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)