Page updated: October 27, 2020
Author: Curtis Mobley
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# Apparent Optical Properties

One of the primary goals of optical oceanography is to learn something about a water body, e.g., its chlorophyll or sediment concentration, from optical measurements. Ideally one would measure the absorption coeﬃcient and the volume scattering function, which tell us everything there is to know about the bulk optical properties of a water body, and those IOPs do indeed tell us a lot about the types and concentration of the water constituents. However, in the early days of optical oceanography, it was diﬃcult to measure in situ IOPs other than the beam attenuation coeﬃcient. On the other hand, it was relatively easy to measure radiometric variables such as the upwelling and downwelling plane irradiances. This led to the use of apparent optical properties (AOPs) rather than IOPs to describe the bulk optical properties of water bodies. A “good” AOP will give useful information about a water body, e.g., the types and concentrations of the water constituents, from easily made measurements of the light ﬁeld.

Apparent optical properties are those properties that (1) depend both on the medium (the IOPs) and on the geometric (directional) structure of the radiance distribution, and that (2) display enough regular features and stability to be useful descriptors of a water body.

To understand this deﬁnition, let us consider the easily measured downwelling plane irradiance ${E}_{d}\left(z,\lambda \right)$ as a candidate AOP. ${E}_{d}$ satisﬁes the ﬁrst part of the deﬁnition of an AOP: it depends on the IOPs and on the radiance distribution. If the absorption or scattering properties of the water body change, so does ${E}_{d}$. If the sun angle changes, i.e., the directional structure of the radiance changes, so does ${E}_{d}$. However, if the sun goes behind a cloud, ${E}_{d}$ can change by an order of magnitude within seconds, even though the water body composition remains the same. ${E}_{d}$ can also ﬂuctuate rapidly (on a time scale of 0.01 s) and randomly near the sea surface due to surface wave focusing of the radiance transmitted through the wind-blown surface. ${E}_{d}$ thus does not satisfy the second part of the AOP deﬁnition: it does not “display enough regularity and stability to be a useful descriptor of a water body.” A measurement of ${E}_{d}$ alone tells us little about the water body itself. A measurement of spectral ${E}_{d}$ might show that the water is blue and thus perhaps low in chlorophyll, vs. green and thus perhaps high in chlorophyll. However, we cannot expect to deduce the chlorophyll concentration from a measurement of ${E}_{d}$ because of its sensitivity to external environmental eﬀects such as sun zenith angle, cloud cover, or surface waves. The same is true of other radiometric variables such as ${E}_{u}$, ${L}_{u}$, or even the full radiance distribution: they satisfy the ﬁrst half of the AOP deﬁnition but not the second half. Radiances and irradiances themselves are not AOPs.

Now consider the ratio of upwelling to downwelling plane irradiances,

 $R\left(z,\lambda \right)=\frac{{E}_{u}\left(z,\lambda \right)}{{E}_{d}\left(z,\lambda \right)}\phantom{\rule{2.6108pt}{0ex}}.$

If ${E}_{d}$ changes because of a change in sun location, cloud cover, or surface waves, ${E}_{u}$ will change proportionately because ${E}_{u}$ arises mostly from upward scatter of the same downwelling radiance that determines ${E}_{d}$. Thus the ${E}_{u}∕{E}_{d}$ ratio should be much less inﬂuenced by changes in the external environment than are ${E}_{u}$ and ${E}_{d}$ individually. The irradiance reﬂectance $R={E}_{u}∕{E}_{d}$ is thus worthy of further investigation as a descriptor of the water body itself. However, we still expect at least a small change in ${E}_{u}∕{E}_{d}$ as sun zenith angle changes, for example, because diﬀerent parts of the VSF will contribute to the upward scattering of downwelling radiance.

The same holds true of the normalized or logarithmic depth derivative of ${E}_{d}$,

 ${K}_{d}\left(z,\lambda \right)=-\frac{dln{E}_{d}\left(z,\lambda \right)}{dz}=-\frac{1}{{E}_{d}\left(z,\lambda \right)}\phantom{\rule{2.6108pt}{0ex}}\frac{d\phantom{\rule{0.3em}{0ex}}{E}_{d}\left(z,\lambda \right)}{dz}\phantom{\rule{2em}{0ex}}\left({m}^{-1}\right)\phantom{\rule{3.26288pt}{0ex}},$

Again, if ${E}_{d}$ changes suddenly, the second form of the derivative shows that the change in magnitude of ${E}_{d}$ will cancel out, leaving the value of ${K}_{d}$ unchanged. ${K}_{d}$ thus satisﬁes the stability requirement for an AOP. ${K}_{d}$ should also depend on the IOPs because changing them will change now rapidly the irradiance changes with depth. The diﬀuse attenuation function ${K}_{d}$ is thus another candidate worthy of consideration as an AOP.

It is easy to think of other such ratios and depth derivatives that can be formed from radiometric variables: ${L}_{u}∕{E}_{d}$, ${L}_{u}∕{L}_{d}$, $-dln{E}_{u}∕dz$, $-dln{L}_{u}∕dz$, and so on. Each of these candidate AOPs must be investigated to see which ones provide the most useful information about water bodies and which ones are least inﬂuenced by external environmental conditions such as sun location or sky conditions. Table 1 lists the most commonly used AOP’s. These will be developed in more detail on the following pages.

 AOP name Symbol Deﬁnition Units diﬀuse attenuation coeﬃcients (K functions) of radiance in any direction $L\left(𝜃,\varphi \right)$ $K\left(𝜃,\varphi \right)$ $-dlnL\left(𝜃,\varphi \right)∕dz$ ${m}^{-1}$ of upwelling radiance ${L}_{u}$ ${K}_{Lu}$ $-dln{L}_{u}∕dz$ ${m}^{-1}$ of downwelling irradiance ${E}_{d}$ ${K}_{d}$ $-dln{E}_{d}∕dz$ ${m}^{-1}$ of upwelling irradiance ${E}_{u}$ ${K}_{u}$ $-dln{E}_{u}∕dz$ ${m}^{-1}$ of scalar irradiance ${E}_{o}$ ${K}_{o}$ $-dln{E}_{o}∕dz$ ${m}^{-1}$ of PAR ${K}_{PAR}$ $-dlnPAR∕dz$ ${m}^{-1}$ reﬂectances irradiance reﬂectance $R$ ${E}_{u}∕{E}_{d}$ nondim remote-sensing reﬂectance ${R}_{rs}$ $s{r}^{-1}$ remote-sensing ratio $RSR$ ${L}_{u}∕{E}_{d}$ $s{r}^{-1}$ mean cosines of the radiance distribution $\overline{\mu }$ $\left({E}_{d}-{E}_{u}\right)∕{E}_{o}$ nondim of the downwelling radiance ${\overline{\mu }}_{d}$ ${E}_{d}∕{E}_{od}$ nondim of the upwelling radiance ${\overline{\mu }}_{u}$ ${E}_{u}∕{E}_{ou}$ nondim

Table 1: Commonly used apparent optical properties. ${R}_{rs}$ is a function of wavelength only; ${K}_{\text{PAR}}$ is a function of depth only; all other AOPs are functions of both depth and wavelength.

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