90
tions from the sea bottom (Phillips and Koerber, 1984),
whereas scattering of particles in water lead to spatial
spreading of the laser beam. These effects can influence
the determination of the ocean depth significantly. For
example, when the sea is deep, the bottom reflection is
usually weak and in some cases, it is also broadened sig
nificantly due to the dispersion of the laser beam in wa
ter. Under these circumstances, the bottom reflection
may be embedded in the tail of the surface reflection
and the separation between the two reflections is lost.
As mentioned, each waveform consists of signal reflec
tions including a surface echo, a volumetric backscatter
from the water, and a weak bottom return. When the
blue-green laser pulse travels in a water column of uni
form turbidity, the backscattered energy decays expo
nentially with increasing depth (Phillips and Koerber,
1984). In contrast, a nonuniform turbidity of the wa
ter column will cause distorted backscatter envelopes
with several spurious peaks. When the sea water is of
moderate depth and the bottom reflectivity is weak, the
bottom reflection may be embedded in the backscatter
envelope. Figure 3 is an example of this type of wave
form. We see that the separation between the surface
and bottom reflections is lost.
The separation between the surface and bottom reflec
tions may also be lost even when the bottom reflection
is very strong. For example, if the sea is shallow, the
strong bottom reflection lies close to the surface reflec
tion. In such a case, the two reflections may merge into
one and the depth information may be lost.
In order to improve the accuracy in sea depth estima
tion, we developed the mathematical model presented in
Section 2 to characterize the surface and bottom reflec
tions in terms of mathematical functions. This model
is then used along with the optimization technique de
scribed in Section 3 to decompose LARSEN waveforms
into two signal components which represent the surface
and bottom reflections. This technique yields accurate
depth estimates independently of the degree of their
overlap.
2 CHARACTERIZATION OF LARSEN
WAVEFORMS
In airborne laser bathymetry, the reflections of the laser
pulse from the sea can be analyzed quantitatively by
representing the received waveforms by mathematical
functions. The purpose here is to reduce a complicated
process which depends on many parameters to a sim
pler one involving a small number of parameters. This
data reduction leads to the required information quickly
and, further, if the parameters of the functions turn out
to be physically meaningful, then by understanding the
influence of each parameter, one can gain insight into
the behavior of the process.
The raw LARSEN waveforms are first preprocessed by
digital filtering to remove noise (Wong and Antoniou,
1990). Specially selected mathematical functions are
then used to characterize the preprocessed waveforms.
In this analysis, we assume that atmospheric effects on
the laser pulse are negligible. This assumption is valid
since the atmospheric temporal dispersion of the pulse
is small and its intensity is only slightly reduced when
compared to that of the transmitted pulse (Hoge and
Swift, 1983).
sea state, etc. as was pointed out earlier. Although
it cannot be easily characterized with high precision,
it tends to have the functional form of the Gaussian
probability density function
ya(t) = A max ■ e- (t " w)2/2 " 2 (1)
where
Amax = maximum amplitude of the Gaussian
function
tmax — position at which the maximum amplitude
occurs
a = standard deviation
This function depends on three parameters which can be
adjusted to match the Gaussian function to the bottom
reflection. Parameter o is particularly useful since it
conveniently describes the different possible pulse widths
of the bottom reflection. The results obtained verify
that the Gaussian function is indeed reasonable in de
scribing the bottom reflection mathematically.
2.2 Characterization of Surface Reflection
Assuming that the blue-green laser pulse travels in a
water column of uniform turbidity, the backscattered
energy decays exponentially with increasing depth. As
a result, the surface reflection is skewed and a more
general shape model is necessary to permit its descrip
tion. We have chosen the exponentially modified Gaus
sian (EMG) function to characterize surface reflections
because this function can yield a large variety of asym
metric profiles and, as will be demonstrated, it can give
good results.
The EMG function is obtained via the convolution of the
standard Gaussian function and an exponential decay
function and is given by
Vemg(*) = fi(t) * /2(0
/i(i) = /i G e- (t - iG)2/2<7 °
is the Gaussian function
m = -e-‘^U(t)
T
is the exponential decay function and U(t) is the unit-
step function
U(t) =
1 t > 0
0 otherwise
The convolution of f\(t) and /2(t) is the integral
VeuoW = — ' r e-' 7 ' • • if (2)
T Jo
Equation (2) shows that the EMG function Uemg^) de
pends on four parameters: the function amplitude (^g)?
the time of maximum amplitude (to), the standard de
viation (<7g) of the parent Gaussian function, and the
time constant (r) of the exponential decay function.
To reduce the complexity of the expression, we normal
ize the function by introducing the variable
j, _ (i ~ *g)
&G
2.1 Characterization of Bottom Reflection
The bottom reflection is affected by a number of major
factors such as sea turbidity, bottom reflectivity, and
which measures the time t in units of the standard de
viation gq and defines the ratio
5,
og_
T
(3)