Chapter 4. Velocity analysis

Transcription

1 1 Chapter 4 Velocty analyss Introducton The objectve of velocty analyss s to determne the sesmc veloctes of layers n the subsurface. Sesmc veloctes are used n many processng and nterpretaton stages such as: Sphercal dvergence correcton MO correcton and stackng Interval velocty determnaton Mgraton Tme to depth converson. There are dfferent types of sesmc veloctes such as: the MO, stackng, RMS, average, nterval (Dx), phase, group, and mgraton veloctes. The veloctes that can be derved relably from T-X data are the MO, RMS, and stackng veloctes. T-X-derved veloctes The T-X curve of a sngle homogeneous horzontal layer s a perfect hyperbola gven by: T (X) = T (0) + X /V, (4.1) where T(X) s the two-way traveltme at offset X, T(0) s the two-way traveltme at zero offset, and V s the layer velocty.

2 In a seres of plane horzontal homogeneous layers, the exact offset (X) and two-way traveltme (T) to the bottom of the th layer are gven by the followng parametrc equatons: X andt pv H, 1 1 ( pv ) H, 1 V 1 ( pv ) where V, and H are the nterval velocty and thckness of the th layer; and p=dt/dx s the parameter of the ray receved at X wth T = T (.e., the ray that s reflected from the bottom of the th layer and receved at an offset X). However, we can express the exact T n terms of X n a polynomal seres expanson as: T (X) k C k X = C0 + C1 X + C X (4.) k 0 The T-X curve gven by seres (4.) converges to the exact T-X curve of the th reflector under two condtons: (1) Ck X k 0 as k (ths means that Ck X k decreases wth ncreasng k or smply that Ck X k < 1 for k > 0). Ths can be accomplshed practcally by settng Xmax/Z < 1, where Xmax s the maxmum offset used and Z s the th reflector depth (see Fgure 4-A1). () Infnte number of terms s used n the seres (.e., k ). If nfnte number of terms are used n equaton (4.), then (Taner and Kohler, 1969): T H ( X 0) C0 t T 1 V 1 (0), (4.3)

3 3 and C t 1 1 1/ V RMS V t 1. (4.4a) C = ( =1 V t ) 4 ( =1 t )( =1 V t ) = T V 4 4 RMS V t 4( =1 V t ) 4 =1 4T 3 V RMS 8 (4.4b) where t s the nterval zero-offset two-way traveltme across the th layer and VRMS s the RMS velocty to the bottom of the th layer. Equaton (4.4a) shows that the RMS velocty (VRMS) can be defned n terms of the true T-X curve as the square root of the recprocal of the coeffcent of the X term n an nfnte seres expanson of the exact T -X curve of multple layers: 1 V RMS. (4.5) C 1 Equaton (4.4a) shows also that the RMS velocty (VRMS) to the th reflector can be defned n terms of the propertes of subsurface layers as: V RMS V 1 1 t t 1/ (4.6) where V s the nterval velocty and t s the nterval zero-offset two-way traveltme of the th layer, and s the number of layers. The stackng velocty (VS) to the th reflector s found from the exact T-X curve by fttng a best-ft hyperbola: T (X) C0 + C1 X, (4.7) where C0 T (0) and C1 1/VS and the approxmaton s due to fttng only -terms of the polynomal n equaton (4.).

4 4 Fgure 4.1 shows how the best-ft Vs s found n practce. Fgure 4. shows the relaton between VS and VRMS relatve to the true T-X curve. otce that we can ft curves to the true T-X curve n two ways: (1) By frst squarng the true T and X and then fttng equatons 4. or 4.7 to the true T -X curve. () By drectly fttng the square roots of equatons 4. or 4.7 to the true T-X. See the companon Excel Sheet named (RMS-VEL(3L).xls) for the effects of truncaton and offset on VRMS calculaton. Exercse (1): (a) Redo sheet E-Short-L3 usng maxmum offset = * L3 depth and compare error to that we get when usng maxmum offset = L3 depth. (b) Repeat the same analyss for L. Why s the error n L more than L3. (c) What do you expect the error n L1? Why? The normal moveout (MO) s the tme dfference between traveltme at a gven offset and zero offset. At small offsets, the MO s approxmated by: TMO(X) X /[VMO T(0)]. (4.8) The MO velocty (VMO) to the th reflector s found from the true T-X curve by searchng for the velocty that wll flatten the true T-X curve from the th reflector (.e., wll make the true T-X curve perfectly horzontal) usng equaton (4.8). o Fgure 4.3 shows how the best VMO s found n practce. Although the MO and stackng veloctes use hyperbolc T-X equatons, they are generally not equal because they are calculated usng completely dfferent methods. However, n practce, they are consdered equal f only small offsets are used.

5 5 The stackng and MO veloctes cannot be related drectly to the propertes of subsurface layers. Only VRMS s related to layers propertes (equaton 4.6). However, at small offsets (maxmum offset reflector depth), the MO and stackng veloctes are approxmately equal to the RMS velocty. Therefore, the equvalence of the RMS, MO, and stackng veloctes at small offsets can be used to relate the stackng and MO veloctes to layer propertes. The nterval (Dx) velocty (V) of the th layer can be calculated from the RMS (or stackng or MO) veloctes at small offsets as follows: V V RMS T (0) T (0) RMS1 V T (0) where VRMS-1 and VRMS are the RMS veloctes to the top and bottom of the th layer, and T(0)-1 and T(0) are the total zero-offset traveltmes to the top and bottom of the th layer. Exercse (): Derve equaton (4.9) from equaton (4.6). 1 T (0) 1 (4.9) Velocty determnaton n practce (1) The T -X method If we approxmate the true T-X curve by a hyperbola of the form gven by equaton (4.7), then a plot of T versus X wll produce a straght lne whose slope and ntercept are 1/Vs and T (0), respectvely. Hence, we can use ths equaton to fnd the stackng velocty Vs from the slope of the best-ft lne to the true T -X curve. Ths s called the T -X method.

6 6 The T -X method s not practcal to use for common sesmc exploraton datasets because t needs pckng of the TWTT of many reflectors at many offsets on many CDPs, whch has the followng problems: 1. It s tme consumng f done by experenced humans because of the huge datasets commonly encountered n sesmc exploraton. Remember that typcal sesmc surveys nclude: o 1,000s-1,000,000s of CDPs o CDP folds between 60 to 480 traces o About 10 major reflectors to be analyzed for velocty. It s prone to errors f done by computers especally n nosy datasets. Therefore, t s manly used wth small datasets of relatvely hgh S/ rato (e. g., expermental or synthetc data). Fgure 4.4 shows an example. () Constant-velocty stacks (CVS) Ths method attempts to fnd the MO velocty to each reflector. Ths method conssts of the followng steps: 1. A small porton of the lne (consstng of 0-5 adjacent CMP gathers) s repeatedly MO-corrected and stacked usng a range of constant velocty values.. The constant-velocty stacks are dsplayed besde each other n panels wth a panel for each attempted velocty.

7 7 3. Followng a certan reflector across the velocty panels, as t s stacked usng dfferent veloctes; we choose the velocty that produces the most laterally contnuous stack of the reflector as the VMO of that reflector. 4. Proceedng n ths way for the other reflectors of nterest n the panel, we can buld up a velocty functon that s approprate for ths porton of the lne. 5. Choose another porton wth dfferent (or overlappng) CMP gathers and repeat steps Interpolate the MO veloctes for portons that were not analyzed. The veloctes found usng ths method are often called stackng veloctes (Vs) because a stacked secton s used, although the contnuty of reflectons depended on the qualty of the MO correcton OT how these reflectons ftted a hyperbola. Important parameters to consder when usng the CVS method are the mnmum, maxmum, and ncrement n the tral MO veloctes. The CVS method s especally useful n areas wth complex structures (Why?). Fgure 4.5 shows an example of CVS panels wth pcks. Fgure 4.6 shows an example of velocty feld along a -D lne wth nterpolaton between CDPs. (3) The velocty spectrum Ths method attempts to fnd the stackng velocty to each reflector. It maps the T-X data of a sngle CMP gather onto the velocty-spectrum plane. In the velocty-spectrum plane, the vertcal axs s T0 and the horzontal axs s Vs. The method conssts of the followng steps:

8 8 (1) Select a CMP gather that has a relatvely hgh S/ rato. The CMP gather should be sorted n offset. Let M be the fold of ths CMP gather. () Determne the mnmum (usually = 0) and maxmum (usually = record length) T0 that you want to analyze. (3) Determne the mnmum, maxmum, and ncrement of Vs to be attempted. (4) Determne the gate wdth, w, around the reference tme T0. Ths s usually equal to the domnant perod of the data. Let be the number of samples n the gate. w s set by the dtrato parameter of the suvelan command n SU. (5) Start wth the mnmum T0 and Vs. (6) ComputeT ( X ) T0 X / Vs, where T0 and Vs are set to the mnmum T0 and mnmum Vs of step (5) and X s the offset of the traces n that CMP gather. (7) The ampltudes n a gate of wdth w centered about T(X) calculated from step (6) are selected from all the traces n the gather. Let Aj be the ampltude of a tme sample wthn ths gate, where 1 and 1 j M. (8) The sum of the ampltudes correspondng to the frst tme sample of the gate on all traces n the gather s computed and squared M j 1 1 A j. (9) Step (8) s repeated for all the tme samples n the gate w. M E S A j 1 j1 (10) The squared sums are added together to gve the stack energy (11) ow, sum up the squared ampltudes of the frst sample on every trace (1) Repeat step (11) for all the other samples n the gate. M A1 j. j1.

9 9 (13) The sums of the squares are added together to gve the prestack energy M E u A j 1 j1. (14) Calculate the semblance E = (1/M)(Es/Eu). ote that: 0 E 1, and that t s larger f the ampltudes n the gate are algned followng a hyperbola whose T0 and Vs are equal to those of the hyperbola you are currently fttng. (15) ow you have one pont on the velocty-spectrum plane, namely (mnmum T0, mnmum Vs, E). (16) Whle fxng T0, ncrement Vs and repeat steps (6)-(14) untl you reach the maxmum Vs. (17) Increment T0 by L = / samples and repeat steps (5)-(16) untl you reach the maxmum T0. L s set by the nsmooth parameter of the suvelan command n SU (18) For a reflecton that has a zero-offset TWTT=T0, ts correct Vs s the one that s assocated wth the maxmum semblance occurrng at that T0. (19) Select another CMP gather and repeat steps (5)-(17). You should end up wth a set of pcks (T0, Vs) for every selected CMP. To fnd the (T0, Vs) sets for the other, unprocessed CMPs, we nterpolate them. Es, Eu, and E are measures of coherency (smlarty) of the sgnal along a hyperbolc curve. Other measures of sgnal coherency are often used such as: The stacked ampltude. The normalzed stacked ampltude. The un-normalzed crosscorrelaton. The normalzed crosscorrelaton. The energy-normalzed crosscorrelaton.

10 10 The coherency measure s usually dsplayed as: Contour plot (Fgure 4.7). Gated row plot (Fgure 4.8). Important parameters to consder when usng the velocty spectrum method are the mnmum, maxmum, and ncrement Vs. The veloctes used n ths method are often called the semblance veloctes (Why?). The velocty spectrum method s more suted for nose-contamnated datasets (Why?). Fgure 4.9 shows an example of velocty spectrum wth pcks and MO-corrected CMP gather. Fgure 4.10 shows more fgures. Factors affectng velocty estmates Velocty estmaton from the velocty spectrum can be lmted n accuracy and resoluton for the followng reasons: Spread length: Adequate resoluton n the velocty spectrum can only be attaned wth spreads that span both near and far offsets because: Usng only near offsets degrades the coherency peaks for deep reflectons due to the low MO assocated wth deep reflectons at low offsets. Usng only far offsets degrades the peaks for shallow reflectons due to the hgh MO assocated wth shallow reflectons at far offsets. CMP fold: Usng very low CMP fold sgnfcantly shfts the coherency peaks n the spectrum due to the loss of hyperbolc character of reflectons.

11 11 S/ rato: The accuracy of the velocty spectrum s lmted when the S/ rato s poor due to the many erroneous peaks generated by algnng random nose. Mutng: It decreases the energy magntude because of the muted (zeroed) ampltudes. If mutng s used, the energy calculaton has to compensate for the effect of mutng. T0 gate length (/): Smaller T0 gates ncrease the computatonal costs, whle coarse ones reduce the temporal (vertcal) resoluton of the spectrum (.e., dstngushng veloctes of closely-spaced reflectons). Attempted stackng veloctes (VSmn, VSmax, VS): The mnmum and maxmum attempted stackng veloctes should be chosen to span all expected stackng veloctes of prmary reflectons across the area (.e., n tme and space). The velocty ncrement should be chosen fne enough to gve the requred resoluton of the spectrum. Departures from hyperbolc moveout: The moveout can depart from a hyperbola due to ansotropy or lateral heterogenetes n the overburden. Usng three-term (quartc) seres fttng mght help n pckng the veloctes. SU uses the followng quartc equaton: T ( X ) X A X 4 1 T0, (4.10) V 1 A X where A1 and A are small ansotropc parameters to be attempted (A1 = A = 0 n the case of sotropy).