Nonlinear approach to spectral ratio method for estimation of seismic quality factor from VSP data

Accepted Manuscript Nonlinear approach to spectral ratio method for estimation of seismic quality factor from VSP data

Pardeep Sangwan, Dinesh Kumar, Subrata Chakraborty, Vidya Mundayat, M.K. Balasubramaniam PII: DOI: Reference:

S0926-9851(18)30452-X https://doi.org/10.1016/j.jappgeo.2019.04.001 APPGEO 3742

To appear in:

Journal of Applied Geophysics

Received date: Revised date: Accepted date:

25 May 2018 27 November 2018 1 April 2019

Please cite this article as: P. Sangwan, D. Kumar, S. Chakraborty, et al., Nonlinear approach to spectral ratio method for estimation of seismic quality factor from VSP data, Journal of Applied Geophysics, https://doi.org/10.1016/j.jappgeo.2019.04.001

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ACCEPTED MANUSCRIPT Nonlinear Approach to Spectral Ratio Method for Estimation of Seismic Quality Factor from VSP Data

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Pardeep Sangwan1,* [email protected], Dinesh Kumar2, Subrata Chakraborty1, Vidya Mundayat1, M K Balasubramaniam1 1 Reliance Industries Ltd., Mumbai India 2 Department of Geophysics, Kurukshetra University, Kurukshetra, India * Corresponding author.

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ABSTRACT

Conventional spectral ratio method for estimating Seismic Quality factor (Q) involves log-linearization

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and least square fitting, which is very sensitive to noise. In this work we have proposed a new approach to spectral ratio method, where the seismic quality factor is estimated by non-linear inversion of spectral

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ratios using Levenberg-Marquardt (LM) method. The efficacy of the proposed methodology has been tested on synthetic and real VSP datasets. Both conventional log-spectral ratio method (LSR)and

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nonlinear optimization method (LM) could effectively estimate the Q model for datasets having good signal to noise ratio. Numerical tests indicated that with increase in noise floor LM method provided more accurate and stable results. Application on real VSP dataset having high S/N ratio revealed that the

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estimated Q models using both the approaches are almost similar. Results demonstrate a reasonable

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correlation of Q model in sand-shale sequence and low Q values around 69 in gas reservoir.

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INTRODUCTION

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KEYWORDS: Nonlinear Inversion, Quality factor, Scattering, Spectral Ratio Method, VSP.

Seismic waves travelling through a dissipative medium undergoes attenuation and dispersion,

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which are attributed to frictional losses or scattering etc. Random heterogeneity as observed in weathered basalts leads to scattering attenuation, while the intrinsic attenuation is a typical property of the medium. It is observed by many researchers (Biot, 1956a, b;Mavko and Nur 1975;White, 1975;Winkler and Nur 1982;Mavko et al. 1998, Jhonson 2001,Pride et al. 2003, 2004) that the petro-physical properties viz. lithology, porosity, type of fluid, saturation, permeability etc. influences the attenuation. Various Q models describing the physical mechanisms of attenuation have proposed the frequency dependence of Q (Toverud et al. 2005). However, the constant Q model (Kjartansson1979) appropriately explains the frequency independent behavior in exploration seismic bandwidth for Q>10(Sun et al. 2009).

ACCEPTED MANUSCRIPT The eventual loss of seismic resolution and shadow zones below highly absorptive zones such as gas clouds is a subject of wide research in seismic till date. Moreover in an absorptive medium AVO analysis (Luh, 1993; Chapman et al., 2006) is adversely affected. After compensating through inverse Q filtering (Hargreaves and Calvert, 1991; Wang 2002, 2006) or during Q migration (Mittet et al. 1995, Wang 2008) remarkable uplift is observed in seismic

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resolution, which enabled more reliable structural interpretation (Kaderali et al., 2007)and AVO analysis

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(Toksöz et al., 1979; Frisillo and Stewart, 1980; Luh, 1993).Both of these techniques of absorption

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compensation require reliable estimates of Q (attenuation) model of the area.

Estimation of quality factor has been attempted (Sams et al., 1997) from various datasets like VSP, surface seismic, sonic log, laboratory tests on cores etc. However VSP data is most reliable for its

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in-situ attenuation estimation owing to geometry of raypath, isolation of target zone and good signal/noise

using VSP data through a sand-shale sequence.

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ratio etc. (Dasupta et al. 1998). In this study we have performed estimation of in-situ seismic attenuation

The published work on comparison of different Q estimation methods (Newman 1982, Jannsen et

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al. 1985,Tonn 1989, 1991, Senlin Yang, 2009, Cheng and Margrave 2013) concludes that Analytical

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signal (maximum method) yields the best results; while the phase modelling and amplitude decay methods are worst. However, in case of datasets having poor signal to noise ration one of the methods is reliable (Tonn 1991). Frequency shift methods suggested by Quan and Harris (1997), Zhang and Ulrych

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(2002) are claimed to be more robust than the traditional spectral ratio methods; however not stable in

the industry.

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noisy case (Wang 2014). Nonetheless Spectral ratio and Frequency shift methods are most widely used in

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Classical log- spectral ratio method involves inversion of log spectral ratios with a straight line fit to solve out for Q in the zone of investigation. Spectral Ratio method is reliable in datasets with good signal/noise ratio (Tonn, 1991). However the spectral information is quite noisy in real data, the log transformation in case of noisy data further diverge the estimation from the true decay. Log-Linear solution is very sensitive to noise, as its log transformation amplifies the noise effects (Otto et.al. 2011).Also noted by Blias (2012) that for a small change in high frequency part of the spectrum, the inversion of log spectral ratios results in erroneous Q values. To avoid these errors, we inverted the spectral ratios non-linearly using the Levenberg Marquardt method. Linear least square inversion of logspectral ratios was also performed. The results for both the inversion methods on VSP data with different noise levels are discussed.

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THEORY / METHODLOGY Attenuation is described by quality factor Q (Knopoff 1958) as in Equation 1; which is the loss of

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= − 2πE

(1) 0

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Q

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1

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energy - ΔE per cycle.

For a constant Q model, the exponential decay of a downgoing wave after time t1 at first reflector can be geometrical spreading and I is receiver response. πft1 Q

)

(2)

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A(f, t 1 ) = S(f) R 1 G(t1 ) I(f) exp (−

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expressed in the form of Equation2, where S represents source spectrum, R1 is reflectivity and G is

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After time t2 at second reflector with reflectivity R2 the decay is represented as in equation 3

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A(f, t 2 ) = S(f)(1 − R 1 ) R 2 G(t 2 ) I(f) exp(−

πft2 Q

)

(3)

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The reflectivity is assumed as white and the response of source and receiver are considered to be time invariant. In conventional spectral ratio method (Bath,1974) spectra of wavefield recorded at two

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different depth levels is divided to obtain the Q. Dividing the Eq. 3 by Eq. 2, the equation of spectral ratio method thus becomes

= SR(f) = Cnq exp (−

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A

A(f,t1 )

πfΔt Q

)

(4)

where SR(f) is the spectral ratio of the target and reference wavelet; Δt(i.e. t2-t1) is the time to reach from the 1st reflector to second reflector; Q represents the quality factor of the medium between the two depth levels; while Cnq is constant which takes into account factors other than Q viz. the reflectivity ratio, geometrical spreading, source & receiver response etc. After linearization using logarithmic transformation the Eq. 4 becomes a straight line (Eq. 5).The quality factor (Q) is then obtained from the slope of line fitted (in least square sense) on log-spectral ratios.

ACCEPTED MANUSCRIPT log (SR(f)) = log (Cnq ) −

πfΔt

(5)

Q

However in case of noisy data, non-linear fitting of exponential decay curve is a better approach than loglinear solution (Menke, 1984).The non-linear inverse problem of Eq.4 can be represented as in equation (6), where m is the model parameters Cnq and Q; dobs is the observed data SR(f) i.e. spectral ratios. dobs = f(m)

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(6)

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The objective function to be minimized for the above non-linear inverse problem can be formulated as obs φ = ∑M − Ykest )2 k=1(Yk

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(7)

where Ykobs is the observed spectral ratio, Ykest is the estimated ratios and M is the number of observations

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in the selected frequency bandwidth. The equation 7 can be written in a matrix form and can be solved

(JT J + λ Diag (JT J)) ΔX = JT ΔY

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using Levenberg-Marquardt method (Levenberg, 1944; Marquardt, 1963;Pujol, 2007) as (8)

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where 𝐽 is the matrix of partial derivatives of function with respect to model parameters (C nq& Q) as shown in Equation 9; ΔX is the perturbation; 𝐽 𝑇 is the transpose of𝐽; λ is the damping parameter;

∂Q

J=

∂SR(f1) ∂Cnq

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∂SR(f1)

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Diag(𝐽 𝑇 𝐽) is the diagonal matrix of 𝐽 𝑇 𝐽.

∂SR(f2)

∂SR(f2)

∂Q

∂Cnq

̇̇̇

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∂SR ̇(fn )

[

∂Q

(9)

̇̇ ̇ n) ∂SR(f ∂Cnq

]

The method is a combination of the steepest descent algorithm and the Gauss–Newton algorithm, when λ is very large it approximates to the steepest descent method. For small value of λ the Eq. 8 approaches to Gauss–Newton algorithm. The iterative equation for the Levenberg-Marquardt method can be written as mk+1 = mk − (JT J + λ Diag (JT J))−1 JT ΔY

(10)

ACCEPTED MANUSCRIPT The initial model to start the optimization is taken from the final solution of conventional LSR method. Convergence of solution is checked iteratively and the best model is obtained for the minimum of objective function.

SYNTHETIC STUDY Numerical tests were carried out on the synthetic data generated through an isotropic 1D viscoelastic

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model. The velocity, density and Q profile of the model used to simulate the VSP data is shown in Figure

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1.Water column is 800m thick and the velocities / densities of the layers represent a sand-shale sequence typical of a sedimentary basin. Q profile ranges from 100 to 200 across depth sandwiched with a 50m

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layer of high absorption (Q=70) at depth of 1400m. The data was generated using an airgun array located at 4m water depth. Figure 2(a) shows the receivers layout at 20m interval along the depth from 800 m to

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3400m. To estimate the interval quality factor, we have selected the reference and target receiver within the investigation zone itself (as illustrated in Figure 2a), so that the reflectivity and other factors affecting

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estimation do not deviate the actual idea behind the study. Downgoing energy modeled with absorption is shown in Figure 2 (b). The simulated data was de-bubbled, flattened the first arrivals and subsequently isolated for spectral calculation as shown in Figure 3(a). After selecting receivers for respective layers the

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spectral ratios were calculated for each zone.

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Figure 1. Velocity, Density and Q profiles of the 1D isotopic layered Model. Q for water layer is 10000.

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Figure 2. (a) Layout of receiver and the identified receivers in a particular zone for Q estimation. (b) Noise free zerooffset synthetic VSPdata

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Figure 3. Synthetic data after pre-processing of VSP synthetic (a) no noise, and with addition of different scales of random noise i.e. 5 %, 10% and 15 % in (b), (c) and (d) respectively

Random noise of 5 %, 10 % and 15 % of maximum amplitude of deepest receiver were generated and added to the noise-free data as displayed in Figure 3(b), (c) and (d) respectively. A total of 50 independent realizations of each noise scale were carried out for the study. Spectral ratios of a particular realization of all these scenarios are displayed in Figure 4 (a) and 4 (b) for the shallowest and deepest zone respectively. It can be noted that the spectral distortion is very severe in high frequency part of spectrum. Further the variation is quite high for the deepest zone where the signal to noise ratio is poor compared to shallow zone.

ACCEPTED MANUSCRIPT Figure 4. : Spectral ratio of (a) shallowest and (b) deepest zone for different noise levels

Linear least square inversion was performed on the log-spectral ratios. Q estimation was done for different noise levels across a range of bandwidths by the non-linear inversion of spectral ratios using Levenberg-Marquardt method. It was noted that any effective bandwidth from 20-100 Hz to 30-60 Hz would provide the reliable estimate for noise free case. For noise-free case the estimated Q values for all zones are almost near to true model for both linear and non-linear approaches.

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However for noisy scenario the effective bandwidth decreased with depth and needs to be

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selected optimally. Figure 5 displays the estimated mean Quality factor using non-linear approach for 50

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independent realizations of different noise levels for different frequency bands starting from 20 Hz, 25 Hz and 30 Hz to 100 Hz. The 90% confidence limits for the results of each frequency band are also plotted. A comparison of Figure 5 (a), (d) and (g) indicates that smaller bandwidth have a large variation in the

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estimation, while the bandwidths greater than 50 Hz have a stable Q value with smaller deviation from the true Q i.e. 200.

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Further in the same frequency band there is shift in mean Q as well as a large range of estimated values with increase in noise level. In case of 15 % noise as shown in Figure 5 (c), (f) and (i), negative Q

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values are estimated for bands having frequency greater than 85 Hz. For higher noise levels higher frequency part was avoided. An effective bandwidth of 20-95 Hz was selected above 1900 m depth for

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stable results, while 20-85 Hz was selected for zones below 1900 m. Results for the deepest zone (of a particular realization of 10 % noise case) using both the approaches are shown in Figure 6. The black

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curve is the observed ratios in the deepest zone, red and blue curves are estimated decay using linear and non-linear solution respectively. The results for LSR and LM were 143.04 and 185.76 respectively, while the actual model value for Q was 200. LM solution is more close to true value in comparison to LSR as

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the log-transformation distorted the actual decay.

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Figure 5. Estimated mean Quality factor (with 90 % confidence limits) using non-linear approach using different frequency bands with different noise levels for the deepest zone. Model Q value is 200. In (a), (b) and (c) the starting frequency is 20 Hz, with increasing noise level of 5%, 10% and 15 % respectively. For (d), (e) and (f) the starting frequency is 25 Hz, while for (g), (h), and (i) the starting frequency is 30 Hz.

Figure 6. Observed spectral ratios of the deepest zone for 10% noise case and decay curves using the solutions of LM and LSR method

Figure 7 (a), (b) and (c) shows the resulting Q profiles in case of 5%, 10% and 15 % noise respectively. The Q profile for 5 % noise (estimated from 20-90 Hz band) is close to true model; except for the deepest zone where the S/N ratio is comparatively low than shallower zones. As noise increased to 10 % and 15

ACCEPTED MANUSCRIPT % the effective bandwidth narrowed with depth; however the results for non-linear approach are more close to true values in comparison to linear inversion. Moreover the less variability of estimated Q values was visible in case of LM than the LSR solution. It can be observed from the 15% noise scenario (Figure 7(c)) that the results are not stable for both the approaches, nevertheless non-linear is less far from the actual model.

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Figure 7. Mean Quality factor estimated using optimum frequency bands with different noise levels of 5%, 10% and 15 %

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in 7 (a), (b) and (c) respectively.

Relative errors of estimated mean quality factor for 50 independent realizations of each noise scale are

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plotted in Figure 8 (a), (b) and (c). It may be noted that the relative error (%) increased with depth for both LM and LSR solutions in each noise case. This is because S/N ratio in this synthetic study decreases

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with depth. For layer number 4, a relatively large error using both inversion schemes could be due to high absorption (Q=70) in thin layer (50 m). Nonetheless, for all depth intervals the relative error in LM

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approach is less than conventional LSR method. As the noise level is increased the divergence of solutions increases, however beyond a certain S/N ratio both of the approaches are very far from the true

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solution

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Figure 8. Layer-wise relative error (%) in mean Quality factor of 50 independent realizations in each noise scenario.

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Field Data example

The VSP data was acquired in the shallow waters of a petroliferous basin located in the East Coast, offshore India. Well was drilled to explore the high amplitude channels and lobes system in the shelf area

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of upper fan portion of Ganges delta. VSP data was acquired with a three gun cluster of air guns of 150 cu in. each at a depth of 5m from sea surface. Receivers were deployed at 15m intervals, from 300 m to

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1800m in the vertical well. Five shots were fired for each depth level and the best was selected depending on the S/N ratio. The data was recorded at 1ms sample interval for 6 seconds. Vertical component of the acquired near-offset VSP data is displayed in Figure 9 (a).After band pass filtering raw data was corrected for source-receiver geometry and then shifted to mean sea level by using the water velocity of 1524.0 m/sec. Data was aligned by applying LMO correction using first break times. The downgoing and upgoing wavefield were separated with median filtering. The direct arrivals were isolated by muting after 100ms of direct arrival and the trace length was extended to 500ms (Figure 9(b)). FFTs were then taken for respective receivers of different zones as shown by dashed horizontal lines in Figure 10. Amplitude

ACCEPTED MANUSCRIPT Spectra variation with depth is shown in the Figure 9(c), where the vertical blue dashed lines indicate the effective bandwidth used for Q estimation.

Figure 9. Raw VSP data at the site is displayed in (a). (b) shows down going energy after wavefield separation and aligned for spectral calculation, while (c) displays the amplitude spectra of corresponding zones.

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For shallower data (i.e. selected above 1500 m) an effective bandwidth of20-95Hz was selected for Q

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estimation, while for deeper zones it was 20-85 Hz. Spectral ratios of the identified receivers of respective

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zones were calculated and Q estimation was carried out using both LSR and LM method.

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RESULTS

In Figure 10 we analyze the resulting Q profiles with well logs i.e. VShale, GR (natural gamma log) overlaid on Q, Vp/Vs ratio and Impedance log. Estimated Q values are found in reasonable range across

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depth at this well except in the zone from 826mto 886m where it is negative. Depth interval from 380m to 436 m has a low Q value of80 represents the unconsolidated zone. Also the interval includes a thin gas

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bearing zone. For thick sand zone from depth 496m to 721m estimated Q value is 208.Q is around 86 at depth interval from 721m to 826m which includes three thin water bearing sands. Negative Q in zone

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from 826 m to 886m is due to local interference effects (O’Doherty and Anstey, 1971; Ganley and Kanasewich, 1980, Hackert, C. L., and Parra, J. O., 2004).From depth 886 m to 1408 m there is increase

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in impedance and decrease in Vp/Vs ratio. However Q profile decreased gradually, which should increase as density and velocity increased in this interval. The decrease in effective Q value may be possible due to

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scattering attenuation introduced by thin layering in this zone. Another zone (1408m to 1470m) of low Q

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i.e. 69 includes a gas bearing sand.

Figure 10. Plot of Gamma log, Vp/Vs , Q factor for LM and LSR method and Vsh.

The estimated low Q values are indicative of gas zone at this investigation site. Also it may observed from the well log curves of Vsh, Gamma log, Vp/Vs ratio and Impedance log that there is some correlation of Q profile with the litho-logical variations. Moreover the selected VSP dataset for this study is having very good signal to noise ratio. It is to be noticed that the estimated Q profile with both nonlinear inversion (LM displayed by blue line) log-linear (LSR as red) approach is almost similar.

CONCLUSTIONS

ACCEPTED MANUSCRIPT Non-linear approach of spectral ratio inversion using VSP data is proposed in this study as a tool for Q estimation. For noise free data, the results of both the approaches of estimation using spectral ratio method are same. However, as the noise level increases, the classical log-spectral ratio method results are deviating from the actual. It is also worth noticing that the effective bandwidth for estimation is decreasing with increase in noise level and should be chosen cautiously. However, in datasets having moderate signal to noise ratio, non-linear optimization of the model parameters is more accurate as

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compared to that of linear optimization. For very poor signal to noise case, both the approaches provide

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erroneous results. Non-linear approach to spectral ratio method can improve the accuracy of the Q model. Local method of non-linear optimization such as LM may trap in local minima if initial guess is very far.

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Non-linear solution in our case might get entrapped in local minima as we have started from the log-linear solution. With global inversion schemes it will avoid any such issues of trapping in local minima.

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Spectral calculation using high resolution time-frequency transforms may further improve the estimation using non-linear approach. The approach may yield stable results from surface reflection seismic data,

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where signal to noise ratio is usually low compared to VSP.

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ACKNOWLEDGMENTS The authors are grateful to Head of Exploration for his support and encouragement. The authors are very

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thankful to Reliance Industries Ltd. for providing necessary resources for carrying out this research work

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Highlights

Conventional spectral ratio method for estimating Seismic Quality factor (Q) involves log- is very sensitive to noise.

A new approach to spectral ratio method is proposed where the seismic quality factor is estimated

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

by non-linear inversion of spectral ratios using Levenberg-Marquardt (LM) method. Numerical tests indicated that with increase in noise floor LM method provided more accurate

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

and stable results.

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Application on real VSP dataset having high S/N ratio revealed that the estimated Q models using

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both the approaches are almost similar.

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