SPWLA 51st Annual Logging Symposium, June 19-23, 2010
JOINT PROCESSING OF FORWARD AND BACKWARD EXTENDED PRONY AND WEIGHTED SPECTRAL SEMBLANCE METHODS FOR ROBUST EXTRACTION OF VELOCITY DISPERSION DATA Jun Ma, Pawel J. Matuszyk*, Robert K. Mallan, Carlos Torres-Verdín, and Benjamin C. Voss, The University of Texas at Austin * on leave from AGH – University of Science and Technology, Krakow, Poland Copyright 2010, held jointly by the Society of Petrophysicists and Well Log Analysts (SPWLA) and the submitting authors.
damage because it associates high- and low-frequency slownesses with near- and far-borehole formation properties, respectively. Consequently, elicited velocity anisotropy can be attributed to either formation damage or in-situ stress, or both. Detailed analysis of Stoneley and flexural dispersion is necessary to properly estimate compressional- and shear-wave velocity data in the presence of borehole washouts, mudcake, sand-shale laminations, borehole deviation, and fractures, among other formation conditions.
This paper was prepared for presentation at the SPWLA 51st Annual Logging Symposium held in Perth, Australia, June 19-23, 2010.
ABSTRACT Various extensions of the Prony method are commonly used to detect and extract frequency-slowness dispersion data from array sonic waveforms. We introduce a new dispersion method that combines the forward and backward extended Prony method with the weighted spectral semblance method to improve slowness dispersion estimation from waveforms with presence of noise.
The Prony method is commonly used to detect and extract velocity dispersion data from array sonic waveforms. However, the method is not reliable in controlling the effect of noise present in real sonic waveforms on estimating dispersive modes.
The procedure works as follows: (1) the forward and backward extended Prony method is used to find the poles of the associated characteristic polynomials and to reconstruct the array waveform data; (2) the 2D matrix of weights is calculated in the frequency-slowness plane according to the slownesses associated with the poles; and (3) the correlation matrix is calculated using weighted spectral semblance, wherein the 2D weight matrix is included.
The Backward or Forward Extended Prony methods (BEPM or FEPM) generate spurious modes (Tang and Cheng, 2004), while the extraction of modes by the Prony fitness function usually results in aliased ‘modes’. Joint process of Forward and Backward Extended Prony Method (FBEPM) has been implemented by many researchers (Minami at el., 1986, Yu, 1990, and Sava and McDonnell, 1995, etc.). We found that this approach, when combined with Tikhonov regularization (Aster et al., 2005), is effective in filtering most of the spurious modes in the case of few propagation modes. In practice, however, not only do multiple modes exist having different energy magnitudes, but presence of noise in the waveforms makes dispersion mode estimation a challenging problem.
The new method is applied to noisy sonic logging field data. Results show that estimation of dispersion curves using the new method is more robust than those obtained with either the backward (or forward) extended Prony method or the weighted spectral semblance method. Our method also outperforms alternative, commonly used approaches in the elimination of aliasing effects.
Another method of extracting slowness dispersion data from array sonic waveforms is the Weighted Spectral Semblance method (WSS) (Nolte and Huang, 1997, Tang and Cheng, 2004). This method is simple and efficient. The weighted average provides reliable estimates by utilizing data from the neighboring discrete frequencies, thereby rendering a smoother dispersion curve.
INTRODUCTION Frequency dispersion analysis of sonic waveforms provides significantly more information on formation elastic properties than conventional slowness-timecoherence methods. The recent introduction of new wide-band sonic acquisition systems for monopole, dipole, and quadrupole sources makes it possible to reliably perform sonic dispersion processing. Indeed, frequency dispersion processing of sonic waveforms enables the assessment of drilling-induced formation
Matrix Pencil is another effective method used in signal processing (Sarkar and Pereira, 1995). The joint process
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SPWLA 51st Annual Logging Symposium, June 19-23, 2010
of Forward and Backward Matrix Pencil (FBMP) was developed for effective dispersion estimation of borehole acoustic arrays (Ekstrom, 1996).
P
Xˆ l ( f ) = ∑ b j z lj−1 ( f ) , l = 1, 2," N ,
(1)
j =1
Where It is well known that the WSS method does not generate spurious estimates but does generate aliases. In contrast, extended Prony methods generate spurious estimates but no aliases. To exploit the advantages of both methods, we introduce the Joint Processing of Forward and Backward Extended Prony and Weighted Spectrum Semblance method (JFBEPWSS), in which the solution of the extended Prony method is added into the weighted spectral semblance as a new weight. The main objective behind the application of this new approach is the reduction of the effect of deleterious biasing noise in the signal. A new strategy is proposed to detect physically consistent signals within noisy recorded sonic waveforms. In addition, the application of a weighting matrix on Prony poles eliminates pervading aliasing in the correlation matrix.
z j ( f ) = e⎣
⎡ −α j ( f ) + ik j ( f ) ⎤⎦ d
, α j ≥ 0 is a damping
factor, k j ( f ) = 2π f s j ∈ R is a wavenumber and s j is +
a slowness. We set P=N/2. The variable frequency was omitted to simplify the equation below. Thus the observed data can be expressed by: (2) X l = Xˆ l + El , l = 1, 2,..., N , where El is the error of the estimation. The key to the Prony's method is that the P exponentials z j are the roots of a characteristic polynomial: P
P
l =0
l =1
Ψ ( z ) = ∑ al z ( P − l ) = z P + ∑ al z ( P − l )
,
P
= ∏ ( z − z j ) = 0, a0 = 1.
(3)
j =1
From Equations (1) and (3), one can obtain a recursion equation for the forward forecast as follows:
The new method is first applied to clean and noiseadded synthetic sonic waveforms considering a dipole source and presence of a mandrel tool. Subsequently, we apply the new method to noisy sonic monopole and cross-dipole logging field data acquired in unconsolidated siliciclastic and carbonate formations.
P
Xˆ l = −∑ a j Xˆ l − j ,
P +1 ≤ l ≤ N,
(4)
j =1
The minimization of the total error N
P
∑ ∑aE
k = P +1 j = 0
METHOD
j
2
(5)
k− j
leads to solving of the following system of linear equations (Zhang, 2002, Liu et al., 2006): Ya = β , (6)
The JFBEPWSS method consists of three parts: • the forward and backward extended Prony method and Prony fitness function, • the new weighting matrix equation associated with the Prony solution, and • the joint weighted spectral semblance.
where a = [1 a1 a2 " aP
]
T
is a vector of unknown
coefficients, and β = [ β 0 0 " 0]
T
is a ( P + 1) × 1
P
vector, where β = ∑ Y (0, j )a j is the estimation of j =0
minimum error energy. The elements Y ( j , k ) of the matrix Y are defined as follows:
Forward Extended Prony Method Wave propagation along the borehole (waveguide) in sonic logging is considered a one-dimensional propagation issue. We assume that there is a sequence of full waveforms x1 (t ), x2 (t ),..., xN (t ) in time domain, recorded by an array of N equally distributed receivers with inter-receiver spacing d. Their Fourier transforms are X 1 ( f ), X 2 ( f ),..., X N ( f ) , respectively. We refer to these as observed data. In the preceding, t is time and f is frequency. The propagation vector is given by ei (κ x −ωt ) , where κ is the complex wavenumber, ω = 2π f is the angular frequency and x is the spatial position. It is supposed that the estimation of the observed data Xˆ l consists of P modes in the frequency domain:
Y ( j, k ) =
N
∑
l = P +1
X l − k X l*− j , j , k = 0,1, 2," , P ,
(7)
where * denotes complex conjugation. The linear system (6) is solved by the least squares method (LSQ) (Liu et al., 2006) under the 0-th order Tikhonov regularization (Aster et al., 2005), or by the total least squares method (TLSQ) (Zhang, 2002). For the LSQ, the number of modes estimated is P, whilst for the TLSQ, P is a number of effective modes estimated. After having calculated the coefficients al , the roots of the characteristic polynomial Ψ are found by BirgeVieta method followed by the Newton-Raphson polishing procedure. These roots are estimations of the dispersion modes z j . Next, we calculate Xˆ l by Equation
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SPWLA 51st Annual Logging Symposium, June 19-23, 2010
mode pairs defined above, and we remark them with the reordered index. (ii) A tolerance ε T is prescribed, which is usually 0.001 < ε T < 1.0 . Any pair for which
(4). Then, these estimates are used in Equation (1) to find the complex amplitudes b j by solving the least squares problem with N × P Vandermonde matrix Q : b = ( Q H Q ) Q H xˆ 1N , −1
ErrjFB > ε T is filtered out as noise. Then the qualified
(8)
T where b = [b1 b2 " bP ] , xˆ 1N = ⎡⎣ Xˆ 1 Xˆ 2 " Xˆ N ⎤⎦ Qlj = z lj−1 , l = 1, 2," , N , j = 1, 2," , P .
T
estimation of dispersion modes from FBEPM can be calculated as: −1 z FB = 12 ⎢⎡ z Fj + ( z Bj ) ⎥⎤ , j = 1, 2," M , M ≤ P. (10) j ⎣ ⎦ (iii) The amplitude of a mode at the nearest receiver is calculated using the following equation:
and
Backward Extended Prony Method
N −1 b FB = 12 ⎡b jF + b jB ( z Bj ) ⎤ , j = 1, 2," M , M ≤ P. j ⎣⎢ ⎦⎥
The processing of the backward extended Prony method is similar to that of the forward extended Prony method. The differences are as follows: • the sequence of observed data is reversed, • in Equation (1), negative values of α j ≤ 0 and
(iv) The estimation of sonic logging data for FBEPM can be expressed by the following formulas: Xˆ FB= 1 Xˆ F + Xˆ B l
k j < 0 are assumed,
•
Having estimated the observed data by the backward and forward Prony methods separately, we have two sets of estimations for dispersion modes, coefficients of the polynomials, and the complex amplitudes. We add subscript F and subscript B for variables related to the forward and backward variants of the method, respectively. To reduce the effect of deleterious biasing noise in the signal and to detect physically consistent signals within noise in the recorded sonic waveforms, we set the algorithm (called Forward-Backward Extended Prony Method (FBEPM) (Sava and McDonnell, 1995) as follows: (i) find the closest corresponding pairs of modes: B j2
−1
1
ErrjFB =
−1
z Fj + ( z Bj )
−1
.
F j
l −1
+ b Bj ( z Bj )
N −l
⎤, l = 1,..., N ⎦⎥
(12)
(13)
s FB ≥ sMIN ; then, if s FB > sMAX , we decrement its value j j
by 1, until s FB ≤ sMAX ; only if sMIN ≤ s FB ≤ sMAX , the j j mode is regarded as the real dispersion mode, and the applied value of n is saved. The same rule is used for BEPM or FEPM. The original sMIN and sMAX are usually chosen by experience, and their proper values can be chosen on the basis of pre-processing with use of the Prony fitness function (see Appendix I).
(9) Another problem is phase shifting for slownesses. By the specification of the method, the phase is not shifted for low frequencies (under 4 kHz). The continuation of each dispersion curve in the frequency-slowness domain is used to correct the improper phase. Thus, the correct estimates of slownesses are obtained and the
Then reorder the pairs with increasing differences FB ErrjFB +1 ≥ Errj , j ∈ 1, 2," P − 1 . At the same time, we
(
j =1
F j
It is assumed that the slowness of a real signal is bounded by limiting values sMIN and sMAX . The default value of n in Equation (13) is 0. If the current slowness s FB < sMIN , then multiplier n is incremented by 1, until j
and BEPM sets and calculate their relative difference: z Fj − ( z Bj )
∑ ⎡⎣⎢b ( z )
)
Strategy for Detecting Physically Consistent Signals in the Slowness Domain
obtained from FEPM
2
1 2
N −l
l
where n is an appropriate integer assuring the continuity of the slowness of the j-th mode. The equations are analogous to the ones obtained in FEPM or BEPM method, but with P replacing M, and, in the case of BEPM, with z replaced by 1/z.
Joint Process of Backward and Forward Extended Prony Method (FBEPM)
F j1
M
⎪ ⎪ FB FB ⎨ s j = ⎣⎡ Im ( ln z j ) + 2π n ⎦⎤ /(ω d ) , ⎪ ⎪⎩ j = 1, 2," M , M ≤ P
at the nearest (forward) or farthest (backward) offset receiver, and the sequence of estimations of the observed data is also reversed in BEPM.
( z , ( z ) ) , j , j ∈1, 2,", P
(
(v) The slownesses and attenuation factors are then estimated from calculated poles using formulas: ⎧α FB = − Re ( ln z FB j j )/d
the mode amplitudes b j denotes its excitations
•
2
=
the estimations of the dispersion modes are z −j 1 , j = 1, 2," , P ,
•
(11)
)
get a group of amplitude pairs b Fj1 , b Bj2 according to the
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SPWLA 51st Annual Logging Symposium, June 19-23, 2010
alias is filtered out at the same time. We applied necessary phase shifting for slownesses using the above strategy, for each method other than WSS.
We consider additionally a slowness weighting matrix: ⎛ − ( sn − sk )2 ⎞ ⎟, WnkS = W S ( sn , sk ) = exp ⎜ (21) ⎜ 2 ( Δs Δn )2 ⎟ s ⎝ ⎠ where n ∈ [ k − Δk , k + Δk ] , 2Δk is the number of neigh-
Weighting Matrix Equation Associated with the Prony Estimation
boring points taken to calculate the weighted average in the slowness domain and Δns ≤ Δk , which controls the amount of discrete points with the effective value of weight, and sk is the estimation point.
We define a set of discrete admissible slownesses: S = {sk : s1 < s2 < ... < sK , sk +1 − sk = Δs} (14) sMAX − sMIN K −1 For discrete admissible frequencies, we give similar definition by replacing S/s by F/f. Having calculated the dispersion modes and proper values of multiplier n (Equation (13)), the new weighting matrix is defined to lighten the real mode in frequency-slowness domain: s1 = sMIN , sK = sMAX , Δs =
VmkX = V X ( f m , sk ) =
εP
(
max ε P , min dskjX j =1,..., P
)
,
Finally, the weighted average is also added in slowness domain, and we define the final combined semblance function joining FBPM and WSS methods: JWSS mk = JWSS ( f m , sk ) =
(15)
k + Δk
∑
n = k −Δk
VmnFBWnkS VSS mn .
(22)
Applying the weighted averaging twice, one uses as much data as possible to compute the new semblance function for each point ( f m , sk ) . In practice, we use the estimated filtered data Xˆ FB , instead of the observed
where X ∈ { F , B, FB} , ε P ∈ (0,1] ( 0.01 ≤ ε P ≤ 0.1 for field data), sk ∈ S , and
l
sk − s Xj
(16)
data X l . Hence, the result will be more reliable and robust. Finally, the points ( f m , sk ) in the frequencyslowness domain with a maximum semblance are tracked out to deliver the estimated dispersion mode curves.
The weighted spectral semblance function (WSS) is defined as follows (Tang and Chen, 2004):
The algorithm of the method can be summarized as follows: 1. pre-processing by the Prony fitness function, 2. application of the FBEPM and calculation of new weighting matrices, 3. calculation of the semblance function VSS, 4. calculation of the semblance function JWSS and extraction of the dispersion curves. For convenience, we use acronyms FBEPM1 and JFBEPWSS1 to denote the methods where equation (6) is solved using LSQ, and FBEPM2 and JFBEPWSS2 to denote the methods where equation (6) is solved using TLSQ.
ds = X kj
.
sk + s Xj
Joint Weighted Spectral Semblance Method
WSS mk = WSS ( f m , sk ) =
m + Δm
∑
j = m −Δm
W jmF ρ jk ,
(17)
where N
ρ jk = ρ ( f j , sk ) =
∑ X ( f ) z( f ) *
l
l =1
j
N
l −1
j
,
N ∑ Xl ( f j ) Xl ( f j ) *
(18)
l =1
is the spectral semblance function, and ⎛ − ( f − f )2 j m F F W jm = W ( f j , f m ) = exp ⎜ ⎜ 2 Δf Δn 2 f ) ⎝ (
⎞ ⎟ (19) ⎟ ⎠ is a frequency weighting matrix. 2Δm is the number of neighboring points taken to calculate the weighted average in the frequency domain, and the quantity Δn f ≤ Δm controls the amount of discrete points
COMPARISON OF THE DIFFERENT METHODS IN THE PROCESSING OF SLOWNESS EVENTS Synthetic Data
We compare different dispersion processing methods for synthetic sonic logging data. The first example is a synthetic sonic dipole, simulated in the presence of a mandrel tool, in a fast formation, and at center frequency 10 kHz (Ma and Torres-Verdín, 2008). Table 1 shows the physical parameters used in the simulation. In the table, SP and SS denote the P-wave slowness and
with the effective value of weight, and f m is the estimation point. Then we define the combined semblance function:
VSS mk = VSS ( f m , sk ) =
m +Δm
∑
j = m −Δm
V jkFBW jmF ρ jk .
(20)
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SPWLA 51st Annual Logging Symposium, June 19-23, 2010
Figure 1: Results obtained for synthetic sonic dipole: (a) waveforms; (b) slownesses extracted by FBMP; (c) slownesses extracted by FBEPM1; (d) wave spectrum; (e) slownesses extracted by WSS; (f) slownesses extracted by JFBEPWSS2.
S-wave slowness, respectively, ρ is the density of media, and R is the outer radius of the corresponding concentric layer.
SP (us/ft) SS (us/ft) ρ (kg/m3) R (m)
Mandrel 115.45 240.00 1180 0.046
Borehole fluid 203.20 0 1000 0.108
lated by BEPM1 (Figure 1c), WSS (Figure 1e) and JFBEPWSS2 (Figure 1f), where analytic dispersion curves are plotted for comparison. Aliasing is present for WSS (Figure 1e) and for the Prony fitness function (Figure 1c), indicated with white elliptic contours. The uppermost dispersion curve (omitting the aliases) is the mandrel mode (indicated in the plots by label T). The dispersion curves which begin from the S-wave slowness of the formation at their lowfrequency cutoffs are the successive flexural modes (indicated in plots by labels FLi). Labels S are used to indicate shear wave modes and labels PLi are used for P-leaky modes.
Formation 96.46 147.46 2389 ∞
Table 1: Summary of slownesses, densities and outer radii of mandrel, borehole fluid, and formation.
Comparison of the dispersion curves extracted by the FBMP, FBEPM1, and JFBEPWSS2 methods indicates that the obtained results are very similar and coincide perfectly with analytic results. The WSS method delivers the poorest results, introducing alias
The results of the simulation are then processed with the different methods to obtain dispersion curves. Figure 1a shows numerically generated waveforms, and Figure 1d shows the corresponding amplitude spectra. The remaining subfigures show the dispersion curves obtained using: FBMP (Figure 1b), FBEPM1 overlain the Prony fitness function calcu-
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SPWLA 51st Annual Logging Symposium, June 19-23, 2010
Figure 2: Results obtained for synthetic sonic dipole perturbed with colored noise: (a) waveforms; (b) slownesses extracted by FBMP; (c) slownesses extracted by FBEPM1; (d) wave spectrum; (e) slownesses extracted by WSS; (f) slownesses extracted by JFBEPWSS2.
modes as well as giving poorer estimates of low amplitude modes for high frequencies. Some of the aliasing is easy to remove by taking into account the scope of the processed slownesses, but when it is located between the real modes, the identification of artificial modes is much more cumbersome.
Field Data
In this section, we apply the dispersion processing methods to several field data sets. For the purpose of comparison, each data set is processed using: the FBMP, FBEPM1, Prony fitness function, and WSS methods, as well as the new JFBEPWSS2 method.
In the second example, we add 47% colored noise in the frequency domain to the synthetic data used in Example 1 and repeat the dispersion processing procedure. The results are presented in Figure 2. Comparing the extracted dispersion curves obtained for the clean and noise-added data, one can see that all results are much worse: all of the methods partially or totally (in the case of low energy modes) misidentify some of the modes. Among all of the methods, JFBEPWSS2 was able to detect the largest number of points along the dispersion curves. Finally, the WSS method produces the smoothest curves; however, this comes at the cost of introducing unavoidable aliasing.
The results are consistently presented in Figures 3-8. Each figure consists of six subfigures, arranged in the same manner. Subfigures (a) and (d) show the waveforms and the corresponding amplitude spectra, respectively. The remaining subfigures show dispersion plots obtained by the selected dispersion methods: (b) FBMP; (c) FBEPM1/Prony fitness function; (e) WSS; and (f) JFBEPWSS2. In each case, the horizontal reference lines indicate estimated P- and Swave slownesses (of the formation) reported in the field logs. For the cases of the WSS and JFBEPWSS2 methods, we plot the semblance func-
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SPWLA 51st Annual Logging Symposium, June 19-23, 2010
Figure 3: Results obtained for field data (monopole source in an unconsolidated slow formation): (a) waveforms; (b) slownesses extracted by FBMP ; (c) slownesses extracted by FBEPM1; (d) wave spectrum; (e) slownesses extracted by WSS; (f) slownesses extracted by JFBEPWSS2.
tracted by the FBMP and FBEPM1 methods is not as clearly defined as that obtained by the WSS and JFBEPWSS2 methods – we can see numerous discontinuities in the dispersion curves. Also, the nondispersive P-mode is more clearly defined by the latter two methods. In Figures 3c and 3e we can see aliasing, whilst in Figure 3f only the real modes are present. Additionally, the plot of the Prony fitness function (Figure 3c) gives a number of artificial local maxima which are related to the changing amplitude of the characteristic polynomial with respect to the slowness. The real modes are indicated by dark red lines which correspond to the real roots of the characteristic polynomial.
tion. For the FBEPM1 method, the Prony fitness function is plotted. We use the following notations to label the extracted modes: P for compressional body wave, S for shear body wave, ST for Stoneley wave, FL for flexural modes, and PL for P-leaky modes. The first example concerns dispersion processing for an unconsolidated siliciclastic soft formation. The waveforms are acquired with the DSI sonic tool, possessing an array of 8 receivers (spaced 0.5 ft). Data include P&S monopole (located 9 ft from the 1st receiver) and upper dipole (located 11 ft from the 1st receiver).
The results for the dipole source are shown in Figure 4. All of the methods are able to detect the presence of two modes: the flexural mode at low frequencies (below 5 kHz) and the P-wave mode (above 5 kHz). We can see similar behavior of the methods: aliasing is removed for the JFBEPWSS2 method, and
The results for the monopole source are presented in Figure 3. All of the methods are able to detect the presence of two modes: the Stoneley mode at low frequencies (below 5 kHz) and the P-wave mode (above 4 kHz). However, the Stoneley mode ex-
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SPWLA 51st Annual Logging Symposium, June 19-23, 2010
Figure 4: Results obtained for field data (dipole source in an unconsolidated slow formation): (a) waveforms; (b) slownesses extracted by FBMP; (c) slownesses extracted by FBEPM1; (d) wave spectrum; (e) slownesses extracted by WSS; (f) slownesses extracted by JFBEPWSS2.
which results in a need for special algorithms to properly correct the phase.
the WSS and JFBEPWSS2 methods deliver smoother dispersion curves. We can also observe the effect of additional weighting in slowness domain, applied in the JFBEPWSS2 method, which results in a more compact plot of the flexural mode.
Additionally, a very weak P-leaky mode is extracted at very low frequencies (below 2 kHz). The most reliable localization of this mode is delivered by the JFBEPWSS2 method. Both the FBEPM and WSS methods also indicate the presence of this mode, but the plots are smeared and thus cannot give accurate information on the localization of the P-leaky mode in the frequency-slowness domain. Aliasing is clearly seen in the plots for the WSS methods, which partially overlap the Stoneley mode curve (see Figure 5e), whilst it is nearly eliminated by the JFBEPWSS2 method (Figure 5f).
In the next example, we present results of dispersion processing of data obtained in a very hard carbonate formation. The same tool is used as in the previous example. Figure 5 shows the results obtained for the monopole. All of the methods clearly extract two modes: the strongest (with respect to the energy) Stoneley mode (for the range of frequencies 2-8 kHz) and the nondispersive shear mode (above 8 kHz). The curve observed in Figure 5c, crossing the S-wave curve, is the continuation of the Stoneley mode curve, but wrapped by −2π phase. This illustrates the deficiency of the Matrix Pencil and Prony-like methods,
Figure 6 shows the results obtained for the dipole. In this case, only the dominating first flexural mode is detected by all methods (for the range of frequencies 1.8-6 kHz). The FBMP and FBEPM1 methods produce curves which exhibit many discontinuities, whi-
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SPWLA 51st Annual Logging Symposium, June 19-23, 2010
Figure 5: Results obtained for field data (monopole source in very fast carbonate formation): (a) waveforms; (b) slownesses extracted by FBMP; (c) slownesses extracted by FBEPM1; (d) wave spectrum; (e) slownesses extracted by WSS; (f) slownesses extracted by JFBEPWSS2.
(above 5 kHz). The WSS method has difficulty in resolving the Stoneley mode, as the mode plot has a very wide ridge towards the very low frequencies. In contrast, the other methods quite accurately determine the Stoneley mode. The P-mode is equally well recognized by all methods, where the extracted slowness matches the slowness reported in the field data. These plots show once again that the new JFBEPWSS2 method is the most effective in eliminating the spurious noise among all the presented methods.
le the remaining methods deliver smooth curves. However, due to additional weighting in the slowness domain and using information from Prony methods, the JFBEPWSS method gives a more compact plot as compared to the WSS method, especially in the low frequency limit. In the third example, we process field data obtained from a siliciclastic fast formation, for which the Swave speed is slightly higher then the borehole fluid wave speed. The waveforms are acquired with the XMAC sonic tool, possessing an array of 8 receivers (spaced 0.5 ft). Data include P&S monopole (located 12 ft from the 1st receiver) and X-X dipole (located 10.5 ft from the 1st receiver).
Figure 8 presents results obtained for the X-X dipole All methods extracted only the strongest first flexural mode. Once again, the JFBEPWSS2 method delivers the sharpest fitness function and most effectively reduces the noise.
Figure 7 shows the results obtained for the monopole source. All methods clearly extract two modes: the low frequency part of the strongest (in terms of energy) Stoneley mode and the non-dispersive P mode
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SPWLA 51st Annual Logging Symposium, June 19-23, 2010
Figure 6: Results obtained for field data (dipole source in very fast carbonate formation): (a) waveforms; (b) slownesses extracted by FBMP; (c) slownesses extracted by FBEPM1; (d) wave spectrum; (e) slownesses extracted by WSS; (f) slownesses extracted by JFBEPWSS2.
tion of the aliasing produced by the WSS method. Adding noise to the synthetic data resulted in a decreased ability of Prony methods to resolve the various modes, but it did not significantly change the results obtained by the WSS method. The new JFBEPWSS method inherits the stability of the WSS method, and thus is able to resolve existing modes without aliasing.
CONCLUSIONS
We developed a new hybrid dispersion processing method, which combines the Weighted Spectral Semblance method with a Prony-type method. This new method is an extension of the WSS method, where the additional information stemming from a Prony method is utilized to deliver additional weighting in the slowness domain, which results in the elimination of aliasing and the sharpening of the fitness function. Results obtained with synthetic and field data compared the new hybrid method against the respective WSS and Extended Prony methods, as well as the Matrix Pencil method.
The benefit of this new method was also illustrated with several field examples. They confirmed that the new hybrid method in each case eliminates aliasing, which is always present in the WSS method. Furthermore, the JFBEPWSS method delivers a sharper fitness function, thereby resulting in more reliable estimates of wave modes.
In the case of synthetic data without noise, all of the presented methods performed equally well, delivering reliable estimates of the modes, with the excep-
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SPWLA 51st Annual Logging Symposium, June 19-23, 2010
Figure 7: Results obtained for field data (monopole source in a fast formation): (a) waveforms; (b) slownesses extracted by FBMP; (c) slownesses extracted by FBEPM1; (d) wave spectrum; (e) slownesses extracted by WSS; (f) slownesses extracted by JFBEPWSS2.
FBPM BEPM FBEPM
ACKNOWLEDGEMENTS
The work reported in this paper was funded by The University of Texas at Austin’s Research Consortium on Formation Evaluation, jointly sponsored by Anadarko, Aramco, Baker-Hughes, BG, BHP Billiton, BP, Chevron, ConocoPhilips, ENI, ExxonMobil, Halliburton, Hess, Marathon, Mexican Institute for Petroleum, Nexen, Pathfinder, Petrobras, Schlumberger, Statoil, TOTAL, and Weatherford. We thank Dr. Xiaoming Tang at Baker-Hughes for his insight to the Prony fitness function.
FBMP WSS JFBEPWSS P S ST FL PL
ACRONYMS
LSQ TLSQ
Least Squares Method Total Least Squares Method
11
Forward Extended Prony Method Backward Extended Prony Method Jointed Forward and Backward Extended Prony Method Jointed Forward and Backward Matrix Pencil Method Weighted Spectral Semblance Method Jointed Forward and Backward Extended Prony Method and Weighted Spectral Semblance Method Compressional wave mode Shear wave mode Stoneley mode Flexural mode P-leaky mode
SPWLA 51st Annual Logging Symposium, June 19-23, 2010
Figure 8: Results obtained for field data (dipole source in a fast formation): (a) waveforms; (b) slownesses extracted by FBMP; (c) slownesses extracted by FBEPM1; (d) wave spectrum; (e) slownesses extracted by WSS; (f) slownesses extracted by JFBEPWSS2.
ence of a Centered, Non-rigid Cylindrical Tool and Transversely Isotropic Elastic Formations: 8th Annual Formation Evaluation Research Consortium Meeting, Austin, Texas, August 14-15, 2008. Nolte, B., and Huang, X. J., 1997, Dispersion Analysis of Split Flexural Waves: Annual Report of Borehole Acoustics and Logging and Reservoir Delineation Consortia, Massachusetts Institute of Technology. Sarkar, T. K and Pereira, O., 1995, Using the Matrix Pencil Method to Estimate the Parameters of a Sum of Complex Exponentials: IEEE Antennas and Propagation Magazine, 37(1), pp. 48-55. Sava, H. P. and McDonnell, J. T. E., 1995, Modified Forward-Backward Overdetermined Prony’s Method and its Application in Modeling Heart Sounds, IEEE Proc.-Vis. Image Signal Process, 142(6), Dec 1995. Tang, X. M. and Cheng, A., 2004, Quantitative Borehole Acoustic Methods, Elsevier Ltd Press, ISBN: 0-08-044051-7.
REFERENCES
Aster R.C., Borchers, B., and Thurber, C. H., 2005, Parameter Estimation and Inverse Problems, Elsevier Academic Press Publications. ISBN: 0-12065604-3. Ekstrom, M. P., 1996, Dispersion Estimation from Borehole Acoustic Arrays Using a Modified Matrix Pencil Algorithm: 1996 IEEE Proceedings of ASILOMAR-29, pp. 449-453. Liu, D. H., Hu, W. L. and Chen, Z. J., 2008, SVDTLS Extending Prony Algorithm for Extracting UWB Radar Target Feature: Journal of Systems Engineering and Electronics, 19(2), pp. 286-291. Minami, K. and Kawata, S., 1986, Prony's Method Based on Eigenanalysis and Overdetermined System Approach: 1986 Proc. of IEEE Int. Conf. on ASSP, Tokyo, pp.1393-1396. Ma, J. and Torres-Verdín, C., 2008, Radial 1D Simulation of Multi-pole Sonic Waveforms in the Pres-
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SPWLA 51st Annual Logging Symposium, June 19-23, 2010
Yu, T., 1990, Improving the Accuracy of Parameter Estimation for Real Exponentially Damped Sinusoids in Noise, IEEE Proceeding, 137(3), Jun 1990. Zhang, X. D., 2002, Modern Signal Processing, Tsinghua University Press ISBN: 7302060037.
Robert K. Mallan received a BS in Physics from East Central University in Oklahoma in 1990, and a MS in Geophysical Engineering at the University of Arizona, Tucson, in 1995. During 1996-2003, he worked for ElectroMagnetic Instruments (Richmond, CA) on the development and application of borehole EM induction logging and magnetotelluric technologies. He is currently pursuing a PhD at the Department of Petroleum and Geosystems Engineering of The University of Texas at Austin. His research is focused on petrophysics based modeling and inversion of electromagnetic and sonic well logging data acquired in dipping and anisotropic formations. Email:
[email protected]
APPENDIX I: Prony Fitness Function
Having calculated the coefficients of the characteristic polynomial Ψ , the values of the polynomial at a scope of slowness for each discrete frequency are calculated. As the slowness is closer to the solution of the equation (3), the value of the polynomial Ψ will be smaller, while if the slowness is farther from the solution of equation (3), the value of the polynomial Ψ will be larger. The normalized Prony fitness function (Tang and Cheng 2004) is defined as: ⎧ FP ( f m , sk ) = exp(− Ψ ( Z ( f m , sk )) 2 ) ⎪ ⎨ for FEPM ⎧2π f m × sk , ⎪ Z ( f m , sm ) = ⎨ ⎩1 (2π f m × sk ), for BEPM ⎩
Carlos Torres-Verdín received a Ph.D. degree in Engineering Geoscience from the University of California, Berkerley, in 1991. During 1991–1997 he held the position of Research Scientist with Schlumberger-Doll Research. From 1997–1999, he was Reservoir Specialist and Technology Champion with YPF (Buenos Aires, Argentina). Since 1999, he has been with the Department of Petroleum and Geosystems Engineering of The University of Texas at Austin, where he currently holds the position of Zarrow Centennial Professor in Petroleum Engineering. He conducts research on borehole geophysics, formation evaluation, and integrated reservoir characterization, and is founder and director of The University of Texas at Austin’s Joint Industry Research Consortium on Formation Evaluation. Torres-Verdín has served as Guest Editor for Radio Science, and as Associate Editor for the Journal of Electromagnetic Waves and Applications, and the SPE Journal and is currently an Associate Editor for Petrophysics (SPWLA) and Geophysics. He is recipient of the 2003, 2004, 2006, and 2007 Best Presentation Award by Petrophysics, is recipient of the 2006 Best Presentation Award and of the 2007 Best Poster Award by the SPWLA, and is recipient of SPWLA’s 2006 Distinguished Technical Achievement Award and SPE’s 2008 Formation Evaluation Award. Email:
[email protected]
(1-I)
where sk ∈ S are the discrete slownesses, defined by
equation (13), f m ∈ [ f MIN , f MAX ] is the discrete frequency, and f MIN and f MAX are the minimum and maximum values of discrete frequency, respectively.
ABOUT THE AUTHORS Jun Ma received Master’s and Ph.D. degrees in Theoretical Physics from Jilin University in China in 1992 and 1998, respectively, with research direction in acoustic well logging in geophysics. He began to study acoustic well logging from 1989. His areas of expertise include acoustic logging theory and numerical simulation methods on 1D, 2D and 3D problems and signal processing. He has worked on his major for Jilin University for more than 15 years before he came to The University of Texas at Austin. He is currently a Research Associate, working on sonic logging associated problems with the Formation Evaluation group. Email:
[email protected]
Benjamin C. Voss holds a Master of Science degree in electrical engineering from the California Institute of Technology (2004). He has been with the Center for Petroleum and Geosystems Engineering at The University of Texas at Austin since 2006, where he works as a software developer, a research engineer, and petrophysicist. Email:
[email protected]
Pawel J. Matuszyk received MSc (2000) and PhD (2005) degrees in Computer Science from AGH – University of Science and Technology in Krakow, Poland. His main interests are numerical methods, especially hp-adaptive finite-element method in application to problems of solid and fluid mechanics. Email:
[email protected]
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