Content area
The method of regularization factor selection determines stability and accuracy of the regularization method. A formula of regularization factor was proposed by analyzing the relationship between the improved SVD and regularization method. The improved SVD algorithm and regularization method could adapt to low SNR. The regularization method is better than the improved SVD in the case that SNR is below 30 and the improved SVD is better than the regularization method when SNR is higher than 30. The regularization method with the regularization factor proposed in this paper can be better applied into low SNR (5<SNR) NMR logging. The numerical simulations and real NMR data process results indicated that the improved SVD algorithm and regularization method could adapt to the low signal to noise ratio and reduce the amount of computation greatly. These algorithms can be applied in NMR logging.[PUBLICATION ABSTRACT]
APPLIED GEOPHYSICS, Vol.8, No.3 (September 2011), P. 233 - 238, 3 Figures. DOI: 10.1007/s11770-011-0294-0
Study on algorithms of low SNR inversion of T2 spectrum in NMR*
Lin Feng1,2, Wang Zhu-Wen1, Li Jing-Ye1, Zhang Xue-Ang1, and Jiang Yu-Long1
Abstract: The method of regularization factor selection determines stability and accuracy of the regularization method. A formula of regularization factor was proposed by analyzing the relationship between the improved SVD and regularization method. The improved SVD algorithm and regularization method could adapt to low SNR. The regularization method is better than the improved SVD in the case that SNR is below 30 and the improved SVD is better than the regularization method when SNR is higher than 30. The regularization method with the regularization factor proposed in this paper can be better applied into low SNR (5<SNR) NMR logging. The numerical simulations and real NMR data process results indicated that the improved SVD algorithm and regularization method could adapt to the low signal to noise ratio and reduce the amount of computation greatly. These algorithms can be applied in NMR logging.
Keywords: nuclear magnetic resonance, T2 spectrum, singular value decomposition, regularization method
Introduction
Inversion of T2 spectrum in nuclear magnetic resonance logging is critical in handling a core part, and exactitude of inversion directly affects the accuracy of geological parameters calculated. Because the signal of NMR logging is weak, the Low signal to noise ratio can be prominently problematic. More precise inversion of the T2
spectrum at low signal to noise ratio (SNR) circumstances has been an important research topic of nuclear magnetic resonance logging.
In recent years, there has been progress on the research of multi-exponential inversion of NMR. A number of different inversions have been published in China and/or worldwide, including Butler et al.s penalty function method, Wang et al.s singular value decomposition (SVD), and etc. (Butler et al., 1981;
Wang et al., 2001) The SVD method which is one of the popular methods for treating the linear inversion problem, is subjected to high demand of signal to noise ratio. In Wang et al.s (2002, 2003) method, after adding the smoothing factor, the lower non-zero singular values are cut off in singular value decomposition of the model matrix in order to obtain steady relaxation spectra from low SNR well logging data. The Improved SVD algorithm proposed by Jiang et al. (2005) could reach the nonnegative constraint by using an iterative scheme and keep the continuity of T2 spectrum .Based on Dines and Lyttles ART algorithm (1979) Yao et al improved the SIRT algorithm and provide good results (Yao et al,, 2003). The regularization method used by Hansen (1990) can weaken the noise interference to some extent. In addition, some stochastic explorative algorithms were proposed by Tan et al. (2007) and Pan et al. (2008). However, the problem is still not completely solved.
*Manuscript received by the Editor April 31, 2010; revised manuscript received June 7, 2011.1. College of Geo-Exploration Science and Technology, Jilin University, Changchun, 130026, China;2. Faculty of Science, Jilin Institute of Chemical Technology, Jilin, 132022, China.
233
Algorithms of low SNR inversion
Lin et al. (2009) has reported an improved algorithm for SVD inversion of T2 spectrum elsewhere to solve this problem. In this paper, a formula for computing regularization factor is presented by the comparative study of SVD and regularization method, which can provide better results and have high computation efficiency when being applied to low SNR NMR logging.
Theory
1
diag .
This method can be applyed only when SNR 80. However, generally speaking, the SNR of current NMR logging data is less than 30, which is a limitation when applying Wangs SVD algorithm to real situations.
In low SNR, there is signicant difference between the solutions with different k. We can get the best k in each case of SNR by using the least square method. The best number of singular values retained is k in the following expression:
X
V
1
1
,
,
,
SNR
T
2
1
Improved SVD algorithm
The discrete linear model of relaxation signal of NMR can be written conci sely, using vector / matrix notation, as
Y = AX. (1)
where the matrix A is a combined n m matrix, X is the combined data vector with m elements of T2 amplitudes; Y is a set of measurements of echoes with elements. The singular value decomposition theorem in linear algebra states that any real nm matrix A (n m) can be written as the product of an orthonormal matrix U, an nm diagonal matrix W with positive or zero diagonal elements, and the transpose of an mm orthonormal matrix V, i.e.
A = UWVT,
where
UTU = I, VTV = I,
n
q
p
pq
n
(
min V .
Where xi is the ith component of the constructed spectrum, xki is the ith component of the spectrum that was inverted with k singular values. The best number of singular values retained corresponds to a range of condition number:
)
/
,
( 1
max
max
k
/ k Z
1 )
k x
x
2
d
d
n
k
i
ki
i
1
n
U ,
m
q
p
pq
,
ip d
U
G
,
1
d
Z
Z
Z .
The best conditions increase with increasing SNR. We can approximately consider tha t they are linear relationship in the range of SNR from 5 to 100. The author proposed the following formula.
b
a
SNR
condbest . (2)
Then an improved SVD algorithm can be written as:
Y
U
,0
,0,
iq
i
1
,
m
i
V ,
]
0
,
,
0
,
,
,
diag[
= 2
1
, k
ip d
V
G
,
1
d
iq
1
W .
The diagonal elements 0
2
Z
Z
Z
Z are the non-zero singular values of the matrix A.
If the small non-zero singular values are removed, the solution will become stable. The ratio of the maximum singular value to the minimum singular value reflects the ill extent of the ill-conditioned matrix, called matrix condition number, expressed as:
k
1 !
!
! k
Z
!
Z
X
V
1
1
diag .
Traditional SVD algorithm is a special case of this improved SVD with a = 1 and b = 0.
,
,
,
a
SNR
b
1
T
2
1
Z
Z /
/
cond 1
Z
Z
max .
It is clear that cutting out the small singular value can make the condition number smaller, hence the
solution will be stable. However, on the other hand, the more the singular values are removed, the more useful information will be lost. To select the appropriate number of singular values retained is crucial to the singular value decomposition method. Wang et al. (2001) proposed a method to cut off small singular value based on SNR. The singular values which less than the ratio of the maximum singular value and SNR are set to zero in this method. So the solution can be described as follows:
Y
U
,0
,0,
min
Regularization method
Regularization method that is also known as Phillps-Twomey method, Tikhonov-Miller regularization method
234
Lin et al.
transforms equation (1) into the following form
(ATA + H) X = AT Y
Where the constant is the regularization factor. H is a constraint matrix. When H is a unit matrix, the regularization method called zero-order regularization has its simplest form:
(ATA + I) X = ATY.
The method of regularization factor selection denitely affects the stability and accuracy of the regularization method. It is very difficult to choose the optimal regularization factor, because regularization factor is affected by integral kernel, shape of solution, noise of data, etc. Although the method of calculating the optimal a has been reported (Hansen 1992), these methods is difcult to be adopted in the practical application. As a most closely with the SNR, so an apriori function of a and SNR is commonly established by test. Such as the following apriori function (Weng et al., 2001):
3
.
1
2
Z
D . (3)
It can get a good solution without iterative optimization that the optimal regularization factor is found by using the formula (3). The problem that iterative optimization is too slow to practical application will be avoided. The parameters a and b in the formula (3) can be tted by numerical experiments. First, we construct some echoes with different signal to noise. Then we do inversion with different condition number. The best condition number which make the inversion be most accurate can be found by using the least square method. Finally, we do linear tting between the best condition number and SNR to get the best value of a and b. The numerical experiments have shown that the result of inversion will be best by using a = 1.45 and b = 16 when the echo number is 500 and the T2 spectral number is 32.
1
opt SNR
| b
a
Numerical experiments
D .
By analyzing the link of the regularization method and singular value decomposition, we can get a formula of regularization factor.
Assume the singular values of the matrix A are (1,
2, , n), the singular values of ATA are (12, 22, , n2) and the singular values of ATA + I are (12 + , 22 + , , n2 + ). The condition number of ATA + I is (12 + ) / (n2 + ). Because 12 >> >> n2, the condition number of ATA + I is approximately equal to 12 / .
Select the best value of that is to select the optimal condition number in fact. Therefore, the essence of regularization method and SVD are identical.
By applying equation (2), regularization factor can be linked to SNR.
The condition number of ATA is
D
Z /
cond
cond
2 1
opt SNR
100
The 32 T2 spectral values are logarithmically spaced between 0.3 and 3000 ms. The 500 echoes are sampled at an echo spacing of 1.2 ms. By adding normal random noise, we construct some echoes with different signal to noise ratio used for inversion.
The algorithm proposed in this paper compared to SVD algorithms
We constructed a typical double-peak T2 spectrum. Then three methods were respectively used to do inversion with different SNR. They were the regularization method with factor in equation (3) and improved SVD (Lin et al., 2009) and traditional SVD algorithm (Wang et al., 2001). The comparison of such solutions is shown in Figure 1.
Clearly, three kinds of algorithms were similar to inversion in high SNR cases. But when SNR less than 30, traditional SVD does not reect the bimodal structure
A
AAT ,
so
2 |
235
Algorithms of low SNR inversion
The constructed spectrum. Solution of the regularization method proposed in this paper. Solution of the improved SVD. Solution of traditional SVD. Fig.1 Comparative results of different SNR. (a) SNR = 100. (b) SNR = 50. (c) SNR = 30. (d) SNR = 20. (e) SNR = 10. (f) SNR = 5.
of the constructed spectrum. The improved SVD algorithm or the regularization method proposed in this paper is able to reect the structure of the double-peak structure of spectrum clearly, even if the SNR is only 5. The algorithm proposed in this paper is more suitable for the actual needs of NMR logging data processing.
Despite the improved SVD algorithm and the
regularization method proposed in this paper is essentially the same, and Figure 1 shows little difference between the results of two methods. It is clear that in Table 1 the regularization method is better than the improved SVD in the case that SNR is below 30 and the improved SVD is better than the regularization method in the case of SNR higher than 30.
Table 1 Sum of squares of deviations
SNR The regularization method proposed in this paper
The improved SVD Traditional SVD
100 1.23687 0.646061 0.91842 50 1.69189 1.00689 1.00689 30 3.03813 1.2174 12.9107 20 4.71819 13.1455 23.8545 10 7.79876 14.4016 25.64 5 10.4307 20.1493 25.6272
Test of the regularization factor proposed in this paper
We also constructed single-peak spectrum, double-peak spectrum, triple-peak spectrum as shown in Figure 2. The best regularization factors can be identied by using dichotomy at every SNR. One can use Table 2 to compare the empirical regularization factor proposed in this paper with the best regularization factors and the empirical regularization factor proposed in Weng (2001). As the
empirical regularization factor was a decision made only by SNR without considering other factors, empirical regularization factor is different from the best factors. Lower SNR leads to greater deviation. More complex patterns of T2 spectrum causes greater deviation. It is clearly shown in Figure 3 that the empirical regularization factor proposed in this paper is closer to the average value of the best factors in low SNR. Therefore, the regularization method proposed in this paper is better for low SNR when T2 spectrum is unknown.
The constructed spectrum. Solution of the regularization method proposed by Weng (2001).+ Solution of the regularization method proposed in this paper. Solution of the best regularization factor.
Fig.2 Simulation of relaxation distribution of different shape and comparative results of different regularization factor.
(a) Single-peak. (b) Double-peak. (c) Triple-peak.
236
Lin et al.
Table 2 The best regularization factor and Empirical regularization factor
SNR The best regularization factor of single-peak
The best regularization factor of double-peak
The best regularization factor of triple-peak
The empirical regularization factor proposed in this paper
The empirical regularization factor proposed by Weng (2001)
5 5.10724 4.00511 2.85838 4.757202 12.34068 10 2.74064 1.99298 1.35248 2.76438 5.011872 20 1.59322 1.06343 0.69452 1.269909 2.035453 30 1.1674 0.736724 0.492705 0.726379 1.201551 40 0.936668 0.570653 0.389157 0.469606 0.826651 50 0.655576 0.46876 0.324916 0.32833 0.618499
References
The essence of regularization method and SVD are identical, because both of them find stable solution by selecting the optimal condition number. The key to achieve good effect is to select the best condition number.
The regularization method is better than the improved SVD in the case of SNR below 30 and the improved SVD is better than the regularization method in the case of SNR higher than 30.
We found that the empirical regularization factor proposed in this paper is closer to the average value of the best factors in low SNR by comparative test. Comparison results suggest that the regularization method with the regularization factor proposed in this paper is better for low SNR when T2 spectrum is unknown.
How to accurately inverse the structure of T2 spectrum with complex structure in low SNR still needs further study.
Butler, J. P., Reeds, J. A., and Dawson, S. V., 1981,
Estimating solutions of rst kind integral equations with nonnegative constraints and optimal smoothing: SIAMJ Numer Anal, 18(3), 381 397.
Chakrabarty, T., Longo, J., 1997, A new method for mineral quantification to aid in hydrocarbon exploration and exploitation: Journal of Canadian Petroleum Technology, 36(11), 15 21.
Dines, K., and Lyttle, J., 1979, Computerized Geophysical
Tomography: Proc. IEEE, 67, 1065 1073.
Hansen, P. C., 1990, The discrete picard condition for discrete ill - posed problems: BIT, 30, 658 672. Hansen, P. C., 1992, Analysis of discrete Ill-posed problems by means of the L-curve. SIAM Review, 34(4), 561 580.
Jiang, R. Z., Yao, Y. P., Miao, S., and Zhang, C. S. 2005,
Improved algorithm for singular value decomposition inversion of T2 spectrum in nuclear magnetic resonance: Acta Peolei Sinica, 26(6), 57 59.
Liao, G. Z., Xiao, L. Z., Xie, R. H., Fu, J. J., 2007,
Influence factors of multi-exponential inversion of NMR relaxation measurement in porous media. Chinese Journal of Geophysics (in Chinese), 50(3), 932 938. Lin, F., Wang, Z. W., Liu, J. H., Ding, Y., and Li, C. C.,
2009, An Improved Algorithm for Singular Value Decomposition Inversion of T2 Spectrum in Nuclear Magnetic Resonance: Journal of Jilin University (Earth Science Edition), 39(6), 1150 1155.
Pan, K. J., Chen, H., and Tan, Y. J., 2008, Multi-exponential inversion of T2 spectrum in NMR based on differential evolution algorithm: Acta Physica Sinica, 57(9), 5956 5961.
Tan, M. J., Shi, Y. L., and Xie, G. B., 2007, NMR T2
Distribution Inversion Based on Genetic Algorithm. Well Logging Technology, 31(5), 413 416.
Timur, A., 1969, Producible porosity and permeability
The best regularization factor. The empirical regularization factor proposed by Weng (2001). + The empirical regularization factor proposed in this paper.
Fig.3 The relationship between the regularization factor and SNR.
Conclusions
237
Algorithms of low SNR inversion
of sandstone investigated through nuclear magnetic resonance principles: Journal of Petroleum Technology, 21, 775 786.
Wang, C. Z., and Li, N., 2002, On the method of analyzing and processing T2 relaxation spectra from NMR log data and relative program design on cif2000 platform: Well Logging Technology, 26(5), 360 363.
Wang, C. Z., and Shang, W. Z., 2003, Application of singular value decomposition in solution of T2 relaxation spectra from nuclear magnetic resonance log data: Oil Geophysical Prospecting, 38(1), 91 94.
Wang, W. M., Li, P., and Ye, C. H., 2001, Multi-exponential inversion of nuclear magnetic resonance relaxation signal: Science in China, Series A, 31(8), 730 736. Wang, X.W., Xiao, L. Z., Xie, R. H., and Zhang, Y. Z.,
2006, Study of NMR porosity for terrestrial formation in China. Science in China (Series G) (in Chinese), 36(4), 366 374.
Weng, A. H., 2001, Study on data processing theories and experiments of NMR logging: PhD thesis Jilin
University, Changchun.
Xiao, L. Z., 1998, NMR Image Logging and NMR in Rock
Experiments (in Chinese): Science Press, Beijing.
Yao, X. G., and Wang, Z. D., 2003, A new Inversion
Algorithm for NMR Rela xation Distribution. Well Logging Technology, 27(5), 373 376.
Lin Feng is a associate professor of Jilin Institute of
Chemical Technology. He is studying for his doctor degree at College of Geo-Exploration Science and Technology of Jilin University (Jilin). His research work mainly focuses on the new method of NMR logging.
238
Editorial Office of Applied Geophysics and Springer-Verlag Berlin Heidelberg 2011