1 Chapter 4 Velocity analysis Introduction • The objective of velocity analysis is to determine the seismic velocities of layers in the subsurface. • Seismic velocities are used in many processing and interpretation stages such as: Ø Spherical divergence correction Ø NMO correction and stacking Ø Interval velocity determination Ø Migration Ø Time to depth conversion. • There are different types of seismic velocities such as: the NMO, stacking, RMS, average, interval (Dix), phase, group, and migration velocities. • The velocities that can be derived reliably from T-X data are the NMO and stacking velocities. • The T-X curve of a single constant-velocity horizontal layer is a perfect hyperbola given by: T2(X) = T2(0) + X2/V2, (4.1) where T(X) is the 2-way traveltime at offset X, T(0) is the 2-way traveltime at zero offset, and V is the layer velocity. • In a series of plane horizontal constant-velocity layers, the exact offset (XN) and two- way traveltime (TN) to the Nth layer are given by the following parametric equations: 2 N pVi H i X N = 2∑ , i =1 1 − ( pVi ) 2 N Hi and TN = 2∑ , i =1 Vi 1 − ( pVi ) 2 where Vi, and Hi are the interval velocity and thickness of the ith layer; and p = dT/dX is the parameter of the ray received at X = XN with T = TN. • However, we can express the exact TN in terms of XN in a Taylor’s series expansion about XN = 0: TN2(X) = C0 + C1 X N2 + C2 XN4 +.... (4.2) • The series (4.2) is an exact representation of the exact T-X curve of the N-th reflector under two conditions: 1. X = 0. 2. Infinite number of terms is used in the series. • Putting X > 0 in equation (4.2) will result in losing some accuracy in the representation of series (4.2) of the exact T-X curve. • Therefore, we can only use this series as long as we keep close to X = 0. A practical measure is X/Z ≤ 1. • If infinite number of terms are used in equation (4.2), then: 2 N 2H i 2 N C 0 = ∑ = ∑ ∆t i = TN (0) 2 , (4.2a) i =1 Vi i =1 N ∑ ∆t i and C1 = Ni =1 = 1 / VRMS N 2 . (4.2b) V 2 ∆t ∑ i i i =1 3 where ∆ti is the interval zero-offset two-way traveltime of the i-th layer and VRMSN is the RMS velocity of the N-th reflector. • Since we cannot use infinite number of terms practically, we have to truncate equation (4.2). This will introduce an error, and C1 will deviate from 1/VRMS2. • In general, the less terms left in the series, the greater is the error. • Truncating equation (4.2) to two terms, we get the following hyperbolic relation between T and X: TN2(X) ≈ TN2(0) + X2/VRMSN2, (4.3a) • Note the two conditions above when estimating VRMSN from equation (4.3a). • The RMS velocity can be defined in terms of the true T-X curve as the square root of the reciprocal of the coefficient of the X2 term in the series approximation of the exact T2-X2 curve of multiple layers: 1 VRMS = . (4.3b) C1 • It can also be shown that VRMS is equal to the square root of the reciprocal of the slope of the tangent to the exact T2-X2 curve at X = 0. That is: 1 V RMS = . (4.3c) 2 [dT / dX 2 ] X = 0 • The RMS velocity VRMSN to the N-th reflector can also be defined in terms of the properties of subsurface layers as (see equation (4.2b)) : 1/ 2 N 2 ∑ Vi ∆t i VRMSN = i =1N (4.4) ∑ ∆t i i =1 4 where Vi is the interval velocity and ∆ti is the interval zero-offset two-way traveltime of the i-th layer, and N is the number of layers. • The stacking velocity Vs is found from the T-X data by fitting a best-fit hyperbola to the true (nonhyperbolic) T-X curve, which takes the form: T2(X) = T2(0) + X2/Vs2, (4.5) where T(X) is the 2-way traveltime at offset X and T(0) is the 2-way traveltime at zero offset. • Figure. • Figure. • Note that when fitting a hyperbola to the true T-X curve, we always choose the constant term to be equal to T2(0) because it is a known condition and must be satisfied. Therefore, this is a constrained fit. • The normal moveout (NMO) is the time difference between traveltime at a given offset and zero offset. At small offsets, the NMO is approximated by: ∆TNMO(X) ≈ X2/[2VNMO2 T(0)]. (4.6) • The NMO velocity VNMO is found from the T-X data by searching for the velocity that will best NMO-correct a certain reflection (i.e., makes it perfectly horizontal). Figure. • The stacking and NMO velocities cannot be related directly to the properties of subsurface layers. • However, at small offsets (offset ≤ depth), the NMO and stacking velocities are approximately equal to the RMS velocity. 5 • Since VRMS is related to layers properties (equation 4.4), the equivalence of the RMS, NMO, and stacking velocities at small offsets can be used to relate VS and VNMO to layer properties directly through Dix formula defined next. • The interval (Dix) velocity (VN) of the N-th layer can be calculated from the RMS (or stacking and NMO at small offsets) velocities as follows: 1/2 V RMSN 2 T ( 0 ) N − V RMSN −1 2 T ( 0 ) N −1 VN = (4.7) T ( 0 ) N − T ( 0 ) N −1 where VRMSN-1 and VRMSN are the RMS velocities to the top and bottom of the N-th layer, and T(0)N-1 and T(0)N are the zero-offset traveltimes to the top and bottom of the N-th layer. • The NMO and stacking velocities are generally not equal because they are calculated using different unrelated methods. However, in practice, they are considered equal if only small offsets are used. • See this Excel Sheet for the effects of truncation and offset on VRMS calculation. Velocity determination in practice (1) The T2-X2 method • If we approximate the true T-X curve by a hyperbola of the form given by equation (4.5), then a plot of T2 versus X2 will give a line whose slope and intercept are 1/Vs2 and T2(0), respectively. • Hence, we can use this equation to find the stacking velocity Vs from the slope of the best-fit line to the true T-X curve. This is called the T2-X2 method. 6 • The T2-X2 method is not practical to use for common seismic exploration datasets because it needs picking of T(X) at every offset, which has the following problems: 1. It is time consuming if done by humans because of the huge datasets commonly encountered in seismic exploration. 2. It is prone to errors if done by machine especially in noisy datasets. • Therefore, it is mainly used with small datasets of relatively high S/N ratio (e. g., experimental or synthetic data). (2) Constant-velocity stacks (CVS) • This method attempts to find the NMO velocity to each reflector. • This method has two variations: (a) The first variation consists of the following steps: 1. A selected CMP gather is repeatedly NMO-corrected using a range of constant velocity values. 2. The NMO-corrected gathers are displayed beside each other in panels. 3. Following a certain event, as it is NMO-corrected using different velocities; we choose the velocity that best flattens (i.e., makes it perfectly horizontal) the event as VNMO of that event. 4. Proceeding in this way for the other events of interest in the CMP gather, we can build up a velocity function that is appropriate for the NMO correction of this CMP gather. 5. Choose another CMP gather and repeat steps 1-4. 6. Interpolate the NMO velocities linearly for CMPs that were not analyzed. 7 (b) The second variation consists of the following steps: 1. A small portion of the line (consisting of several adjacent CMP gathers) is repeatedly NMO-corrected and stacked using a range of constant velocity values. 2. The constant-velocity stacks are displayed beside each other in panels. 3. Following a certain event, as it is stacked using different velocities; we choose the velocity that produces the most laterally continuous stack of the event as VNMO of that event. 4. Proceeding in this way for the other events of interest in the panel, we can build up a velocity function that is appropriate for this portion of the line. 5. Choose another portion with different (or overlapping) CMP gathers and repeat steps 1-4. 6. Interpolate the NMO velocities linearly for portions that were not analyzed. • The velocities found using variation (b) are often called stacking velocities (Vs) because a stacked section is used. • Important parameters to consider when using the CVS method are the minimum, maximum, and increment in the trial NMO velocities. • The CVS method is especially useful in areas with complex structures. 8 (3) The velocity spectrum • This method attempts to find the stacking velocity to each reflector. • It maps the T-X data of a single CMP gather onto the velocity-spectrum plane. • In the velocity-spectrum plane, the vertical axis is T0 and the horizontal axis is Vs. • The method consists of the following steps: (1) Select a CMP gather that has a relatively high S/N ratio. The CMP gather should be sorted in offset. (2) Determine the minimum and maximum T0 that you want to analyze. (3) Determine the minimum, maximum, and increment of Vs to be attempted. (4) Determine the gate width, w, around the reference time T0. This is usually equal to the dominant period of the data. (5) Start with the minimum T0 and Vs. (6) Compute T ( X ) = T0 + X 2 / Vs2 , where T0 and Vs are set to T0 and Vs of step (5) 2 and X is the offset of the traces in that CMP gather. (7) The amplitudes in a gate of width w centered about T(X) calculated from step (6) are selected from all the traces in the gather. (8) The average of the amplitudes corresponding to the first time sample of the gate on all traces in the gather is computed and squared. (9) Step (8) is repeated for all the time samples in the gate w. (10) The squared averages are added together to give the stacked energy (Es). (11) Sum up the squared amplitudes of the first sample on every trace. (12) Repeat step (11) for all the other samples in the gate. (13) The sums of the squares are add together to give the un-stacked energy (Eu). 9 (14) Calculate the semblance NE = Es/Eu. (15) Now you have one point on the velocity-spectrum plane, namely (minimum T0, minimum Vs, NE). (16) While fixing T0, increment Vs and repeat steps (6)-(14) until you reach the maximum Vs. (17) Increment T0 by w/2 and repeat steps (5)-(16) until you reach the maximum T0. (18) For an event occurring at a given T0, the corresponding Vs is the one that is associated with the maximum semblance occurring at that T0. (19) Select another CMP gather and repeat steps (5)-(17). • You should end up with a set of picks (T0, Vs) for every selected CMP. • To find the (T0, Vs) sets for the other, unprocessed CMPs, we linearly interpolate them. • Es, Eu, and NE are measures of coherency (similarity) of the signal along a hyperbolic curve. Other measures of signal coherency are often used such as: Ø The stacked amplitude. Ø The normalized stacked amplitude. Ø The un-normalized crosscorrelation. Ø The normalized crosscorrelation. Ø The energy-normalized crosscorrelation. • The coherency measure is usually displayed as: Ø A contour plot. Ø A gated row plot. • The time increment is selected as half the dominant period to avoid: 10 Ø extra costs of processing if we choose less than half, and Ø mixing of closely–spaced reflections if we choose more than half. • Important parameters to consider when using the velocity spectrum method are the minimum, maximum, and increment in the trial stacking velocities. • The velocities used in this method are called often the semblance velocities. • The velocity spectrum method is more suited for noise-contaminated datasets. Factors affecting velocity estimates • Velocity estimation from the velocity spectrum is limited in accuracy and resolution for the following reasons: Ø Spread length: § Adequate resolution in the velocity spectrum can only be attained with spreads that span both near and far offsets. § Using only near offsets degrades the coherency peaks at later times due to the low NMO associated with deep reflections. § Using only far offsets degrade the peaks at shallow times due to the high NMO associated with shallow reflections. Ø Stacking fold: Using very low folds significantly shifts the coherency peaks in the spectrum due to the loss of hyperbolic character of reflections. Ø S/N ratio: The accuracy of the velocity spectrum is limited when the S/N ratio is poor due to the many erroneous peaks generated by aligning random noise. 11 Ø Muting: It decreases the semblance magnitude because of the zeroed (muted) amplitudes. If muting is used, the semblance calculation has to compensate for the effect of muting. Ø Time gate length: Smaller time gates increase the computational costs, while coarse ones reduce the temporal resolution of the spectrum (i.e., distinguishing velocities of closely-spaced reflections). Ø Velocity sampling: The velocity range should be chosen to span the expected stacking velocities of primary reflections across the area (i.e., in time and space). The velocity increment should be chosen fine enough to give the required resolution of the spectrum. Ø Departures from hyperbolic moveout: The moveout can depart from a hyperbola due to anisotropy or lateral heterogeneities in the overburden. Using three-term series fitting might help in picking the stacking velocities.
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