Take the 2-minute tour ×
Mathematica Stack Exchange is a question and answer site for users of Mathematica. It's 100% free, no registration required.

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

enter image description here

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.}
share|improve this question

migrated from math.stackexchange.com Dec 10 '13 at 9:43

This question came from our site for people studying math at any level and professionals in related fields.

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.