## Modelling the underwater light field

The nature of the underwater light field resulting from a given incident light field is determined by the inherent optical properties of the aquatic medium. In principle, therefore, it should be possible, if we know the inherent optical properties, to calculate the properties of the underwater light field. Such a calculation procedure could be used, for example, to explore in more detail than would be practicable by measurement, the exact way in which the nature of the light field depends on the inherent optical properties. It could be used as a substitute for in situ measurement in environments in which such measurement might be difficult. It could also be used for predicting the effects on the underwater light field of anticipated changes in the optical properties of the water, resulting perhaps from the activities of man, such as, for example, the discharge of wastewater into a water body.

In reality, the complex behaviour of the photon population in the water caused by the combined effects of scattering and absorption precludes the establishment of an explicit analytical relation between the properties of the field and those of the medium. There are, however, computer modelling procedures by means of which it is possible, by making physically realistic assumptions about the ways in which the behaviour of light at any point in a water body is determined by the scattering and absorption properties of the medium, to calculate the nature of the light field that will be set up throughout the whole water column.

The simplest calculation procedure, and the one that has been used most, is the Monte Carlo method. Its application to the behaviour of solar photons in the ocean was pioneered by Plass and Kattawar (1972), and it has also been put to extensive use by Gordon and coworkers.477,484 The principles are described in some detail in Kirk (2004). A brief account of the principle of the method will be given here.

The behaviour of individual photons within a scattering/absorbing medium is stochastic in nature. The lifetime of, and the geometrical path followed by, any given photon are governed by its random encounters with absorbing molecules or algal cells and with scattering particles. The inherent optical properties of the medium - the absorption and scattering coefficients, the volume scattering function - are not only (in accordance with their definitions) measures of the proportion of an incident flux absorbed, scattered and so on per unit depth by a thin layer of medium, but are also related in simple ways to the probability that any given photon will, within a certain distance, be absorbed, scattered or scattered at a certain angle.

The Monte Carlo method takes advantage of the statistical nature of photon behaviour and uses a computer to follow the fate of a large number of photons, one at a time, passing into an imaginary body of water (corresponding, if desired, to some real body of water) with specified optical properties. Random numbers are used in conjunction with appropriate cumulative frequency distributions (based on the optical properties) to choose pathlengths between each interaction with the medium, to decide whether the interaction is one of scattering or absorption, to select the scattering angle and so on. Photons are introduced from above the surface, at an angle or a selection of angles appropriate for the lighting conditions (direct Sun, overcast sky, etc.) under consideration. Refraction at the surface is allowed for. If it is wished to take account of waves, then the surface slope/wind frequency distribution data of Cox and Munk (1954) can be used. Each photon is followed, its trajectory being recalculated after each scattering event until it is absorbed or scattered up through the surface again. Within the imaginary water body there is a series of depth markers. For every single trajectory of each photon the computer records which depth markers the photon passes, in what direction (upwards or downwards) and at what angle. When a large number (say 107) of photons has been processed, the computer can calculate from the accumulated data the values at each depth of irradiance and scalar irradiance (upwards and downwards), reflectance, average cosines (downwards, upward, total) and radiance distribution (at a set of azimuth angles or averaged over all azimuth angles). In this way a complete description of the underwater light field is obtained.

It will very often be the case that something much less than a complete description of the light field is required: perhaps just the vertical attenuation coefficient for irradiance in the photosynthetic waveband (Kd[PAR]), or the subsurface reflectance (R[0,lj) in certain wavebands, or the visual clarity of the water body expressed in terms of the Secchi depth ZSD. In such cases, rather than carrying out the full computer simulation, much simpler calculations can be carried out using certain empirical relationships - expressing Kd, or R, or Zsd as approximate functions of the inherent optical properties - that have arisen out of computer simulation of the light field in waters of various optical types (see §6.7).

Mobley (1989, 1994) has developed an alternative numerical modelling procedure for simulating the underwater light field, which uses invariant imbedding techniques to solve the radiative transfer equation. This is computationally quicker than the Monte Carlo method and is available commercially in the Hydrolight software package.

As we noted above, one of the practical applications of modelling of the underwater light field is the prediction of the effects of discharge of wastewater on the optical character of the receiving water body.709,711 Laboratory measurements of absorption and scattering on the waste-water and the receiving water, together with data on anticipated dilution rates, will make it possible to calculate the values of a and b in the water body before and after discharge of the wastewater. With this information, together with an appropriate scattering phase function, the light field that would exist (under some standard meteorological conditions) in the water body with and without the added wastewater can be calculated by Monte Carlo modelling. Alternatively, if it is only certain aspects of the field, such as Kd[PAR]), R[0,1] or ZSD, which are required, the empirical relationships referred to above may be used. In this way an industrial water user, for example, can assess what effect its effluent will have on certain key indicators of optical water quality in a surface water body, before that effluent is ever discharged.