GNGTS – Atti del 19° Convegno Nazionale / 05.05
R. de Franco
Istituto di Ricerca sul Rischio Sismico, Milano
INTERVAL VELOCITY AND THICKNESS ESTIMATE
FROM WIDE ANGLE REFLECTION DATA
A method to estimate interval velocities and thickness in a horizontal isotropic
layered medium from wide-angle reflection travel time curves is presented.
The method is based on a relationship between the squared reflection travel
time differences and the squared offset differences relative to two adjacent reflectors.
The envelope of the squared-time-offset-difference curves, for rays with equal ray
parameter, is a straight line, whose slope is the inverse of the square of the interval
velocity and whose intercept is the square of the interval time.
The method yields velocity and thickness estimates without any knowledge of
the overlying stratification. It may be applicable to wide angle reflection data when
either information for upper crust and/or refraction control on the velocity is not
available. Application to synthetic and real data show that the method, used together
with other methods, allows us define to a reliable 1D-starting model.
The proposed method may be classed among the exact approaches to solve
the interval velocity and thickness in horizontal stratified media. This method is
mainly based on the physical property of offset-time differences (OTD) for reflected
rays, from adjacent discontinuity, with equal ray parameter. The equal ray parameter
adjacent reflection approach was proposed by Claerbout (1978) and GonzalezSerrano and Claerbout (1984) to measure interval velocity, from CMP gathers.
The method is based on a simple consideration of the reflection travel times,
static corrected, relative to two adjacent reflectors. In the following derivation the
common shot gather configuration coordinates are assumed.
I consider a vertical isotropic inhomogeneous medium (planar reflectors) with
constant velocity inside each layer. Beginning from the surface , the thickness and
the velocities of the beds are respectively h1,…,hn and V1,…,Vn. Let us assume that
two adjacent reflection curves are known in a discrete numbers of points respectively
K and L, (Xkn–1, Tkn–1), (Xln, Tln), where the upper index indicates the reflector and the
lower is the data index with k = 1,…,j,…,K and l = 1,…,i,…,L.
For two rays having the same ray parameter (ray with indexes j and i; Fig. 1),
the reflection travel time, Tin, relative to n-th reflector is a function of the n-1-th
reflection travel time, Tjn–1 , the corresponding recording offsets, Xin and Xjn–1, the
interval velocity, Vn , and the thickness, hn ,
Ti n  T jn 1 
2
( X in  X nj 1 ) 2 / 4  hn2
Vn
(1)
Using the offset differences, Xij = Xin – Xjn–1 , and travel time differences, Tij =
– Tjn–1 , between the two adjacent reflection curves, from the equation (3) we
obtain
Tin
GNGTS – Atti del 19° Convegno Nazionale / 05.05
T 
2
ij
X ij2
Vn2
4hn2
 2
Vn
(2)
Then, given a layer with velocity Vn and a thickness hn , the offset-timedifferences, for equal ray parameter reflections, describe an exact equivalent
reflection travel time curve (hyperbola) with offsets Xij and times Tij (Fig. 1).
Fig.1 - (a) Reflection travel times from adjacent reflectors. (b) Ray geometry and model. Thick lines
represent reflected rays having the same ray parameter. Ray tracing and travel times are calculated
for the velocity model shown in Fig. 1b.
It is demonstrated, that the Tlk curves, calculated for any ray parameter, are
simultaneously tangent (Fig. 2) to a one layer reflection curve, relative to the reflected
raypath fractions inside the n-th layer, described by the equation 2. In the equivalent
way, in X2 – T2 domain, the Tlk2 curves are simultaneously tangent to a straight
line (eq. 2). In X2 – T2 domain, the straight line with coefficient 1/Vn2 and intercept
tn2  4hn2 Vn2 , is the common tangent (or envelope, Fig. 2b and Fig. 2c ) to all the Tlk2
curves and it enables the determination of interval velocity and thickness of the n-th
layer.
Given the previous kinematic consideration, in a planar stratified medium, the
starting point for the application of the proposed method is to verify the basic
hypothesis of ray parameter equality and choose the two reflection travel time curve
intervals associated with these rays (Fig. 2a). The plot of Tlk2 curves is drawn using
the two selected travel time curves (Fig. 2b). The straight-line envelope of Tlk2
curves is searched iteratively calculating the minimum slope least square straight line
GNGTS – Atti del 19° Convegno Nazionale / 05.05
interpolating the maxima of each curve (Figs. 2b, 2c). The procedure may be
expressed in the following minimisation form:


min V max l (Tlk2  X lk2 V 2 ) .
(3)
Fig. 2 - (a) Inverse of Ray parameter versus offset curves for the two reflections reported in Fig. 1.
Dots indicate rays with equal ray parameter as in Fig. 1. (b) T2 curves and straight-line envelope. (c)
T2 – X2 /6.42 curve family versus X2 referred to the travel times reported in Fig. 1. The crosses
indicate the points used for envelope estimate (thick line). (d) The same as Fig. 2c considering errors
on the picking of reflection travel times. Dots indicate the points used for the envelope estimate (thick
line).
GNGTS – Atti del 19° Convegno Nazionale / 05.05
Squared interval slowness, squared interval time and relative errors are
obtained calculating the least square straight line maxima best fit, than the interval
velocity and thickness are estimated and relative uncertainties are evaluated
propagating the errors in squared interval slowness and squared interval time.
Fig. 3 - Wide angle data set. (a) Observed travel times (dots). (b) Obtained model and raytracing. The
layer velocities and errors (in km/s) are shown, in parenthesis the solutions obtained by Sain and Kaila
(1994) are also reported. The calculated travel times are superimposed in the Fig. 3a (continuous
line).
Fig. 2d shows the solution considering the previous two adjacent reflection
travel time curves, for the synthetic model in Fig. 1, with errors on the travel time
picks ranging between –0.07 s and 0.07 s. In order to test the method with real data
and to have a comparison with other approaches, I examined the data used by Sain
and Kaila (1994) in the application of their method based on the representation of the
reflection with non-hyperbolic series (Fig. 3a and 3b).
REFERENCES
Claerbout J. F. 1978. How to derive interval velocities using a pencil and straight edge. Stanford
Exploration Project, report 14.
Gonzalez-Serrano A. and Claerbout J.F. 1984. Wave-equation velocity analysis. Geophysics, 49,
1432-1456.
Sain K. and Kaila K.L. 1994. Inversion of wide-angle seismic reflection times with damped least
square. Geophysics, 59, 1735-1744.