Comparison of Propagator Decomposition in Seismic ...... 9] Gabor, D., 1946, Theory of Communication, J. of the Institute of Electrical Engineers 93 (III), 429-457.
Comparison of Propagator Decomposition in Seismic Imaging by Wavelets, Wavelet-Packets, and Local Harmonics Ru-Shan Wu and Yongzhong Wang
Institute of Tectonics, University of California, Santa Cruz, CA 95064
ABSTRACT Kirchho migration operator is a highly oscillatory integral operator. In our primary work [1] (Wu and Yang, 1997), it has been shown that the matrix representation of Kirchho migration operator for homogeneous background in space-frequency domain is a dense matrix, while the compressed operator in beamlet-frequency domain, which is the wavelet decomposition of the Kirchho migration operator, is a highly sparse matrix. Using the compressed matrix for imaging (beamlet migration), we can retain the wide e ective aperture of a full-aperture operator, and hence achieve higher resolution and image quality with reduced computational cost. However, as well known, wavelets work best for zero-frequency stationary signals. But, for the Kirchho migration operator, it is both space-varying in the near- eld region and high-frequency stationary in the far- eld zone. Therefore, wavelets are not very ecient for this kind of operator. In this research, we rst summarize the results of maximum sparsity adapted wavelet-packet transform (MSAWPT) for the decomposition and compression of Kirchho migrator [2] (Wang and Wu, 1998), and then further study the decomposition and compression of Kirchho migration operator by local harmonics (i.e., local cosines/sines). It was found in [2] (Wang and Wu, 1998) that the MSAWPT can generate a more ecient representation for the imaging operator than the standard discrete wavelet transform (DWT) and the compression capability of MSAWPT is much greater than that of DWT. In this paper, we also observed that for low frequency operator, the compression capability of uniform local cosine bases is equivalent to that of standard wavelets and is weaker than that of adapted wavelet-packets; while, for high frequency operator, uniform local cosine bases are more powerful than both the standard wavelets and adapted wavelet-packets. Furthermore, for local cosine transform, a good parameter setting (i.e., type and smoothness of the bell function, edge extension, overlapping radius, folding style, and window size) can generate a higher compression ratio. Keywords: Adapted Wavelet-Packet Transform, Local Cosine Transform, Wavelet Transform, Operator Decomposition, Operator Compression, Seismic Imaging, Kirchho Migration
1 INTRODUCTION The main computational burden of seismic imaging is the backpropagation calculation. So, one wishes to get highly sparse matrix for migration operators to maximumly improve the computation eciency. Kirchho migration operator is a highly oscillatory integral operator. In our primary work [1] (Wu and Yang, 1997), we have shown that the matrix representation of Kirchho migration operator for homogeneous background in spacefrequency domain is a dense matrix, while the compressed beamlet-operator, which is the wavelet decomposition of the Kirchho migration operator, is a highly sparse matrix. Using the matrix in beamlet domain for propagation, we can get the decomposition of the wave eld into multi-scale beams in di erent locations with di erent
propagating directions. Using the compressed matrix in beamlet domain for imaging, we can obtain high quality images with high eciency. However, as well known, wavelets work best for zero-frequency stationary signals. But, for the Kirchho migration operator, it is both space-varying in the near- eld region and high-frequency stationary in the far- eld zone. Therefore, wavelets are not very ecient for this kind of operator. This motivates us to further study the operator decomposition and compression by other multiresolution schemes. In [2] (Wang and Wu, 1998), the adapted wavelet-packet transform (AWPT) for migration operator decomposition has been explored and a new maximum sparsity AWPT algorithm was introduced. It is found that AWPT can generate a more ecient matrix representation of Kirchho operator and therefore achieve greater compression capability than the standard DWT. In this paper, we rst summarize the results of AWPT and then further study the decomposition and compression of Kirchho migration operator by local harmonics (i.e., local cosines/sines). A comparison among the standard DWT, AWPT and the local harmonics decompositions of the migration operator is done in the section follows and conclusions are made in the nal section.
2 TRANSFORM METHODS 2.1 Basic Principle for Operator Decomposition The Kirchho integral can be expressed as [3]
Z @u g ? u @g ; u(z ) = ds @n @n s
(1)
where u(z ) is the extrapolated wave eld for forward problem, or the image function for inverse (imaging) problem, s is the surface for 3D problem, and the curve for 2D problem, on which the wave eld u (pressure) is known, n is the outgoing normal of the surface (curve), and g is the Green's function. For the sake of simplicity, we use the Kirchho integral on a at surface (or a straight line). In this case, the Kirchho integral can be reduced to a Rayleigh integral Z @g : u(z ) = ?2 dsu(z0 ) @n
s
(2)
First, we discuss the forward propagation problem in a homogeneous background. For the 3D case, the Green's function in space-frequency domain can be written as
g = e4r ;
(3)
g = 4i H0(1) (kr):
(4)
ikr
+
and for the 2D case, it becomes
p
In the above equations, i = ?1, k = !=v0 with ! as the circular frequency and v0 as the wave propagation speed, and r is the distance between the source and the receiver. H0(1) is the zero-order rst kind of Hankel function. If we replace the Green's functions in above formulae by their conjugates, the surface or curve integrals will be changed to imaging (backpropagation) integrals. Kirchho integral operator can be written as the following matrix equation
U (z ) = PU (z0); (5) T where U (z0 ) is the known wave eld on the plane at depth z0 , U (z ) = (U1 (z ); U2(z ); ; UN (z )) is the wave eld on the plain at depth z with Ui (z ) = U (xi ; z; !); i = 1; 2; ; N , and N is the number of sampling points along
x-direction. Applying an orthogonal transform (represented by the operator W ) to Eq. (5), we have
W [U (z )] = W [PU (z0)] :
(6) In this paper, W is selected as orthogonal local cosine transform (In our primary work [1] (Wu and Yang, 1997), W is selected as orthogonal wavelet transform; in [2] (Wang and Wu, 1998), W is selected as orthogonal adapted wavelet-packet transform). Because of the orthogonality of W , we have W T = W ?1 , i.e., the transpose of W is equal to the inverse of W . So, Eq. (6) can be rewritten as i.e., in beamlet domain
where
and
WU (z ) = WPW T WU (z0);
(7)
U~ (z ) = P~ U~ (z0 );
(8)
U~ (z ) = WU (z );
(9)
U~ (z0 ) = WU (z0 );
(10)
P~ = WPW T :
(11) Eq. (11) is the two-dimensional wavelet transform, adapted wavelet-packet transform or local cosine transform of migrator P , i.e., P~ is the decomposed matrix in beamlet domain of P . Eq. (8) is the migration process (imaging) in beamlet domain, where U~ (z ) and U~ (z0 ) are wavelet transforms, adapted wavelet-packet transforms or local cosine transforms of wave elds on the planes at depth z and z0 , respectively. In the case of imaging, U~ (z ) is the corresponding image eld. The image in space domain can be obtained by applying the corresponding inverse transform to the image in beamlet domain: h
i
U (z ) = W ?1 U~ (z ) :
(12)
2.2 Maximum Sparsity Adapted Wavelet-Packet Transform (MSAWPT) 2.2.1 Wavelet Packet Basis [4]
Wavelet packet basis is the generalization of the compactly supported wavelet basis constructed by Daubechies [5]. It can still maintain all the advantages of the standard wavelet basis. Its remarkable feature is that it has the
exibility of further decomposing optimally the signal or operator in the high- frequency bands. Let '(x) stand for the scaling function and (x) stand for the corresponding wavelet in the standard wavelet transform, then, we de ne
W0 (x) = '(x); W1 (x) = (x):
First, for the xed scale, we express the wavelet-packets generated by '(x) as follows
(13) (14)
p
W2f (x) = 2
X
p
W2f +1 (x) = 2
k
hk Wf (2x ? k);
X
k
(15)
gk Wf (2x ? k);
(16)
f = 0; 1; 2;
where gk and hk are the high-pass and low-pass lters, respectively, and satisfy X
n
hn?2k hn?2l = kl ;
X
k
p
hk = 2; gk = (?1)k h1?k :
Second, for the variable scale, wavelet-packets are characterized by three attributes: position p, scale s, and frequency f , we can write them as
Wpsf (x) = 2?s=2 Wf (2?s x ? p); p 2 Z; s 2 Z; f 2 Z+ :
(17)
To see more clearly the decomposition characteristics of wavelet-packets, we de ne the family of subspaces in
L2 (R) :
It is obvious that
sf = spanfWpsf (x) : p 2 Z g; s 2 Z; f 2 Z+ :
(18)
s0 = Vs ; s1 = Ws ; where Vs , Ws is the scaling function space and wavelet space at scale s, respectively. From Vs = Vs+1 Ws+1 , it follows that s0 = s0+1 s1+1 ; moreover, we have s+1
sf = s2+1 f 2f +1 ;
furthermore,
(19)
Ws = s = s s = s s s s = = sk k sk k sk+1k ? : 1
+1 2 +2 4
+1 3 +2 5
+ 2
+ 2 +1
+2 6
+2 7
+ 2
1
From the above formulae, we see that wavelet-packet spaces sf represent a further decomposition of the wavelet space Ws . This characteristic implies that wavelet-packets are more powerful to capture the detailed information of signal or operator than standard wavelets. The standard wavelet transform can be regarded as a special case of wavelet-packet transform.
2.2.2 MSAWPT A wavelet-packet tree constitutes an overabundant set of basis functions such that a great number of subsets can form orthonormal bases. However, not all the bases are ecient for getting the maximum sparsity for operator decomposition. Therefore, we construct a new algorithm to pick out the \best basis" from all the possible wavelet packets based on a cost functional.
To search for the wavelet-packet best basis to achieve maximum sparsity for the operator (matrix), a cost functional is de ned to evaluate the sparsity of a matrix in the speci ed basis. There are several kinds of cost functionals appeared in the literature [6]. Here, we use the so-called \THRESHOLD" [6] as the cost functional, i.e., cost functional e(P )=the number of coecients whose absolute values exceed a threshold , where P is the transformed matrix. Our MSAWPT [2] (Wang and Wu, 1998) can be described schematically as follows:
e(Pfs )
??
?@
s
e(P2sf+1 )
s
@@
s
e(P2sf+1+1 )
Figure 1: Cost functionals at the parent node and the two corresponding child nodes. Decision will be made on whether the matrix should be further decomposed at the node (s; f ) based on these cost functionals. First, we begin the decomposition of the operator (matrix) from the root of decomposition tree for each row and calculate the cost functional at every stage. If the cost functional at the parent node (s; f ) is greater than the summation of the cost functionals at the corresponding two child nodes (s + 1; 2f ) and (s + 1; 2f + 1), i.e. (see Fig. 1), e(Pfs ) > e(P2sf+1 ) + e(P2sf+1+1 ), then we continue decomposing the matrix, otherwise, we stop at the parent node (s; f ); , and so on. During the searching, if the decomposition stops at some nodes, these nodes obviously consist of the wavelet-packet coecients in the best basis and once the MSAWPT is carried out, the best basis is found and also the wavelet-packet transform using the best basis is completed. After the above algorithm has been implemented for rows of the original matrix in space-frequency domain, the best row basis has been found. In this paper, because of the symmetry inherent to the Kirchho migration operator, we assume the best column basis to be the same as the best row basis. Second, we perform the wavelet-packet transform to the columns of the row-transformed matrix in the speci ed best column basis. Note that the best basis found by MSAWPT is only a locally optimum basis, while, this new algorithm can also automatically nd the best decomposition depth (level). The globally best basis can be searched using the Coifman-Wickerhauser's best basis pursuit algorithm [7]; however, such an algorithm requires signi cantly more computation time and memory, and it searches the best basis only if the decomposition depth (level) is given. As for the computation complexity of MSAWPT, as we have observed in our numerical tests, the real operator decompostion complexity of MSAWPT is around O(N 2 ).
2.3 Local Cosine Transform (LCT)
2.3.1 Brief Description of Local Cosine Basis Local cosine bases were constructed by Coifman and Meyer in 1991 [8]. The local cosine transform has much in common with the windowed or short time Fourier Transform ( WFT or STFT) [9] which corresponds a decomposition into phase space atoms. The Coifman-Meyer local cosine basis [8, 10] involves an arbitrarily smooth bell function. Like wavelet-packet, local cosine basis can be characterized by position p, scale s, and wavenumber k as follows
psk (x) =
r
1 x?p 2 2s b(x)cos k + 2 2s
(20) h
i
where b = b(x) is a bell function which is a smooth function supported in the compact interval ? ; + with + ? . This interval contains [ ; ] which will be called as its nominal support . So, we can control the nominal window width by scale index s, the nominal left endpoint of the window by position index p, and the wavenumber by index k. It is worth noting that the odd half-integer wavenumbers are used in the cosine function, the goal of this action is to change LCT into the fast DCT-IV. The symmetry properties of the bell function, i.e., 0
0
(
b(x)2 + b(2 ? x)2 = 1; x 2 h[ ? ; + ] ;i b(x)2 + b(2 ? x)2 = 1; x 2 ? ; + ; 0
(21)
0
ensure orthogonality and make fast algorithm possible. So, all of these local cosine packets from Eq. (20) form an orthonormal basis for L2 (R) space. Two-dimensional local cosine basis can be generated by the tensor products px sx kx (x) py sy ky (y) and their nominal supports are the cartesian product rectangles of the nominal supports of the x and y factors.
2.3.2 Implementation A. Bell Function
Generally, bell function over I = [ ; ] is de ned by
b(x) = S (x ? )C (x ? ); x 2 R; ? and C (x ? ) cos n x? . In this paper, we select
(22)
0
where S (x ? ) sin n
? x?
8
1 : 2 And n (x) = n?1 (sin 2 x). Using induction we can show that S (x ? ) has 2n ? 1 vanishing derivatives at x = ? and x = + , and C (x ? ) has 2n ? 1 vanishing derivatives at x = ? and x = + . So, we say S (x ? ) or C (x ? ) is an arbitrarily smooth cuto . 0
0
0
0
B. Folding
Rather than calculating inner products with the sequences psk , we can preprocess data so that the standard fast DCT-IV algorithm may be used. This may be realized by folding the overlapping parts of the bells back into the intervals. This can be implemented by folding the signal f across , onto the intervals [ ? ; ) and ( ; + ], using the bell function b(x) de ned above, we can set
f+ (x) = b(x)f (x) + b(2 ? x)f (2 ? x); if < x + ; f? (x) = b(2 ? x)f (x) ? b(x)f (2 ? x); if ? x < : Then the folding replaces the signal f = f (x) with the left and right parts, i.e., f?(x) and f+ (x). Unfolding reconstructs f (x) from f+ (x) and f? (x) by the following formulae: f (x) =
b(x)f+ (x) ? b(2 ? x)f? (2 ? x); b(x)f+ (2 ? x) + b(2 ? x)f? (x);
if < x + ; if ? x < :
(24)
(25)
C. Edge Extension
When the bell shifts to the leftmost endpoint or the rightmost endpoint of the signal, we can NOT directly obtain f+(x) or f? (x) from the above formulae because of the lack of data in the leftmost or rightmost overlapping zone. Usually, we have four extension ways: (1) zero-extension; (2) symmetry-extension; (3) smoothnessextension; (4) periodization-extension. In this work, based on the features of imaging propagators, we choose the zero-extension. D. DCT-IV After the folding, we can apply the fast DCT-IV to the well-prepared data to obtain the local cosine transform coecients.
3 COMPARISON OF NUMERICAL RESULTS In this section, we show the decomposition and compression results of the Kirchho migration operator using di erent transform methods, i.e., DWT, MSAWPT and LCT. As a reference, Fig. 2 shows the matrix representations of a full-aperture migrator in a homogeneous medium in space domain for di erent frequencies (left: 5:9Hz and right: 25Hz ). This matrix will migrate a wave eld of 128 points with sampling interval x = 25m to a depth of z = 25m. The velocity of wave propagation in the homogeneous medium is 2000m=s. From Fig. 2, we see that the Kirchho migration operator P in space domain is a dense matrix, i.e., the o -diagonal elements fall o slowly when moving away from the diagonal. Kirchoff: 5.9Hz, real
Kirchhoff: 25Hz, real
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Figure 2: Matrix representations of Kirchho migration operators in space domain: dx = dz = 25m; v = 2000m=s; N = 128. Left: 5:9Hz , Right: 25Hz . Only real parts of the complex operators are plotted. We choose Daubechies 4 (i.e., Daub4, 4 vanishing moments, compact support= [0; 7]) and Coi et 5 (i.e., Coif5, 10 vanishing moments, compact support= [0; 29]) as the mother wavelets for all transformations. Their scaling functions and corresponding wavelets are shown in Fig. 3. From Fig. 3, we can see that Daub4 is short and sharp, while Coif5 is long and smooth; moreover, Coif5 is more symmetrical than Daub4. The decompositions of Kirchho migration operators for di erent frequencies using di erent mother wavelets by DWT and MSAWPT are compared in Figs. 5-6, and the corresponding best decomposition trees, which is quite
Daub4 scaling function
Daub4 wavelet
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Figure 3: Scaling functions (left) and wavelets (right) : Daub4 (top) and Coif5 (bottom).
Figure 4: Best wavelet-packet decomposition trees for 5:9Hz (left) and 25Hz (right) for Daub4 (top) and Coif5 (bottom).
DWT, Coif5, 5.9Hz, real
MSAWPT, Coif5, 5.9Hz, real
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Figure 5: Comparison of matrix representations of Kirchho migration operators in beamlet domain (Coif5). The left panels are for DWT, and the right panels are for MSAWPT. The top panels are for 5:9Hz , and the bottom panels are for 25Hz . Only real parts of the complex operators are plotted.
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Figure 6: Comparison of matrix representations of Kirchho migration operators in beamlet domain (Daub4). The left panels are for DWT, and the right panels are for MSAWPT. The top panels are for 5:9Hz , and the bottom panels are for 25Hz . Only real parts of the complex operators are plotted.
MSAWPT, Coif5, 5.9Hz, real
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Figure 7: Matrix representations of the complex Kirchho migration operators in MSAWPT domain (Coif5). The left panels are for real parts, and the right panels are for imaginary parts. The top panels are for 5:9Hz , and the bottom panels are for 25Hz .
Uniform LCT, 5.9Hz, real; N=128
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Figure 8: Matrix representations of the complex Kirchho migration operators in Uniform LCT (ULCT) domain with N = 128. The left panels are for real parts, and the right panels are for imaginary parts. The top panels are for 5:9Hz , and the bottom panels are for 25Hz .
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Figure 9: Matrix representations of the complex Kirchho migration operators in ULCT domain with N = 256. The left panels are for real parts, and the right panels are for imaginary parts. The top panels are for 5:9Hz , and the bottom panels are for 25Hz .
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Figure 10: Matrix representations of the complex Kirchho migration operators in ULCT domain with N = 512. The left panels are for real parts, and the right panels are for imaginary parts. The top panels are for 5:9Hz , and the bottom panels are for 25Hz .
di erent from the standard DWT decomposition tree, are shown in Fig. 4. In the implementation of MSAWPT, we choose = one percent of maximumly absolute wavelet-packet coecient as the threshold in cost functionals. Fig. 5 is the matrix representations of Kircho migrators in Coif5 by DWT (left) and MSAWPT (right) with the corresponding decomposition tree in Fig. 4 (lower two panels). We see that all the matrices in beamlet domain are much sparser than those in space domain (see Fig. 2) regardless of DWT or MSAWPT. Moreover, for low frequency (5:9Hz ), the upper-left part with high oscillating components, where it is not compressed well by DWT, can be automatically re ned by MSAWPT; and for high frequency (25Hz ), the lower-right part with high oscillating components, which is not compressed well by DWT, can be also automatically re ned by MSAWPT. The re nement in appropriate region makes the energy more concentrated and therefore the operator matrix sparser than the standard DWT decomposition. That means the operator can be represented more eciently by MSAWPT. The same phenomena can be also observed from Fig. 6, which is the decomposition comparison between DWT and MSAWPT using Daub4. However, the interscale coupling between large and small scales for Daub4 is obviously stronger than that for Coif5 even in wavelet-packet best bases. More careful observation shows that for high frequency (25Hz ), the sparsity of Kirchho operator is always worse than that for low frequency (5:9Hz ) regardless what basis (Daub4 or Coif5) or what transform (DWT or MSAWPT) is used. This means an ecient representation for operator also depends on the properties of the operator itself. Kirchho migrator is a complex operator. We decompose its real part and imaginary part independently. Fig. 7 shows the decompositions of both real parts and imaginary parts for 5:9Hz and 25Hz in Coif5 by MSAWPT. We see that the imaginary parts are decomposed more eciently. Notice that the Kirchho operator is complex, while the transform operator (see Eq. (6), represented by W) is real. According to Eq. (6), to make the inverse transform of the image function in beamlet domain valid, we have to assume the best bases to be the same for both real parts and imaginary parts. From many numerical tests, we nd that imaginary-part matrices can be better sparsi ed by MSAWPT than real-part ones. Therefore, we select the best basis for the imaginary-part matrix as the decomposition basis for the real-part matrix. The decompositions of Kirchho migration operator for di erent frequencies (Fig. 8, Top: 5:9Hz , Bottom: 25Hz ) by local cosine transform (here, only uniform LCT is used) are shown in Fig. 8. It is shown that the major energy of the operator concentrates in the neighboring region of the diagonal and these matrices are all highly sparse. However, the matrices for imaginary parts seem to spread out wider than the matrices for real parts. In Fig. 9 and Fig. 10, where the number of sampling points is N = 256 and N = 512, respectively, similar results can be observed. In implementing LCT, we select the cut-o function from [11] ( also, see formula (23) ) to construct the bell function, the overlapping radius is one half of the nominal support length, and the xed folding, zero-padded extension are used. In Figs. 8-10, the nominal support size of the row/column is one-eighth of the operator length. Furthermore, from many numerical tests, we have found that a good parameter setting of LCT can generate a higher compression ratio. Table 1: Comparison of compression ratios for di erent frequencies and di erent mother wavelets: DWT vs. MSAWPT. Wavelet Type Frequency DWT MSAWPT Daub4 5.9Hz 19.87 24.88 25Hz 6.43 7.56 Coif5 5.9Hz 36.36 61.57 25Hz 7.84 17.08 We obtain the compressed matrix with a cut-o level of one percent of the maximal coecient in absolute value. The comparison of compression ratios between DWT and MSAWPT for di erent frequencies and di erent mother wavelets is shown in Table 1, and Table 2 shows the comparison of compression ratios among uniform
Table 2: Comparison of compression ratios for di erent frequencies and di erent operator lengths : Uniform LCT vs. DWT, MSAWPT. Here, the results of DWT and MSAWPT are based on Coif5. Frequency Uniform LCT DWT MSAWPT 5.9Hz (N=128) 35.73 36.36 61.57 25Hz (N=128) 21.31 7.84 17.08 5.9Hz (N=256) 128.76 124.09 239.32 25Hz (N=256) 86.29 34.68 39.52
LCT, DWT and MSAWPT for di erent frequencies and di erent operator lengths. From Table 1, we see that the compression ratios by MSAWPT are much higher than those by DWT in any cases, and from Table 2, regardless of N = 128 or N = 256, it can be observed that for low frequency (5:9Hz ), the compression capability of local cosine bases is equivalent to that of standard wavelets and is weaker than that of adapted wavelet-packets; however, for high frequency (25Hz ), local cosine bases are more powerful than both the standard wavelets and adapted wavelet-packets (the compression ratio of the entire complex operator is de ned as the arithmetic mean of the compression ratios of real part and imaginary part).
4 CONCLUSIONS In this paper, we study the decomposition and compression of Kirchho migration operator by the uniform LCT and compare with the adapted wavelet-packet transform and the standard DWT. From the numerical results, it is found that the MSAWPT can generate a more ecient decomposition of Kirchho operator than the DWT, and the compression ratios by MSAWPT are much higher than those by DWT. MSAWPT can automatically search the high oscillating components and further decompose these components along the tree to achieve the maximally possible sparsity. It can adaptively pursue a locally optimum basis and nd the best decomposition depth (level) with the computation eciency O(N 2 ), which is the same as that of 2-D DWT. As for uniform LCT, in low frequency case, the compression capability of uniform local cosine bases is equivalent to that of standard wavelets and is weaker than that of adapted wavelet-packets; while in high frequency case, uniform local cosine bases are more powerful than both the standard wavelets and adapted wavelet-packets. Furthermore, for local cosine transform, good parameter settings (i.e., type and smoothness of the bell function, edge extension, overlapping radius, folding style, and window size) can generate higher compression ratios.
5 ACKNOWLEDGEMENTS This research has been funded by the WTOPI (Wavelet Transform On Propagation and Imaging for seismic exploration) Consortium at University of California, Santa Cruz. The authors are very grateful to the sponsors. The authors would also like to thank Dr. Hai Deng at the University of Texas, Austin for the valuable discussion. The facility support from the W. M. Keck Foundation is acknowledged, too.
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