Inversion of the Lidar signal

The name LIDAR originates, analogous to RADAR, from the first application of this technology, the determination of the distance to large solid bodies (laser range finder). The evolution of the LIDAR technique reaches from the measurement of the distance to diffuse targets, for example clouds (cloud ceilometer), to the determination of the range resolved turbidity of the atmosphere (visibility sensor). The modern LIDAR systems have been reduced so far, that they can be built into a van or even a car.

The transmitter emits short pulses of laser light in the order of some nanoseconds. These packages of light will interact with particles of the atmosphere, like water droplets or dust and will be partially scattered back into the receiver with a delay time given by the travelling time: two times the range (to and from the scattering event) with the speed of light. The number of received photons at fixed time intervals gives the range resolved density of scatterers. The detected intensity will be focused on a photo diode, transformed into an electrical current and finally after some amplification converted into an electrical voltage or digital information respectively. This can be described with the single scattering lidar equation:


U(R): digitized signal
R: measuring range
k: system constant (amplification, transmitted energy etc.)
x(R): optical overlap function (field of view of the receiver from the transmitted beam)
b(R): volume back scattering coefficient at range R
t(R)2: two way damping due to turbidity of the atmosphere, i.e. transparency

The transparency is given by:

the back scattering coefficient can be written as:


s(r): the local extinction coefficient
P: back scatter phase function (depends on shape, size, material of the scatterers but maybe also on the range)

The last two equations from above, t(R) and b(R), but more precisely the two components s(r) and P, represent the influence of the medium: the number of scatterers and their properties respectively. The lidar equation can be written in a manner that the influence from the atmosphere is separated from the measured values:

Differentiating the logarithm of the signature S(R) leads to a well known ordinary differential equation of typ Bernoulli or homogeneous Ricatti:

which has the solution:


R0: minimum distance of measurement
s(R0): estimated extinction coefficient at the corresponding minimum distance

This equation shows unfortunately wrong results caused by the difference in the denominator, which may become very small and even negative (noise, slightly wrong estimated extinction). A stable solution of the differential equation can nevertheless be given by changing the limits of the integral. This solution is known as the Klett algorithm (Klett, J.D.: Stable Analytic Inversion Solution for Processing Lidar Returns, Applied Optics 20, 211 1981):


Rm: maximum distance of measurement (noise level)
s(Rm): estimated extinction coefficient at the end of the measurement range

The drawback now is that the extinction coefficient has to be estimated at the end of the measurement path, which is rather a contradiction of remote sensing. Also the signature value at the end of the path S(Rm) must contain some usefull information about the atmosphere, as this value influences all calculated extinction coefficients.

In the case of an fully automatic process of calculating the extinction coefficient this means:

Finally after determining the local extinction coefficients s(R) one can calculate the local visibility or visual range by


e: contrast threshold (2% normal visual range, 5% meteorological visual range)

Click on the picture to update the figures

The top left intensity plot shows the local visual range (colour coded) again as a 24 hours plot. A coloured patch will be plotted only if the visibility is below a system specific threshold, which depends on the sensitivity of the system (given by the output laser energy, the receiver area, the detector efficiency but also on the number of accumulated shots), but also on the distance to the cloud or fog bank, as the intensity of the received signal and therefore the signal-to-noise ratio follows 1/R2.

The plot on the right side shows the actual visual range versus altitude, and the 24 hour plot on the bottom the mean visibility (blue) with error bars (red). Strong fluctuations are are due to fast changes in the atmospheric conditions like travelling cloud patches, strong vertical gradients like for example ground fog, but also due to situations next to the instrument range (far away or very thin layer).

A major difference between a local visibility measurement for example using a transmissometer and a remote sensing system like here is that one must keep an eye on where the turbidity happens (altitude in the above intensity plot), when it happens (time axis). One should also keep in mind, that here the vertical visibilty is measured for the first time, which may be an unusual measure.

The vertical red bar indicates again the actual time mark.

DLR, Institut für Physik der Atmosphäre
Abteilung LIDAR

P.O.B. 1116 - 82230 Weßling
Contact: Jürgen Streicher
Tel.: ++498153 28 1609
Fax: ++498153 28 1608

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