# Calculating Second-Order Tikhonov Regularization Parameter in Mathematica

I am trying to map slowness of underwater sound velocity in a river using some tomographic device. The location of each acoustic receiver/transmitter is shown in the picture below

To find the slowness m={S11,S12,S13, ...S112,...,S71,S72,S73, ... , S712} from the observations d={t_R1L1,t_R1L2,...,t_R4L4} I need to solve the regularized least-square equation min||Gm-d||^2+a^2||Lm||. I, therefore, established matrices

  G={{0, 12.1524, 2.7609, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15.6297, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 15.6297, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
5.65726, 9.97241, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15.6297, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 15.6297, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
14.3434, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 14.912, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 15.6283, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
15.6283, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4.50649, 11.1218, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 15.6283, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
15.6283, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11.0604, 2.88671, 0, 0, 0,
0, 0, 0, 0}, {0, 0, 19.1877, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
2.16882, 17.9406, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4.58072, 15.5287,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6.99261, 13.1168, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 9.40451, 10.7049, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
11.8164, 8.293, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14.2283, 0.877225,
0, 0, 0}, {0, 0, 17.3355, 7.70369, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
10.5772, 15.6647, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2.61622,
18.2809, 5.34477, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12.9361, 13.3058,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4.97514, 18.2809, 2.98585, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 15.295, 10.9468, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 7.33406, 14.2199}, {0, 0, 0, 0, 14.4946, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 9.24233, 9.34103, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2.41548,
16.1679, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18.5834, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 14.172, 4.41137, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7.34514,
11.2382, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17.054, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0}, {0, 0, 0, 0, 11.8019, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
15.1311, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15.1311, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 15.1311, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15.1311,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15.1311, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 13.5034, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 2.1671, 10.7843,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16.6048, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 7.58393, 9.02091, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16.6048,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9.34735, 7.25749, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 16.6048, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11.1108,
1.36223, 0, 0, 0}, {0, 0, 0, 0, 1.29176, 15.5556, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 5.29103, 16.3088, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
4.53791, 17.0619, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3.78479, 17.815,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3.03167, 18.5681, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 2.27856, 19.3212, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
1.52544, 16.2157}, {0, 0, 0, 0, 0, 0, 18.5392, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 19.4035, 2.06375, 0, 0, 0, 0, 0, 0, 0, 0, 0, 19.9029,
1.56439, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20.4023, 1.06503, 0, 0, 0, 0, 0,
0, 0, 0, 0, 20.9016, 0.565678, 0, 0, 0, 0, 0, 0, 0, 0, 0.433035,
20.9679, 0.0663213, 0, 0, 0, 0, 0, 0, 0, 0, 0, 19.7006, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 14.1187, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 16.3486, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9.75723,
6.5914, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16.3486, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 4.73928, 11.6094, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16.3486, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14.59, 0, 0, 0, 0, 0, 0, 0}, {0, 0,
0, 0, 0, 0, 1.65553, 11.4627, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
15.1901, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15.1901, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 15.1901, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15.1901,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15.1901, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 7.70533, 3.70495, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0.447913,
15.4943, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10.2405, 8.21959, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 17.5151, 0.944913, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 18.4601, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6.32976, 12.1303,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13.6044, 4.85561, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 15.1623}, {0, 0, 0, 0, 0, 0, 0, 0, 6.77971,
17.4671, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14.9997, 11.4365, 0, 0, 0, 0, 0,
0, 0, 0, 5.00342, 18.2163, 3.21656, 0, 0, 0, 0, 0, 0, 0, 0,
13.2234, 13.2128, 0, 0, 0, 0, 0, 0, 0, 0, 3.22712, 18.2163, 4.99286,
0, 0, 0, 0, 0, 0, 0, 0, 11.4471, 14.9891, 0, 0, 0, 0, 0, 0, 0, 0,
0, 17.4914, 6.76916, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0,
0, 0, 0, 0, 18.2098, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16.1093, 3.74475,
0, 0, 0, 0, 0, 0, 0, 0, 0, 13.0672, 6.78682, 0, 0, 0, 0, 0, 0, 0,
0, 0, 10.0251, 9.8289, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6.98307, 12.871,
0, 0, 0, 0, 0, 0, 0, 0, 0, 3.941, 15.913, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 17.7183, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0,
14.3166, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15.6093, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 15.6093, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9.15931,
6.45003, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15.6093, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 15.6093, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11.7252, 0,
0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 2.36708, 11.8831, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 15.5369, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15.5369,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14.6006, 0.936286, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 15.5369, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15.5369,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12.7613}}


and

L=Flatten[
Table[
M = ConstantArray[0, {7, 12}];,
M[[i, j]] = -4;
M[[i, j + 1]] = 1;
M[[i, j - 1]] = 1;
M[[i + 1, j]] = 1;
M[[i - 1, j]] = 1;
Flatten[M],
{j, 2, 11},
{i, 2, 6}],
1];


Given that {{U, V}, {Lamb,M}, X} = SingularValueDecomposition[{G, L}] , where G=U.Lamb.Conjugate[Transpose[X]] and L=V.M.Conjugate[Transpose[X]],

Why the generalized singular values of G and L, gama, become indeterminate or zero? And, how can I fix this issue?

gama=lambda/mu


where

lambda=Sqrt[Diagonal[Transpose[Lamb].Lamb]]
={0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.}


and

mu=Sqrt[Diagonal[Transpose[M].M]]
={0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.}

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