realization of sinuses [3–4]. Here we find, for a surface containing coherent capillary-like waves, that the enhanced backscattering may be observed at larger separation angles, possibly impacting passive remote sensors as well. For passive sensors this effect is likely to be small (less than 5%) and is reduced, on average, the larger the separation angle between source and receiver (Fig. 3).

A complete analysis of the coherent wave effect will require three-dimensional realizations of the sea surface together with distributions of the inherent optical properties. Such data does not exist at present.