Groundwater Geophysics Phần 2 ppt

40 Wolfgang Rabbel

Fig. 2.11. Ray geometry to be considered to derive the Plus-minus method of

Hagedoorn (1959). The meaning of the “plus-time” t+

(x) can be understood by re￾garding (1) the refracted rays connecting the geophone at x with shot points A and

B and (2) the positions of the refracted wave fronts when they reach the same

point P beneath the geophone. P is the foot point of the interface normal pointing

towards geophone position x. The refracted wave fronts emerging from shot points

A and B arrive at P at the same travel time as they arrive at points QA and QB on

the up-going parts of the respective refracted rays. Since, in addition, P-x-QA and

P-x-QB form congruent rectangular triangles, the following travel time equations

apply to respective segments of the refracted rays: τ2(x,xA)= τ2(x,QA)+τ2(QA,xA),

τ2(x,xB)= τ2(x,QB)+τ2(QB,xB), τ2(xA,xB)=τ2(P,xA)+τ2(P,xB)= τ2(QA,xA)+τ2(QB,xB),

τ2(x,QA)=τ2(x,QB). Therefore, t+

(x) can be identified as t+

(x) = 2⋅τ2(x,QA)

=2⋅τ2(x,QB)

This deviation will cause some inaccuracy in cases were the refracting

interface is not locally plane. To image the interface segment vertically be￾low x, the slopes of the refraction time derivatives would have to be com￾puted at points x±

=x±z⋅tan(iC±δ) for shot points A and B, respectively. Al￾ternatively, the ray diagram (Fig. 2.12) shows that the same range of

interface points will be covered by both shot observations if right and left

hand segments of the respective travel time curves are considered, namely

τ2(x+

,xA) for x ≤ x+

≤ (x+s+

) and τ2(x-

,xB) for (x-s-

)≤ x-

≤ x where:

= 2 ⋅ ⋅ cos( ± 2 ⋅δ )/ cos( ± δ ) ±

C C s r i i (2.13)

The velocity v2(x) of the bottom layer can be computed from averaging the

slopes of these curve segments.

2 Seismic methods 41

Fig. 2.12. Horizontal shift of segments of reversed refraction travel time curves

belonging to the same of interface segment (cf. Eqs. 2.12 and 2.13)

2.2.4 Consistency criteria of seismic refraction measurements

The above consideration of the two-layer case shows that there are a

number of criteria of field layout and travel time analysis needed for inter￾pretation of seismic refraction measurements. These requirements listed

below are valid and equally important also for multi-layer situations with

lateral velocity heterogeneity and more complicated interfaces:

1) The seismic profile has to be covered with multiple source points

and continuous receiver lines so it is possible to observe or construct

reversed branches of refracted arrivals (Fig. 2.13). Optimum is a com￾bination of overlapping and reverse shots.

2) The seismic velocity of refracting layers can be determined by av￾eraging the slopes of reversed corresponding travel time branches.

This procedure can be applied not only to plane interfaces but also to

smoothly curved interfaces if the reversed branches apply to the same

underground segment.

3) The consistency of travel time observations from different shot

points has to be checked by application of the principle of reciprocity

42 Wolfgang Rabbel

to each type of arrival [τj(xP,x0)=τj(x0,xP)]. Since the principle of recip￾rocity applies in a strict sense, field layout should provide adequate

shot-geophone arrangements so the interpreter can take advantage

from it.

4) The intercept times of refracted arrivals observed for left and right

hand spread from the same shot point agree if the refracting interface

is plane. This criterion of travel time consistency may be violated in

case of curved or disrupted interfaces (Fig. 2.14).

5) Overlapping refracted travel time branches of the same interface

should appear parallel. This criterion can be applied for consistency

checks and for combining observations from different source points

into one long travel time branch (Fig. 2.15). This latter procedure is

required for some interpretation algorithms such the wave front

method (see below). Note that this “parallelism of refraction travel

time branches” applies only to plane homogeneous layers in a strict

sense. Deviations may occur in case of curved interfaces and in case

that the velocity increases with depth inside the refracting layer.

Fig. 2.13. Examples of typical overlapping acquisition schemes for seismic refrac￾tion measurements. Top: one-sided spread (end-on spread); middle and bottom:

split spread

2 Seismic methods 43

Fig. 2.14. Intercept times for plane and disrupted refracting horizons

Fig. 2.15. Construction of a long refracted travel time branch (dashed line) from

overlapping observations (solid lines). The underlying assumption of the “parallel￾ism of travel time branches” (top figure) applies to smooth interfaces and negligible

velocity increase inside the refracting layer. In this case refracted waves of adjacent

source points travel along the same path (bottom figure)

44 Wolfgang Rabbel

2.2.5 Field layout of seismic refraction measurements

In order to perform seismic field measurements the geometrical ar￾rangement of source and geophone points has to be defined. This involves

the specification of

- the length of the geophone spread for each source point,

- the geophone spacing along the spread,

- the spacing of shot points, and

- the overlap of geophone spreads.

The particular choice depends mainly on the features of the geological

structure to be investigated, namely:

- the seismic velocity of the overburden,

- the depth and seismic velocity of the refracting target layer, and

- the heterogeneity of the these properties.

The above considerations and formulae of the two-layer case provide

the means to estimate the requested properties. They can be applied for

multi-layer cases, too, if the upper layer is regarded as sort of an average of

the hanging wall of the target layer.

The remarks below apply to the common situation that near surface

seismic refraction measurements are performed by deploying a large num￾ber of equally spaced geophones along spreads compared to which the

number and spacing of shot points are more sparse or wide, respectively.

This assumption is clearly somewhat arbitrary because the role of shot and

receiver points may be exchanged, which is a common situation for seis￾mic measurements at the sea bottom, for example. For sea bottom meas￾urements a number of seismometers is deployed and shots are fired at high

rates from a moving boat.

Spread length. Refracted arrivals can be identified most securely be￾yond the crossover distance xK where they appear as first breaks. In prac￾tice, as a rule of thumb, the slope or the corresponding apparent velocity of

a refracted arrival can be determined if it is observed over a distance of one

wave length or longer. For impulse type signals a wave length λ can be de￾fined via:

λ = v ⋅T (2.14)

where v is the propagation velocity and T is the apparent period or time

duration of one oscillation cycle (extending over one positive plus one

negative deflection). From a ray geometrical point of view the spread has

to be long enough so the same underground segments can be imaged by

shot and reverse shot observations. Applying Eqs. 2.12 or 2.13 leads to the

conclusion that the spread S should extend longer than: