3.2.2. Continuous Wavelet Transform (CWT)

3.2.2.1. Seismic attribute based on CWTWhile the Fourier transform decomposes a seismic signal from sinus or

cosines waves to a different frequency, the wavelet transform decomposes a signal into dilated and translated wavelets. The wavelet transforms is a method to provide a flexible time-frequency window that automatically is narrowed when observing high frequency phenomena and is widens when observing low-frequency environment. The integral wavelet transforms or wavelet transforms (WT) is decomposed signal by using dilated and translated wavelet. A family wavelet in time frequency analysis is obtained by scaling or dilated by s and translating by u the wavelet function (t). The

wavelet transforms of x ( t ) at time u and scale s is:

W x(u , s )=∫−∞

+∞

x ( t )ψu , s¿ ( t )dt

(1)

and the inverse wavelet transforms is:

x ( t )= 1Cψ

∫0

−∞

∫−∞

+∞

W x(u , s )ψu , s ( t )dudss2 (2)

with Cψ is called admissibility conditions (Nurcahya, et.al, 2003).

One of difference between complex trace attributes based on Hilbert transform and WT is on the domain. The complex trace attributes based on Hilbert transform is in time domain, but the complex trace attributes based

on wavelet transform is in time frequencies domain. The complex trace based on WT or CWT with a typical mother wavelet is

W x(u , s )=Re {W x(u , s )}+ i Im {W x (u , s )} (3)

where W x(u , s ) is the wavelet transform of a seismic signal. Variations of seismic attributes based on CWT which can be generated by equation (3) are: decomposition seismic data in certainly frequency bandwidth, instantaneous amplitude and instantaneous time-frequency derivative of amplitude. CWT method also could be used to extract instantaneous amplitude in low frequency bandwidth that shows the increasing of wave energy because of the existing of diffusive wave in poro-elastic medium that filled by fluid or hydrocarbon (Goloshubin, 2006).

3.2.2.2. Instantaneous amplitude Mathematically, the instantaneous amplitude based on CWT is

formulated as:

A( t )=√x2 ( t )+ y2( t ) (4)

where

x ( t )=ℜ[ 1Cψ ∫0+∞

∫−∞

+∞

W x (u , s )ψu, s (t )du dus2 ] and

y ( t )=ℑ[ 1Cψ ∫0+∞

∫−∞

+∞

W x (u , s )ψu , s (t ) du dus2 ]

so the magnitude of the instantaneous amplitude is :

A(u , s)=|W x(u , s )|1/2=√W x(u , s )⋅W x

¿ (u , s ) (5)

3.2.2.3. Time- frequency derivative of Instantaneous amplitude (GAMP)

Time-frequency derivative of Instantaneous amplitude (GAMP) is formulated as:

W x(∇ )(u , s )=|∇W x (u , s )| (6)

with

∇= i ∂∂ u

+ j ∂∂ s

∂∂ sW x

¿ (u , s )=− 1s|s|∫−∞

+∞

x( t ) ∂∂uψ¿ ( t−us )dt

The attribute is primarily used to figure the value of seismic wave attenuation. The attenuation of seismic wave is influenced by rock porosity (either matrix or fracture) saturated by fluid. The high value of the attribute shows the high porosity of the rock. Porous sandstone saturated by fluid will

∂∂uW x

¿ (u , s )=− 1s|s|∫−∞

+∞

x ( t )(ψ¿( t−us )+ t−us ∂∂uψ¿( t−us ))dt

showed by high value of GAMP attribute. Therefore, the attribute could be used to estimate the distribution of sandstone, especially for sand-shale of geological environment. Figure-2.1 shows the example of correlation between section of Time-frequency derivative of Instantaneous amplitude (GAMP) and petrophysics data. High GAMP almost relates with porous sand. Figure-2.2 shows the example of sand distribution from a reservoir that is delineated using time-frequency derivatives of instantaneous amplitude.

Figure-2.1.The example of correlation between section of time-frequency derivative of Instantaneous amplitude (GAMP) and petrophysics data

Figure-2.2. The example of sand distribution from a reservoir that is delineated using time-frequency derivatives of instantaneous amplitude

3.2.2.4. Low-frequency AnalysisLow frequency anomaly of seismic reflection that comes from

reflection of porous rock saturated by fluid has been describes using poro-elastic theory, diffusive wave and result of well test (Goloshubin, 2004). Simply, reflection coefficient of seismic wave that pass through porous rock saturated by fluid is formulated as:

R=R0+(1+ i )√ κρμ ωR1 (7)

where ω is frequency of seismic wave, ρ is rock density, κ rock permeability and η is fluid viscosity. The reflectivity equation also could be described as:

R=√(R0+√ κρη ωR1)2

+ κρηωR1

2(cos (arctan ( κρηωR1

R0+√ κρη ωR1 ))+i sin(arctan (κρηωR1

R0+√ κρη ωR1 ))) (8)

The reflectivity value will be maximum if the value of κρηωR1

approximate zero or, on the other word, frequency of seismic is very low. Figure-2.3 shows seismic reflection response from laboratory test among dry reservoir zones and saturated by water or oil (Korneev, 2004).

Figure-2.3. Seismic reflection response from laboratory test among dry reservoir and saturated by water or oil (Kornev, 2004)

Figure-2.4. A seismic line (a) and low frequency (<15 Hz) (b) from Ai Pim Western Siberia oil (Goloshubin, 2006)

Figure-2.4 shows a seismic line and low frequency (<15Hz) from Ai Pim Western Siberia oil field was used to image two different types of oil-saturated reservoirs. Black dots show where there is oil; white dots show where there is no oil. The AC11 is the pore sandstone reservoir (11-15m), and the Ju0 is fractured shale reservoir (15-20m) well data indicate that the upper reservoir (marked AC11, Goloshubin, 2006).

3.2.3. Extended Elastic Impedance (EEI)

3.2.3.1. EEI TheoryWhitcombe refined the definition of elastic impedance to remove the

dependence of dimensionally on the angle θ (normal incidence). Reconized some of rock cannot be predicted from existing seismic gathering due to limitation on incidence angle range (0-30) in the elastic impedance. (Whitcombe, D.N., Connolly, P.A., Reagan, R.L., Redshaw, T.C, 2002). The equation sin2θ needs to exceed unity to estimate some petrophysical ;however,it is impossible that the reflectivity values exceed unity without negative (and therefore unrealizable) impedance contrast (Hicks, G.J., Francis, A, 2006).

Therefore Whitcombe et.al (2002) introduced the extended elastic impedance to solve the elastic impedance limitation. The extended angle range from 0-30 degrees which is defined mathematically over a 0-90○ angle (0-0.25) range which corresponds to sin2θ by substituting sin2θwith tan χ . The variable θ is now a new function called χ (chi angle or project angle) which varies between -90○ and +90○ (Figure 1 and figure 2)

Figure 1. Extended elastic impedance angles can range from -90○ and +90○ , at which values sin2θis physically impossible (adopted from Humpson-Russell help system)

Extended elastic impedance provides a framework to work with pre-stack AVO but in terms of impedance instead of reflectivity. For the EEI analysis, EEI logs are generated for each well as a function angle and correlated with the target petro-physical logs. For each petro-physical log, cross correlation of EEI logs at different angle is computed and a plot is then made for the correlation coefficient as a function of angle. EEI can be defined as:

EEI χ=α 0β0[( αα 0 )p

( ββ0 )q

( ρρ0 )r] (9)

where

p= (cos χ+sin χ )

q=−8K sin χ

r=(cos χ−4K sin χ )

α 0, β0and ρ0 are the average for the respective property used as normalization factors for P-velocity, S-velocity and density respectively. K is the average of (α/β)2 in the time/depth interval. The EEI logs at different angles correspond to different rock properties. (Whitcombe et.al, 2002)

Figure 2. The EEI functions for various χ values for particular well. Note the inverse correlation between EEI ( χ =+90○ ) and EEI (χ =-90○). (Whitcombe et.al 2002)

The distinct difference between the extended elastic impedance and normalized version of elastic impedance is the change variable. EEI is a function of χ (an angle in abstract construction) and EI is a function of θ (an angel in a physical experiment). (Francis, A., Hicks, G.J,2006). This can lead to EEI much more efficient than EI method and supposed to give different outcomes than standard EI inversion method. It is important to notice that new variable ( χ) allows calculation of impedance value beyond physically observable range of angle θ (including imaginary angles necessarily recorded in the gathers). A clear example of this situation happens when shear impedance corresponds to sin2θ = -1.25. It is obvious, negative angle is

not physically recordable but can be projected from angle gathers by linear extrapolation (Hicks, G.J., Francis,A,2006).

To show that the EEI at χ = 0 is similar to EI log at θ =0, which is simply the acoustic impedance (AI). Whitcombe et al. (2002) provides a simple robust application for deriving lithological and fluis sensitive seismic impedance volumes. According to his perspective under certain approximation, the EEI log at various chi angles proportional to different rock elastic parameters

The chi angle (χ) can be selected to optimize the correlation of the EEI curves with petrophysical reservoir parameter such as Vshale, Sw and porosity or with an elastic parameters such as bulk module, shear module and lame constant and so on (Whitcombe et al. 2002).Therefore, EEI logs for specified angles from these parameters can be produced by using EEI equation which is suited for tie well data directly to seismic data (Figure 3). Directness of EEI method is the main advantages which provide an EEI volume attributes that correspond to pethophysical parameters of interest.

Two term linearization of Zoeppritz equation for reflectivity (Aki & Richards), equation can define as:

(𝜃) = 𝐴 + 𝐵𝑠𝑖𝑛2𝜃 (10)

Regarding to Whitcombe method when sin2θ replaced by 𝑡𝑎𝑛χ, so equation 10 represented as equation 11 which allows angle to vary from -90○ and +90○

(𝜃) = 𝐴 + 𝐵𝑠𝑖𝑛2𝜃⇒(χ) = 𝐴 + 𝐵𝑡𝑎𝑛χ (11)

(A= intercept, B= Gradient)

Figure 3. Comparisons between elastic parameters and equivalent EEI curves for particular well, representing the high degree of correlation. The EEI function is defined as a function of the angle χ, not the reflection angle θ (Whitcombe et al.2002)

3.2.3.2. EEI WorksBefore we doing inversion based on Extended Elastic Impedance, we

need to doing cross plot to determine the best well data can be using to separate lithology on target area. On pictures below we can see some cross

plot between well data on Pancing-1X. Well Pancing-1X is being used because it is the only well which drilled through Intra Lama interval.

Figure. 4. Cross plot P-Impedance Vs density with gamma ray as color key

From Figure. 4 above we can see that P-Impedance cannot be using to separate between shale and sand. So we need to find another solution to separate lithology on target area. As we can see on Figure 5 and Figure 6 below, Mu-Rho property and Vp/Vs property is better than P-Impedance in case to separate lithology on target area.

On cross plot figures, yellow color is sand area, green color is shale area, and black area is coal area. On Figure 5 cut off value for Vp/Vs ratio for sand area is between 1.30 - 1.60. From Figure 6, cut off value for sand area based on Mu-Rho is above 30 Gpa * g/cc but shaly sand also have same value. Another cross plot result can be seen on Appendix A.

Figure. 5. Cross plot Vp/Vs ratio Vs P-Impedance with gamma ray as color key

Figure. 6. Cross plot Mu-Rho Vs Lambda-Rho with gamma ray as color key

The implementation of Extended Elastic Impedance is to find the best chi angle ( χ) for every well data. EEI log curve can be obtained by using HRS EEI Modelling Trace Maths Script by Kevin Gerlitz (2004). That script based on Whitcombe et al (2002) and using P-Wave log, S-Wave log, and Density log to calculate EEI Log.

The next step is doing cross correlation between EEI log and every well data such as P-Impedance, S-Impedance, Vp/Vs, Mu-Rho, Lamda-Rho and etc. Cross correlation is done by using HRS Cross-Correlate Reflectivity Trace Maths Script by Kevin Gerlitz (2004). Results of cross corelation can be seen on figures below.

Figure. 7. Cross correlation between Vp/Vs ratio and EEI angle (chi)

From Figure 7, Vp/Vs ratio have good correlation with EEI log on 62 degree chi angle. Correlation value from Vp/Vs ratio on 62 degree chi angle is 0.93 of 1.00. On Figure 8, Mu-Rho have best correlation value with EEI log on -45 degree chi angle. On the other hand, P-Impedance also have good correlation with EEI log (Figure 9), but from cross plot P-Impedance cannot be used to separate lithology on target area.

Figure. 8. Cross correlation between Mu-Rho and EEI angle (chi)

Figure. 9. Cross correlation between P-Impedance ratio and EEI angle (chi)

After we get all best chi angle for all well parameter, we calculate the associated elastic parameter reflectivity seismic volumes using :

A+B tan( χ∗pi /180)

With A and B is intercept and gradient volumes based on AVO volumes. Model based inversion process is being done by these reflectivity volumes.

The inversion process is being done from Intra Lama 1 interval until Intra Lama 2 interval. Inversion results can be seen on Chapter 4.1.4.2.