Dimensionality Reduction for Master Event Location Improvement

MULTIVARIATE DIMENSIONALITY REDUCTION
IN CROSS-CORRELATION ANALYSIS
( Optimizing master event templates for CTBT monitoring with dimensionality
reduction techniques as applied to real and synthetic data )
Ivan Kitov, Seismic-Acoustic Officer, IDC/SA/SM
Mikhail Rozhkov
International Data Centre Division
Preparatory Commission for the Comprehensive
Nuclear-Test-Ban Treaty Organization, Provisional Technical Secretariat
Vienna International Centre
P.O. Box 1200, A-1400 Vienna, AUSTRIA

International Data Centre Page 2
Dimensionality Reduction for Master event Location
Improvement
1. In master event location, a matched-filter like technique based on cross-
correlation with pre-defined waveform template, a crucial role plays a template
design. Reduction of templates number for certain region under monitoring is
extremely important both for interactive and real-time processing as it may
dramatically reduce the time of resulting product delivery and may improve low
magnitude event detection threshold and location.
2. A number of dimensionality reduction methods have been explored to minimize
the number of master events needed for cross correlation based seismic event
detection and location, including multidimensional data model approaches
(hypercomplex and tensorial). The primary method considered is Principle
Component Analysis (PCA), which is widely accepted as a superior method of
matrix factorization or Singular Value Decomposition (SVD). For regional
seismic events, Harris (2006) used this in designing a subspace detector for the
cross correlation based event location. Other methods of dimensionality reduction
explored either theoretically or analytically included Robust PCA, Kernel PCA,
Incremental PCA (IPCA), Empirical Subspace Detector (SSD) (Barrett and
Beroza, 2015) and Independent Component Analysis (ICA).

Single Mode
Decompositions

DR Results
 In most cases, the IPCA method performs better than simple PCA for the
majority of window lengths using the underground nuclear explosion data sets.
The waveforms recorded from the aftershock sequence are more complex in
nature and thus could not reduce the design set down as much, however IPCA
still outperformed simple PCA. It is also noteworthy that the SSD method of
empirically determining the set of reduced waveforms as the simple stack and
time derivative of the stack performed as well as the IPCA for some data sets.
This may be a viable alternative to IPCA when using large input data sets where
compute time may be a concern.
 Proper alignment of input waveforms is paramount to accurate representation of
the output dimensions to describe the underlying event source process. The
Single Link clustering has shown to be the most robust for achieving this time
alignment of the signals
 A Quality Measure has been calculated for each analysis to assist in determining
the best dimensionality reduction method and window length. This quality
measure is simply taken from calculating the maximum cross correlation
coefficients across all event pairs in this benchmark test.

Semipalatinsk (Balapan) Nuclear
explosions recorded at Warramunga Array,
Australia. a) Waveforms aligned using
Single Link clustering and data set
amplitudes normalized to unit energy. Top
panel shows all waveforms. Bottom panel
shows mean waveform with one standard
deviation bound in gray region. b)
Summary of results varying DR method
and waveform window length for cross
correlation threshold of 0.7. Top panel
shows the number of output dimensions
for each method and window length
corresponding to a). Bottom panel shows
corresponding Quality Measure, calculated
as described:
DR Results

c) Time series of output dimensions for DR
method and threshold shown in title. Selected
time window indicated by extent of horizontal
axis. d) Colormap of cross correlation
coefficients between original input traces and
output dimensions shown in c). A Quality
Measure has been calculated for each analysis
to assist in determining the best
dimensionality reduction method and window
length. As mentioned, this quality measure is
simply taken from calculating the maximum
cross correlation coefficients across all event
pairs, then taking the minimum of all these
values. Basically, this measure shows the
lowest cross correlation value for a given
method. The higher this value, the better the
method of dimensionality reduction.
DR Results

All in all, IPCA shows best results (less number
of templates in master event set)
Left, top to bottom: Balapan explosions recorded
at EKR, Eskdalemuir Array, Scotland
Balapan explosions at WMQ, Urumqi, China.
Lop Nor, China, explosions at Borovoye Array,
BRV, Kazakhstan.
Balapan explosions at BRV.
Semipalatinsk (Degelen) explosions at BRV
Below: 2012 Sumatra M8.6 Earthquake
Aftershock sequence recorded at Songina Array
(SONM), Mongolia.
DR Results

Cross correlation: applying PCA and ICA
CC of first 5 Real Principal and Independent Components
with 106 Real UNE records with cumulative ICs on right
CC of first 5 Synthetic Principal and Independent Components
with 106 Real UNE records with cumulative ICs on right
Good performance in both cases

Global Cross Correlation Grid: REB seismicity
Density (# per 1deg x 1deg cell) of shallow events in the Reviewed Event Bulletin.
Waveforms from many seismic sources can be used as templates for cross
correlation. For earthquakes, cross correlation most effective for close events.

Global Cross Correlation Grid: 25,000 nodes, depth 0-700 km
Global Grid of Master Events is designed for finding and location
of seismic events based on cross-correlation (CC). The whole
globe is subdivided uniformly by cells surrounding the grid points
The IMS array stations consider the hypotheses of seismic event
occurrence within these cells based on matched filter detection
with the pre-established Master Event record (template).

Global Cross Correlation Grid: primary stations
The International Monitoring System (IMS) network includes 50 primary seismic
stations, which are divided into seismic arrays (circles) and three-component (3-
C) seismic stations (triangles). Auxiliary seismic arrays (circles) are also shown.

Global Cross Correlation Grid
Testing, February 12, 2013
REB: 134 events
Grid: 25,000 nodes
Group 1 = WRA, TORD, MKAR, ILAR, GERES, PDAR, CMAR, SONM, AKASG, BRTR, GEYT
Group 2 = ASAR, ZALV, YKA, ARCES, TXAR, KSRS
Group 3 = USRK, FINES, NVAR, NOA, MJAR
Defining parameters:
Templates: simplest 1D synthetic waveform for all arrays, theoretical time delays
Detections: SNRmin = 0.5; SNR_CCmin=2.5; CCmin = 0.2; FKSTATmin = 2.5; AZRESmax= 20.0º;
SLORESmax = 2.0 s/º;
Events: dTorigin = 6s; NSTAmin= 3; AZGAPmax= 330º
RESULTS: Total arrivals and XSEL hypotheses: 22,900,402 arrivals; 107,969 events;
After conflict resolution:
SNR_CC>2.5 SNR_CC>3.0 SNR_CC>3.5 SNR>2
XSEL 6,141 events 766 122 2351
REB Matched 92 90 77 101
DPRK 2013: time - 02:57:50.799 , d=24.92 km, OTres=0.1s; nsta=9:
Stations : AKASG, BRTR, CMAR, GERES, GEYT, ILAR, MKAR, SONM, WRA

Global Cross Correlation Grid: more templates
• V0.1: All master templates are synthetics same at all stations, a
version of f-k analysis
• V0.2: Master templates are station/master specific synthetics in
1D velocity model
• V0.3: Master templates are station/master/source (e.g. explosion)
specific synthetics calculated for 2D velocity structure (e.g.
ak135+CRUST 2.0)
• V1.1: Real master templates are used where possible
• V1.2: Grand master events are applied where possible
• V2.0: The set of principal components are optimized where
possible as obtained by the PCA applied to the complete set of
actual and historical data
• V3.0: Synthetic + real master templates based on principal
components with classification algorithms trained on actual data
PROGRESSIVE
IMPROVEMENT

Global Grid Location with constructed templates
Table below: location results with (1) xgrid1 – AK135 synthetics as template, (2) xgrid2 –
first PC of synthetic record set, (3) xgrid3 – first PC of UNE set, (4) xgrid4 – DPRK-2013
genuine records at each element of 9 arrays is a template set. REB – location by IDC.
Location with synthetic PC is the same as location with genuine DPRK records.
Left: Ndef – defining
stations in Global Grid
location. Numbers to the
right of Ndef column –
number of events located
corresponding to the
number of defining
stations.

CC Master directional diagram for array stations AKASG and WRA for xgrid1-4 cases on left (from left
to right). Color code corresponds to cross-correlation coefficient.
Prototype CC Global Grid: monitoring results

Global Grid DPRK-2013 (1 day processing) locations for 4 cases of templates: (a) AK135 synthetics master, (b) PCA
synthetic master, (c) PCA real master, and (d) DPRK-2013 master. All masters except for case 4 were produced by
replicating single template at each array station element implementing predicted time delays for given master
geographical position.
Prototype CC Global Grid: monitoring results
best location
best location
a) b)
c) d)

Multidimensional
Models

Hypercomplex and high-order (HC/HO) master event design
for CTBT monitoring
Multidimensional mathematical
description of seismic network
data representation as
hypercomplex and tensor
constructions is studied for
further understanding the
value it can bring in tasks of
relative location for CTBT
monitoring.
We studied the benefits and the caveats of using hypercomplex and
multilinear Singular Value Decomposition methods applied to such
constructions, and discovered effective measures for correlating
hypercomplex data structures.

HC/HO Processing Background

In given approach, we consider a hypercomplex and high-order (tensorial)
representation of multichannel seismic data as they are recorded by the multichannel
seismic installations, such as IMS 3-C seismic stations and arrays with further
processing within the context of the corresponding multidimensional model for
further cross-correlation-based detection and location. Hypercomplex number
systems are the natural cases of representing a 3-C digital seismogram samples
requiring, however, special attention to the underlying axiomatics. Dealing with the
composite observations (3C arrays) may demand higher than 4 dimensional algebras,
or some specific grouping of them, so tensor representation of seismic wavefield
looks natural in this case. Data processing then would be conducted not on separate
waveform projections but on relatively full multidimensional object and tensor
operations on the data from the 3C arrays would utilize joint volumetric (sensor) and
spatial (array) information. Further dimensionality reduction of tensor data produces
lower order principal components, a basis for the multidimensional waveform
templates. Highly effective master events built with the hypercomplex and
multilinear SVD provide a good example of introducing multidimensional data
models into CTBT practice.

To our knowledge, first evidence of using hypercomplex algebraic systems in
geophysics can be referred to the set of publications in Doklady Academii Nauk,
USSR back in 1983. An associative commutative hypercomplex system of order 4
(or bicomplex system) was introduced by G. Shpilker (1983-1988), and then
expanded to the systems of theoretically unlimited order (ascending to the works of
Segre (1892) and Ketchum (1928) on commutative infinite dimensional algebra) in
order to solve incorrect (or ill posed) problems in Hadamard sense. D’ Alembert,
Lame and Maxwell equations were considered and a common fundamental
hypercomplex solution was found for them. With this, in contrast to classical
Hamiltonian quaternion extensions, octonions and sedenions, Shpilker’s algebra was
associative though not transitive and can also be considered as a superalgebra in its
specific meaning. A seismological application was considered by M. Rozhkov (1986)
in a study of hypercomplex wave field restoration with data recovered by
homomorphic analysis.
In a modern world quaternions are getting more and more popular: computer vision
and colour image processing, missile flight control, robotics (with quaternion neural
networks), exploration geophysics (vector sensor arrays), quantum mechanics,
special and general relativity.

Quaternionic factorization

Example quaternion principle components. Left: input 3C seismogram, Center:
pure quaternion (with zero real part) i, j, and k components of first principal
component, Right: first complex quaternion component of first 3 principle
components.
Hypercomplex (here – classical Hamilton quaternion)
representation example of 3C seismogram
𝑖2 = 𝑗2 = 𝑘2 = 𝑖𝑗𝑘 = −1 ; 𝑖 = 𝑗𝑘 = −𝑘𝑗 ; 𝑗 = 𝑘𝑖 = −𝑖𝑘; 𝑘 = 𝑖𝑗 = −𝑗𝑖
q = 𝑞0 + 𝑖𝑞1 + 𝑗𝑞2 + 𝑘𝑞3 v
q ,
s


Observations
There are certain problems applying straight correlation approach as
in case of real vector. For instance, event for pure input quaternion
vector (i, j, k) matrix singular vectors are not pure in general case (s,
i, j, k). Also, the scalar product of quaternion arguments is
quaternionic, and so the angle between the quaternion vectors. So
we needed to find some appropriate and intuitively understandable
measure for event comparison. To come up with it, we studied the
properties of real and complex quaternion parts relationships for
different quaternionic products (scalar, cross, and convolution)
tested on 3-component seismic quarry blast data. We performed the
quaternion dimensionality reduction through Q-SVD matrices
truncation. Input data is a quaternion representation of 3C seismic
records.

The test is a cross-correlation between the:
Input Q-data vs input Q-data, with the search of the signal best correlating with the other
signals.
Reduced Q-data after SVD-based restoration vs input Q-data.
The measures were taken:
1. A norm of a quaternion cross-correlation coefficient (Q-CC) taken in its classical form.
2. A scalar or pure quaternion of the Q-CC (s(Q-CC), or norm (pure(Q-CC))
3. An angle in a quaternion space of the Q-CC
Quaternion phase correlation between two shifted (in time-event space) color images of
600x3C seismograms. A pick on correlation surface clearly indicates the shifts.
Observations

Dealing with the tensor representation of seismic wavefields can simplify in
certain sense the multidimensional approach to data processing, in particular, to
the data set which was the same as used for the single component processing.
Multichannel data corresponding to a seismic event from, for example, the Aitik
and Kiruna quarries in Sweden can be rearranged as a 3-mode tensor, where first
mode is time, or sample number, second mode is station, or sensor number, and
third mode is direction of ground motion (Z, N and E). Then, a complete test data
set would consist of a 4-mode tensor with the event number corresponding to the
4th dimension. Considering a 3-component seismic array as a multitude of
observations with tensor description (not the tensor field in general sense), the
corresponding data tensor, formally, can be regarded as a tensor product of 3
vector spaces, each with its own coordinate system. Then we could apply tensor
operations to the data recorded by such arrays gaining certain benefits from
utilizing joint volumetric (sensor) and spatial (array) information. Further
dimensionality reduction of tensor data produces a low order principal
components, a basis for the multidimensional waveform templates. Note that a
first-mode, or first-order tensor is a vector, a second-order tensor is a matrix, and
tensors of higher orders are higher-order tensors.
Seismic Tensorial Representation

General approach. High-dimensional tensors or so called multi-way data are
becoming more and more popular in areas such as biomedical imaging,
chemometrics and networking. Traditional approaches to finding lower
dimensional representations of tensor data include flattening the data and applying
matrix factorizations such as principal components analysis (PCA) or employing
tensor decompositions such as the CANDECOMP/PARAFAC (canonical polyadic
decomposition with parallel factor analysis) and Tucker decompositions, which
may be regarded as a more flexible PARAFAC model. Tucker decomposition,
which we use in this work, decomposes a tensor into a set of matrices and one
small core tensor. Then the eigen-images can be extracted for resizing the input
tensor to lower dimensions. There are more approaches to the multimodal
dimensionality reduction such as multidimensional Discreet Fourier Transform
(DCT) mostly used in image processing (JPEG, for instance), 2D SVD (based on
low rank approximation of the matrix), and tensor interpolation (for example, Hotz,
et al, 2010, Tensor Field Reconstruction Based on Eigenvector and Eigenvalue
Interpolation). In this presentation, we provide results for all the above mentioned
methods with the accent on the Tucker tensor decomposition made with the
alternating least squares (ALS) method.

Here we will only highlight the approach to the Tensor Subspace Projection for
Dimensionality Reduction. For any tensor
𝒜 = 𝒮 ×1 𝑈(1) ×2 𝑈(2) … ×𝑁 𝑈(𝑁), or
𝐀(𝑛) = 𝐔(𝑛) ∙ 𝐒 𝑛 ∙ (𝐔 𝑛+1 ⊗ 𝐔 𝑛+2 … ⊗ 𝐔 𝑁 ⊗ 𝐔 1 ⊗ 𝐔 2 ⊗ ⋯ ⊗ 𝐔 𝑛−1 ) 𝑇
For typical image and video tensor objects although the corresponding tensor space is
of high dimensionality, tensor objects typically are embedded in a lower dimensional
tensor subspace (or manifold), in analogy to the vectorized face image embedding
problem where vector image inputs reside in a low-dimensional subspace of the original
input space (M. Turk, et al 1991). Thus, it is possible to find a tensor subspace that
captures most of the variation in the input tensor objects and it can be used to extract
features for recognition and classification applications. To achieve this objective, 𝑃𝑛 <
𝐼𝑛 orthonormal basis vectors (principle axes) of the 𝑛 - mode linear space ℝ𝐼𝑛 are
sought for each mode 𝑛 and a tensor subspace ℝ𝐼1 ⊗ ℝ𝐼2 … ⊗ ℝ𝐼𝑁 is formed by these
linear subspaces

Visualizing the tensor factor
reduction through multilinear
SVD. Top: a standard “detail”
359x371 image (up) is x-y
shifted 20 times and packed as
a tensor, then the last slice is
extracted with tensor
decomposition (4D tensor
processing). Below: same
image decomposed (3D tensor)
and reduced according to the
eigenvalue meaningfulness. It
corresponds to 3D filtering,
thus extracting needed
structure (noise or signal in
case of seismic applications,
see figures to the right).
Tensor Subspace Projection for Dimensionality Reduction.

A matrix representation of this decomposition can be obtained by unfolding 𝒜 and 𝒮 as:
𝐀(𝑛) = 𝐔(𝑛) ∙ 𝐒 𝑛 ∙ (𝐔 𝑛+1 ⊗ 𝐔 𝑛+2 … ⊗ 𝐔 𝑁 ⊗ 𝐔 1 ⊗ 𝐔 2 ⊗ ⋯ ⊗ 𝐔 𝑛−1 ) 𝑇
where ⊗ denotes the Kronecker product and 𝒮 is a core tensor of size:
𝑅1 × 𝑅2 × ・・・ × 𝑅𝑁 .
The decomposition can also be written as
𝒜 =
𝑖1=1
𝐼1
𝑖2=1
𝐼2
…
𝑖𝑁=1
𝐼𝑁
𝒮(𝑖1𝑖2, … , 𝑖𝑁 ) × 𝐮𝑖1
1
° 𝐮𝑖2
2
° … °𝐮𝑖𝑁
𝑁
i.e., any tensor can be written as a linear combination of 𝐼1 × 𝐼2 × ⋯ × 𝐼𝑁 rank-1
tensors. This decomposition is used in the following to formulate multilinear projection
for dimensionality reduction.

Implementation. While the traditional PCA notation to describe algorithm can be
written as:
[PC, SCORE, LATENT] = PCA(A)
where A is a matrix with column vectors representing observations, in tensor notation
it can be written as:
[MPROJ, PC, LATENT] = TPCA(𝒜,d),
where 𝒜 is a tensor with dimensions [p_1, p_2, …,p_D,n], d is a target dimensions
[d_1,…d_D],
MPROJ is a tensor with dimensions [d_1, …d_D,n] after change of basis (analogous
to SCORE for regular PCA), PC is a D-by-1 cell array containing principal
components along each mode. PC{i} has dimension p_i-by-d_i, and LATENT is D-
by-1 cell array containing the d_i eigen values along each mode, ordered from largest
to smallest.

Typical seismic event
(blast at Aitik mine)
recorded at ARCES.
Stations with only
vertical components
were not used in the
analysis
Data for Seismic Tensorial Representation
Aitik mine, aerial view

3-component records
at a single ARCES
station. Bandpass
filter is 3-6 Hz. 160
(upper figure) and 27
(lower) second time
windows presented.
Data for Seismic Tensorial Representation

1. Given n tensor observations (number of seismic events in our case), the program
flattens each along mode I to obtain a matrix,
2. Take outer product of the matrix (p_i-by-p_i),
3. Do the classical PCA on the sum of outer products, retrieve the first d_i principal
components (p_i-by-d_i), then scale the original tensor in the PC basis along
each mode.
Following this procedure we could produce 3C array projected components of the
reduced tensor composed of 57 seismic channel (19 stations x [N, EW, NS]),
corresponding to the tested reduction to the dimension of 1 to n.
A figure is a snapshot of one out of ten
3D components of the reduced tensor
seismogram. A reduced tensor set (3D
for single component and 4D for more
components) inherits the properties of
the input training data set.
Seismic Tensorial Decomposition in short

The multidimensional discrete (MD) cosine transform (DCT-II and
DCT-III for inverse) is popular compression structures for MPEG-4,
H.264, and HEVC (high efficiency video coding), and is accepted as
the best suboptimal transformation since its performance is very close
to that of the statistically optimal Karhunen-Loeve transform
𝑋(𝑘1, 𝑘2, … , 𝑘𝑟) =
𝑛1=0
𝑁1−1
𝑛2=0
𝑁2−1
…
𝑛𝑟=0
𝑁𝑟−1
𝑥(𝑛1, 𝑛2, … , 𝑛𝑟) ∙ cos
𝜋 2𝑛1 + 1 𝑘1
2𝑁1
… cos
𝜋 2𝑛𝑟 + 1 𝑘𝑟
2𝑁𝑟
,
where 𝑘𝑖 = 0,1, … , 𝑁𝑖 -1 and 𝑖 = 1,2, … , 𝑟. Inverse truncated MD DCT
returns the reduced tensor array with the required number of
components used for cross correlation template construction.
Other Tensorial Reductions

MD variants of the various DCT types follow from the one-
dimensional definitions:
1. They are a separable product (equivalently, a composition) of
DCTs along each dimension, and
2. Computing a multidimensional DCT by sequences of one-
dimensional DCTs along each dimension is known as a row-
column algorithm.
3. The inverse of a multi-dimensional DCT is just a separable product
of the inverse(s) of the corresponding one-dimensional DCT(s), e.g.
the one-dimensional inverses applied along one dimension at a time in
a row-column algorithm.

A tensor containing records of N events of length M from the arrays of M Z -
sensors can be thought as a 3D cube sliced to N slices, and the reduced tensor –
to R slices where R corresponds to the meaningful part of eigenevent. 4D cube
can represent the same of 3C array, or a network of L-arrays.
Tensorial processing of seismic array

Regional eigenstructure of the tensor composed by the 4-station IMS
regional subnet composed of stations ARCES, NORSAR, FINES and
HFS. Regional seismicity (mining explosions at Kiruna and Aitik) is
under study.

Global eigenstructure of the tensor composed by the 8-station IMS
subnet composed of stations AKASG; FINES; MKAR; NVAR;
TXAR; USRK; WRA; ZALV. Global seismicity is under study.

Top: eigenvectors composed of 8 IMS arrays (left) and 1 ARCES array using
same eigenstructure.

Tensor Templates Tests
A number of tests were conducted with the obtained reduced
tensor components in order to evaluate performance of the cross
correlation detector with these components used as templates.
Complete sets of components for the 3C array were produced with
4 methods:
 Tensor interpolation,
 DCT,
 2DSVD and
 HOSVD.
3D templates were reduced to 1D vectorized case ([Z,NS,WE])
and a well established system of tests was applied.

First test was based on cross correlation (CC) of the developed
templates with continuous waveforms measured from the set of 122
events and determining the detection rate based on SNR threshold : the
percentage of detections having SNRCC>3.5.
In second test, we used 46 events not included into the training set of
122. Ten first reduced tensor components were used for detection, i.e.
40 components were tested altogether as presented on figures below:
(1) 2D SVD, (2) MD DCT, (3) HOSVD, and (4) tensor interpolation.
Different template lengths were used from 10 to 30 seconds.

1. The best results were produced by the MD DCT and interpolation
templates for average SNR; the minimum SNRCC over all tested
signals for a given template, which has to be above 3.5; and the
average CC.
2. The overall difference in detection rates is not large.
3. Similar tests were also carried out with the eigenimages (instead
of reduced back-projections) produced for the HOSVD (PC cell
array in TPCA algorithm). It was found that the algorithm
destroys the proper channel alignment in sensor triads and move-
outs related to different stations of the array and the test results
were not impressive. In case of single 3-C station, it does not
make any difference since all the channels in a training set are
aligned by default and there is no need to keep the move-outs so
the regular SVD/PCA case works fine. More work is ahead.

4 methods, 10 templates in each method, variable time window length
4 methods, 10 templates in each method, 5 filter bands:
2-4 Hz, 4-8 Hz, 3-6 Hz, 6-12 Hz, 8-16 Hz.
Tensor Templates Tests (1) 2D SVD, (2) MD DCT, (3) HOSVD, and (4) tensor interpolation

HOSVD tests non-standard truncated reconstruction of the reduced tensor set
After all methods were tested with different template lengths and filters, we have
taken the window (10 seconds) and the filter (3-6 Hz), which work the best. Then we
returned to the high order tensor decomposition and tried our own truncated
reconstruction of the reduced tensor set. The detection rate was the same as for the
best methods above, and the smallest SNRCC was even larger (4.2 against 3.5). More
work has to be conducted to find the optimal multidimensional template design.

There are different methods to compute DCT (Bi, Zeng, 2004). We follow the
tensor reduction procedure implemented by Hua Zhou, 2013, in his tensor
regression software (TensorReg_toolbox). We use a 4D DCT applied to 4-order
tensor composing the 3-component seismic array data. The dimensionality
reduction based on tensor interpolation is also based on the tensor regression
software by Zhou, 2013.
A TensorToolbox software by T. Kolda (Sandia National Lab) is also required for
this processing.
2DPCA discussed in (Lu et al, 2006) is also tested in this work through the
TensorReg as well. We are not discussing the results in this presentation since we
could not obtain good enough performance with this method.

Observations
1. Multidimensional eigen-structures are not inter-aligned, so straight eigen-images
cannot be used for templates.
2. There can be different approaches to the cross-correlation of multidimensional
structures, we are still in a process of defining the best one compliant with nature of
data and its dimensionality.
Approaches used:
1. To create eigen-image inheriting original timing (for example, for array having
same time delays for central sensor), we use the reduced eigen-structure projection
to original vector space.
2. Another approach is extracting single eigen-components and use them in
conventional cross-correlation processing.
3. As it was shown for the quaternion structures, an angle based on norm and scalar
product can be used to measure the divergence of the observed event from the
designed master event. Same can be said for tensors but using frobenius norms.
However, in certain cases well-established mathematical apparatus of phase
correlation developed for colour object processing can be used for
multidimensional seismic structures as well.

Dimensionality Reduction for Master Event Location Improvement

Recommended

Recommended

More Related Content

Similar to Dimensionality Reduction for Master Event Location Improvement

Similar to Dimensionality Reduction for Master Event Location Improvement (20)

More from ivanokitov

More from ivanokitov (7)

Recently uploaded

Recently uploaded (20)

Dimensionality Reduction for Master Event Location Improvement

Editor's Notes