which determines the shape of the function. 
By using T and S T in Eq. (2), we can rewrite the EMG 
function as 
VEMG{i) = • S T ■ e ( s ^ 2_r ) • f e —— d£ (4) 
J—oo \J2"K 
where 
and 
z = T — S T 
— + Sr- T 
°G 
The integral I in Eq. (4) can be approximated by a 
polynomial expression (Abramowitz and Stegun, 1964) 
r — j A ( z ) • * < 0 
\ 1 — A(z) ■ B(q) z > 0 
where 
- wr' n 
B(q) = j2b.q< 
1 = 1 
1_ 
q i + pkl 
and p, bi, . .., b 5 are constants given in Table 1. The 
method used here for the evaluation of the EMG func 
tion has been found to be accurate and reliable in de 
scribing the actual field data. 
Table 1: Constants in the polynomial approximation for 
I in Eq. (4). 
p = 
0.231641900 
6 3 — 
1.781477937 
&1 = 
0.319381530 
b 4 = 
-1.821255978 
6 2 = 
-0.356563782 
b 5 = 
1.330274429 
If we want to determine the shape of the EMG function, 
it is important to use the ratio S r in Eq. (3) rather 
than the absolute values of a G and r. Figure 4 shows 
the shape of different EMG functions with various S T 
values. As can be seen from the figure, a decrease in S T 
causes an increase in the asymmetry of the pulse. On 
the other hand, when S T becomes very large, the EMG 
function will retain the shape of the Gaussian function. 
2.3 Analysis of Simulated Waveforms 
We now wish to simulate the LARSEN waveforms rep 
resented by the function y p (t) which is formed by over 
lapping the EMG curve in Eq. (4) with the Gaussian 
curve in Eq. (1), namely 
yr(t) = VG(t) + y EMail) (5) 
Since S T determines the shape of the EMG function, we 
would like to investigate the effect of S T on the shape of 
yj(t). As can be seen in Fig. 5, at sufficiently small val 
ues of S T , the two peaks can no longer be distinguished. 
In this respect, a decrease in S T has a similar effect as 
a decrease in peak separation. For a constant S T , the 
loss of resolution between two peaks is most pronounced 
when the trailing peak is relatively weak. On the other 
hand, loss of resolution between two peaks may occur 
in the case of a stronger trailing peak. At smaller peak 
separations, the component peaks may overlap to such 
an extent that two peaks are fused into one. 
3 DECOMPOSITION OF LARSEN WAVEFORMS 
In this section, we use a nonlinear least-squares opti 
mization technique to facilitate the decomposition of 
each LARSEN waveform into the surface and bottom 
reflections. To do this, we require initial estimates of 
the model parameters of yr(t) in Eq. (5) in order to ini 
tialize the optimization process. By using the scheme 
presented below, good initial estimates can be obtained 
which assure the rapid convergence of the optimization 
process. 
3.1 Initial Estimation of Parameters 
First we consider the initialization of the three param 
eters of the Gaussian function, and then we describe a 
graphical method to initialize the four parameters of the 
EMG function. 
If the bottom peak in the LARSEN waveform is not em 
bedded in noise or fused into one with the surface peak, 
we can identify the bottom peak and obtain an accu 
rate estimate of its position using a digital narrowband 
differentiator (Wong and Antoniou, 1990). By using 
this estimate together with a simple numerical proce 
dure, the parameters of the Gaussian function can be 
estimated directly from the waveform. However, if the 
bottom peak cannot be identified using the digital differ 
entiator, the three Gaussian parameters obtained from 
the results of the previous waveform can be used as ini 
tial estimates for the current waveform. These estimates 
are fairly accurate since there is, usually, a high degree 
of spatial correlation among neighboring depths. 
In order to evaluate the four EMG parameters ho, t G , 
<tg, and t in Eq. (2) from the waveform, a method in 
volving the use of graphically measurable parameters of 
the EMG function can be employed. In this approach, 
the four measurements A a , B a , W a and t P illustrated 
in Fig. 6 are required. Parameter a is a fraction of the 
peak height. W a is computed as t B — t A , and A a and 
B a are computed as tp — t A and t B — t P , respectively. 
With a specified a, it is possible to determine the EMG 
parameters by calculating the second central moment 
¡i2 of the EMG function (Foley and Dorsey, 1983). For 
a = 0.1, we have 
Wl(BJA a + 1.25) 
4i7 (6) 
With W a and B a /A a known, the parameter a G is eval 
uated as 
W a 
a ° = 3.27(B a /A a ) + 1.2 (?) 
Parameter r is related to and cr G as 
V2 = <t g + r 2 (8) 
Once p 2 an d cr G are determined from Eqs. (6) and (7), 
r can be obtained from Eq. (8) as 
T = \]n2 - °G (9) 
Given t P , t G can be determined from cr G and B a /A a as 
t G =t P - <7 G [-0.193(B Q /A a ) 2 + 1.16'2(B a /A a ) - 0.545] 
(10)