Megameter propagation and correlation of T-waves from Kermadec Trench and Islands

On 18 June 2020 and 4 March 2021, very energetic low-frequency underwater T-wave signals (2 to 25 Hz) were recorded at the Comprehensive Nuclear-Test-Ban Treaty (CTBT) International Monitoring System (IMS) hydrophone stations in the Pacific Ocean (Stations HA11 and HA03) and the South Atlantic Ocean (Station HA10). This work investigates the long-range (megameters) propagation of these T-waves. Their sources were three powerful submarine earthquakes in the Kermadec Trench and Islands, located at approximately 6000, 8800, and 15100 km from Stations HA11, HA03, and HA10, respectively. Arrival time and back azimuth of the recorded T-waves were estimated using the Progressive Multi-Channel Correlation algorithm installed on the CTBT Organization (CTBTO) virtual Data Exploitation Centre (vDEC). Different arrivals within the duration of the earthquake signals were identified, and their correlations were also analyzed. The data analysis at HA03 and HA10 revealed intriguing T-wave propagation paths reflecting, refracting, or even transmitting through continents, as well as T-wave excitation along a chain of seamounts. The analysis also showed much higher transmission loss (TL) in the propagation paths to HA11 than to HA03 and HA10. Moreover, strong discrepancies between expected and measured back azimuths were observed for HA11, and a three-dimensional (3D) parabolic equation model was utilized to identify the cause of these differences. Numerical results revealed the importance of 3D effects induced by the Kermadec Ridge, Fiji archipelago, and Marshall Islands on T-wave propagation to HA11. This analysis can guide future improvements in underwater event localization using the CTBT-IMS hydroacoustic sensor network.info:eu-repo/semantics/publishedVersio

Condition number estimation of preconditioned matrices.

The present paper introduces a condition number estimation method for preconditioned matrices. The newly developed method provides reasonable results, while the conventional method which is based on the Lanczos connection gives meaningless results. The Lanczos connection based method provides the condition numbers of coefficient matrices of systems of linear equations with information obtained through the preconditioned conjugate gradient method. Estimating the condition number of preconditioned matrices is sometimes important when describing the effectiveness of new preconditionerers or selecting adequate preconditioners. Operating a preconditioner on a coefficient matrix is the simplest method of estimation. However, this is not possible for large-scale computing, especially if computation is performed on distributed memory parallel computers. This is because, the preconditioned matrices become dense, even if the original matrices are sparse. Although the Lanczos connection method can be used to calculate the condition number of preconditioned matrices, it is not considered to be applicable to large-scale problems because of its weakness with respect to numerical errors. Therefore, we have developed a robust and parallelizable method based on Hager's method. The feasibility studies are curried out for the diagonal scaling preconditioner and the SSOR preconditioner with a diagonal matrix, a tri-daigonal matrix and Pei's matrix. As a result, the Lanczos connection method contains around 10% error in the results even with a simple problem. On the other hand, the new method contains negligible errors. In addition, the newly developed method returns reasonable solutions when the Lanczos connection method fails with Pei's matrix, and matrices generated with the finite element method

Condition numbers: SSOR preconditioned Pei’s matrix for various <i>d</i>.

<p>The condition numbers of SSOR preconditioned <b>T</b>_<i>Pei</i> using the Lanczos connection method, the modified Hager’s method, and Octave are listed. In this table, error values defined as </p><p></p><p><mi>E</mi><mi>r</mi><mi>r</mi><mi>o</mi><mi>r</mi><mo>=</mo></p><p></p><p><mo stretchy="true">∣</mo>Calculatedconditionnumber<mo>−</mo>Octave<mo stretchy="true">∣</mo></p>Octave<p></p><mo>×</mo><mn>100</mn><mo stretchy="false">(</mo><mo>%</mo><mo stretchy="false">)</mo><p></p><p></p> are also indicated.<p></p><p>Condition numbers: SSOR preconditioned Pei’s matrix for various <i>d</i>.</p

Preconditioned Conjugte Gradient Algorithm.

<p>Pseudocode of the PCG. In the algorithm,⟨⋅, ⋅⟩ denotes vector inner product.</p><p>Preconditioned Conjugte Gradient Algorithm.</p

Preconditioned Conjugte Gradient Algorithm in the modified Hager’s method.

<p>Pseudocode of the PCG in the modified Hager’s method. In the algorithm,⟨⋅, ⋅⟩ denotes vector inner product.</p><p>Preconditioned Conjugte Gradient Algorithm in the modified Hager’s method.</p

Condition numbers: Diagonal scaled Poisson–FEM matrix.

<p>The condition numbers of the diagonal scaled Poisson–FEM matrices using the Lanczos connection method, the modified Hager’s method, and Octave are listed. In this table, error values defined as </p><p></p><p><mi>E</mi><mi>r</mi><mi>r</mi><mi>o</mi><mi>r</mi><mo>=</mo></p><p></p><p><mo stretchy="true">∣</mo>Calculatedconditionnumber<mo>−</mo>Octave<mo stretchy="true">∣</mo></p>Octave<p></p><mo>×</mo><mn>100</mn><mo stretchy="false">(</mo><mo>%</mo><mo stretchy="false">)</mo><p></p><p></p> are also indicated.<p></p><p>Condition numbers: Diagonal scaled Poisson–FEM matrix.</p

Condition numbers: SSOR preconditioned Poisson–FEM matrix.

<p>The condition numbers of SSOR preconditioned Poisson–FEM matrices using the Lanczos connection method, the modified Hager’s method, and Octave are listed. In this table, error values defined as </p><p></p><p><mi>E</mi><mi>r</mi><mi>r</mi><mi>o</mi><mi>r</mi><mo>=</mo></p><p></p><p><mo stretchy="true">∣</mo>Calculatedconditionnumber<mo>−</mo>Octave<mo stretchy="true">∣</mo></p>Octave<p></p><mo>×</mo><mn>100</mn><mo stretchy="false">(</mo><mo>%</mo><mo stretchy="false">)</mo><p></p><p></p> are also indicated.<p></p><p>Condition numbers: SSOR preconditioned Poisson–FEM matrix.</p

The algorithm of Hager’s method.

<p>Pseudocode of Hager’s method. In the algorithm, [<i>i</i>] denotes the <i>i</i>th vector element of a vector.</p><p>The algorithm of Hager’s method.</p

Condition numbers: Diagonal scaled tridiagonal matrix.

<p>The condition numbers of diagonal scaled <b>T</b>_<i>Tri</i> using the Lanczos connection method, the modified Hager’s method, and Octave are listed.</p><p>Condition numbers: Diagonal scaled tridiagonal matrix.</p