Supplement 7.1: The Hydrographic Lidar Equation (2/3)

Special cases

On the previous page we deduced the following relation for a signal from a layer between the depths z₁ and z₂:

P_{z_{1} \to z_{2}} \propto exp (- \int_{0}^{z_{1}} c d z) \cdot \frac{η}{c} (\frac{1}{{(z_{1} + n H)}^{2}} - \frac{exp - (c (z_{2} - z_{1}))}{{(z_{2} + n H)}^{2}})

Homogeneous optically thick media

The attenuation coefficient c and the signal efficiency η of a homogeneous medium (a deep water column, a thick oil film, or another substance) are constant and depth invariant.

One defines as the optical depth a distance which corresponds to the inverse attenuation coefficient: $z = 1 / c$ . By inserting this relation into Lambert‘s law $P (z) = P_{o} exp (- c z)$ it follows that it denotes the depth where the light intensity has decreased to a fraction exp(-1)=0.36, or to 36% of its value at the surface.

An optically thick medium is many optical depths thick. Applied to the lower edge of a layer between z₁ and z₂ it follows: $z_{2} » 1 / c$ , or $c z_{2} \to \infty$ .

With $z_{1} = 0$ und $c z_{2} \to \infty$ it follows for a homogeneous optically thick medium from the relation above:

P \propto \frac{η}{{(n H)}^{2} c}

This is the integral of the hydrographic lidar equation from the surface down to a depth from which no more signal intensity can be detected. These signals are measured with a laser fluorosensor, i.e., an instrument without time-resolving capability, or a time-resolving lidar following integration of the signal return. Apart from the flight altitude H and other instrumental quantities (which are not considered here and are represented by the proportionality) it includes the relation of signal efficiency versus attenuation coefficient only.