92 
The only parameter left is ha- However, if we substi 
tute <jg, r, and ta from Eqs. (7), (9), and (10) into 
Eq. (2) and assume that at f = ip the amplitude of the 
surface peak is yEMo(ip)i and hence ha can be calcu 
lated. Strictly speaking, the amplitude of the surface 
peak may be modified by the bottom reflection and, 
therefore, it cannot truly represent the function value 
Uemg(1 p)- Nevertheless, our requirement at this point 
is to obtain a reasonable initial estimate of the function 
parameters so as to initialize the optimization proce 
dure discussed in the next section. At that time, the 
function parameters are optimized to yield satisfactory 
results. Equations for the computation of the EMG pa 
rameters using other values of a are also available in 
(Foley and Dorsey, 1983). 
3.2 Optimization of Parameters 
A smoothed waveform can be decomposed into two sig 
nal components representing the surface and bottom 
reflections by fitting the waveform to a “mathemati 
cal model” which involves a number of adjustable pa 
rameters. The model in our case is given by Eq. (5). 
The basic approach is to construct an objective func 
tion that will measure the difference between the data 
and yr(t) for a particular selection of the seven parame 
ters of ya(t) and ypA/c(f ) hi Eq. (5). The parameters of 
yr(t) are then adjusted to minimize the objective func 
tion, thereby yielding the parameter values that corre 
spond to the best fit. The adjustment process is thus a 
problem in minimization in many dimensions. 
Let E be the objective function and x be the vector of 
the parameters of yp(i). E is a function dependent on 
the parameters x and the independent variable t of the 
form 
E = y{e(x, tj), e(x, t 2 ), • • •, e(x, f m )} (11) 
where t 1 ,t 2 , • • •, t m are values of t at the m sample 
points. The residuals e(x, tk) are given by 
e(x, t k ) = yr(x, tk) - y(t k ) k = 1, 2, • • •, m(12) 
where y(tk) represents the smoothed waveform. Our 
objective is to minimize E by varying the elements of 
x. 
The objective function E can assume several forms. In 
the problem at hand, the sum of squares of the residuals 
e(x, tk), namely 
m 
E =Y1 e2 ( x ’ *k) 
k= 1 
was found to give good results. 
The strategy used to minimize E is referred to as the 
Levenberg-Marquardt method and was proposed by Lev- 
enberg (Levenberg, 1944) and enhanced by Marquardt 
(Marquardt 1963). This method takes advantage of 
the fast convergence of the steepest-descent method far 
from the minimum point and the fast convergence of 
the Newton-Raphson method as the minimum point is 
approached. In this method, a correction Ax in x is 
computed as 
Ax = -[AI + J r J]- 1 J T e (13) 
’ #ei 
de i 
dei 
dx\ 
dx, 
dx n 
deic 
det 
dek 
dx\ 
dx, 
dx n 
de m 
. de m 
. de m 
- dx\ 
dxi 
dxn 
e = [ei(x) • • • e*(x) • • • e m (x)] T 
and I is the n x n identity matrix. The scalar A can 
be adjusted to control the sequence of iterations. For 
a sufficiently large value of A, the matrix AI + J T J in 
Eq. (13) is positive definite and hence a reduction in E 
can be assured even when x is far from the minimum 
point. On the other hand, when A —► 0 Eq. (13) tends to 
the standard Gauss-Newton step and, as a result, the 
convergence is rapid when x is close to the minimum 
point. 
A minimization algorithm based on the above principles 
due to Marquardt is as follows: 
1. Compute E(x). 
2. Set A to a small positive value (say 0.001). 
3. Solve Eq. (13) for Ax and evaluate E(x + Ax). 
4. If E(x -f Ax) > E(x), increase A by a factor v 
(say 10) and go to step 3. 
5. If E(x + Ax) < E(x), update the trial solution 
x <— x T Ax. 
6. If \Axj\/(/3 + |Xj|) < e, where j = 1, 2, ..., n, 
output x and stop. 
Else, decrease A by a factor v and go to step 3. 
Suitable values for e and /3 are 10 -5 and 10~ 3 , respec 
tively. 
3.3 Results and Discussions 
Some typical results obtained using the Levenberg- 
Marquardt minimization method are now described. Fig 
ure 7(a) shows a situation where the water column has 
nonuniform turbidity. The backscattered envelope is 
distorted by several spurious peaks which cannot be 
fully eliminated by smoothing. In addition, the l’eflected 
pulse from the sea bottom is weak and broadened as 
shown in the figure owing to beam dispersion during 
the transmission of the laser pulse in water. Figure 7(b) 
illustrates the case where the surface and bottom reflec 
tions strongly overlap and merge into a single peak in 
the received waveform. The optimization algorithm has 
resolved the waveform into two reflections and the sea 
depth can, therefore, be easily determined from the peak 
positions of these two reflections. 
The use of optimization techniques in the estimation of 
sea depth offers two major advantages. First, the over 
lapping surface and bottom reflections in the waveform 
are mathematically resolved into two separate compo 
nents. Therefore, both the detection and resolution 
problems can be solved simultaneously. Since the 
method is insensitive to changes in the degree of overlap, 
reliable depth estimates can be obtained for a diverse 
range of circumstances. 
On the other hand, the optimized EMG parameters, 
which reflect the physical structure of the backscattered 
signal, are also useful in investigating the optical prop 
erties of the sea. Recent studies (Lee and O’Neill, 1984; 
Muirhead, 1986) have shown that the physical structure 
of the received laser waveforms, especially the backscat 
tered envelopes within the waveforms, are related to the 
scattering and absorption of the laser beam in water. 
For example, the amplitude of the backscattered signal 
is indicative of the degree of scattering while its decay 
with time is related to absorption. With the knowl 
edge of the optical parameters of the ocean, the oper 
ating limits of the LIDAR system can be characterized 
and adjustments can be made for maximizing its per