RMS Velocity

If the homogeneous medium is replaced by a stack of horizontal layers then the traveltime equation can be replaced by a series expansion in the offset coordinate:

 

t²(x) = C₀+C₁x²+C₂x⁴+...   , (1.4)

where C₀=t(0)², C₁=1/v_RMSN², and the other coefficients are complicated terms. Here, the RMS velocity v_RMSN for the stack of N layers is defined by

 

$$\cong \frac{\sum_i^{N} v_i^2 \Delta t_i}{\sum_{i=1}^{N} \Delta t_i} ,$$ v_RMSN² (1.5)

where $\Delta t_i$ is the vertical 2-way time through the ith layer, and t(0) in equation 1.3 becomes equal to $\sum_{k=1}^{N} \Delta t_i$ the 2-way vertical traveltime to the Nth layer. Note that v_RMSN depends on the the number of layers N, so we say that the Nth reflector has a different v_RMSN value than, say, the N-4 reflector.

Plugging equation 1.5 into 1.4 and truncating after 2 terms yields the traveltime equation for short offsets and a multilayered media:

 

$$\sqrt{t(0)^2 + x^2/v^2_{RMS_N} }.$$ t(x) = (1.6)

The approximation in the above equation is valid for small values of offset x(Yilmaz, 1987). Once the v_RMSN velocity is found for each reflector then the interval velocities v_n can be found by the Dix formula:

$$\frac{ v^2_{RMS_n}t(0)_n-v^2_{RMS_{n-1}}t(0)_{n-1}} {t(0)_n - t(0)_{n-1} },$$ V²_n = (1.7)

where t(0)_n denotes the vertical reflection traveltime to the nth layer. The validity of this formula is easily proven by using the definition of v_RMSNand simple algebraic manipulation.

In summary we have learned some new words.
1. STATION: Recording site connected to a group of geophones. Data from this site is transmitted along one channel in recording line.
2. GEOPHONE Group: Geophones serially connected, and summed signal is fed to one channel.
3. SPREAD LENGTH: Length of recording line.
4. ZO SECTION: Seismic traces obtained by recording seismic energy with a geophone that has zero offset from the source.
5. CSG: Common shot gather of traces recorded with a common shot.
6. CMG: Common midpoint gather of traces where each of the traces have the same midpoint position between the shot and receiver.
7. SHOT AND GROUP INTERVALS: Intervals between shot points and stations.
8. APPARENT VELOCITY: Velocity of an arrival as measured along a horizontal line.
9. ENDLINE AND MIDPOINT RECORDING GEOMETRIES: Shot at end of recording line = endline. Shot at midpoint of recording line = midpoint.
10. The fold number is the number of traces in a CMG. It is the number of traces in a gather with a common reflection point.