Magnetotelluric tensor decomposition: Part I, Theory for a basic procedure

Geophysics ◽  
1998 ◽  
Vol 63 (6) ◽  
pp. 1885-1897 ◽  
Author(s):  
F. E. M. (Ted) Lilley
Keyword(s):  
Pure Shear ◽  
General Theorem ◽  
Tensor Decomposition ◽  
Impedance Tensor ◽  
Model Calculations ◽  
Tensor Decompositions ◽  
The Real ◽  
Mohr Circle ◽  
Basic Procedure ◽  
D Model

The problem of expressing a general 3-D magnetotelluric (MT) impedance tensor in the form of a 2-D tensor that has been distorted in some way is addressed first in terms of a general theorem. This theorem shows that when the real and quadrature parts of a tensor are analyzed separately as distinct matrices, all that is necessary to make a matrix with 2-D characteristics from one with 3-D characteristics is to allow the electric and magnetic observing axes to rotate independently. The process is then examined in terms of the operations of twist and pure shear (“split”) on such matrices. Both of two basic sequences of split after twist, and twist after split, produce a typical 3-D matrix from one initially 1-D, with the parameters of split contributing 2-D characteristics to the final matrix. Taken in reverse, these sequences offer two basic paths for the decomposition of a 3-D matrix, and are seen to be linked to the initial theorem. The various operations on matrices are expressed diagrammatically using the Mohr circle construction, of which it is demonstrated two types are possible. Mohr circles of an observed MT tensor display all the information held by the tensor, and the two types of circle construction respectively make clear whether particular data are well suited to modeling by either split after twist, or twist after split. Generally, tensor decompositions may be displayed by charting their progress in Mohr space. The Mohr construction also displays the invariants of a tensor and shows that tensor decomposition can be viewed as a process of determining an appropriate set of invariants. An expectation that the origin of axes should be outside every circle categorizes as irregular any tensors which, in either the real or quadrature part, do not satisfy a [Formula: see text] criterion. The theory of the present paper applies equally to procedures for distorting 1-D and 2-D model calculations for the purpose of matching observed 3-D data.

Magnetotelluric tensor decomposition: Part II, Examples of a basic procedure

Geophysics ◽  
1998 ◽  
Vol 63 (6) ◽  
pp. 1898-1907 ◽  
Author(s):  
F. E. M. (Ted) Lilley
Keyword(s):  
Tensor Decomposition ◽  
Companion Paper ◽  
Impedance Tensor ◽  
The Real ◽  
Common Features ◽  
Basic Procedure ◽  
Decomposition Procedure ◽  
Valid Data ◽  
Erroneous Data ◽  
Good Agreement

The decomposition procedure suggested for magnetotelluric (MT) data in a companion paper is illustrated by examples. Adjacent pairs of stations are examined to test for common features in their MT response. The method analyzes the real and quadrature parts of an impedance tensor separately, and the results for the real and quadrature parts are compared, and in some cases combined, at the end. Values are obtained which are interpreted directly in terms of nearest 2-D principal impedances, local geologic strike, and regional geologic strike. The requirement for valid data derived in the companion paper, that [Formula: see text] together with [Formula: see text], is shown to be a useful criterion in the winnowing of erroneous data. The examples presented typically show the direction interpreted as local geologic strike to be well determined, period independent, and consistent between real and quadrature MT matrices. In these examples, however, the direction interpreted as regional geologic strike is period dependent and not in good agreement between real and quadrature MT matrices. These results suggest caution is desirable in the use of period‐independent galvanic distortion models, which should be checked first with regional strike determinations.

scholarly journals Magnetotelluric analysis using Mohr circles

Geophysics ◽  
1993 ◽  
Vol 58 (10) ◽  
pp. 1498-1506 ◽  
Author(s):  
F. E. M. Lilley
Keyword(s):  
Mechanical Stress ◽  
Three Dimensional ◽  
Dimensional Structure ◽  
Impedance Tensor ◽  
Two Dimensional ◽  
Skew Angle ◽  
The Real ◽  
Mohr Circle ◽  
Quantitative Expression

The Mohr circle, most commonly met in the analysis of mechanical stress, is used to depict magnetotelluric impedance information, taking the real and quadrature parts of magnetotelluric tensors separately. The magnetotelluric concepts of two‐dimensionality, three‐dimensionality, skew and anisotropy are then all given quantitative expression on a diagram, as are various magnetotelluric invariants. In particular, a new invariant, the “central impedance,” becomes evident in a discussion of effective impedances. Some insight is gained into impedance rotations, and an anisotropy angle is defined, analogous to skew angle. Mohr circles are also tested to depict the effects of the shear and twist operations on a regionally two‐dimensional structure. Generally, the application of shear or twist results in an impedance tensor with a Mohr circle of typical three‐dimensional form.

scholarly journals Evolution of communities of software: using tensor decompositions to compare software ecosystems

2019 ◽  
Vol 4 (1) ◽  
Author(s):  
Oliver A. Blanthorn ◽  
Colin M. Caine ◽  
Eva M. Navarro-López
Keyword(s):  
Dynamic Network ◽  
Tensor Decomposition ◽  
Software Projects ◽  
R Software ◽  
Tensor Decompositions ◽  
Software Ecosystems ◽  
Software Libraries ◽  
Dependency Networks ◽  
Software Packages ◽  
Over Time

AbstractModern software development is often a collaborative effort involving many authors through the re-use and sharing of code through software libraries. Modern software “ecosystems” are complex socio-technical systems which can be represented as a multilayer dynamic network. Many of these libraries and software packages are open-source and developed in the open on sites such as , so there is a large amount of data available about these networks. Studying these networks could be of interest to anyone choosing or designing a programming language. In this work, we use tensor factorisation to explore the dynamics of communities of software, and then compare these dynamics between languages on a dataset of approximately 1 million software projects. We hope to be able to inform the debate on software dependencies that has been recently re-ignited by the malicious takeover of the npm package and other incidents through giving a clearer picture of the structure of software dependency networks, and by exploring how the choices of language designers—for example, in the size of standard libraries, or the standards to which packages are held before admission to a language ecosystem is granted—may have shaped their language ecosystems. We establish that adjusted mutual information is a valid metric by which to assess the number of communities in a tensor decomposition and find that there are striking differences between the communities found across different software ecosystems and that communities do experience large and interpretable changes in activity over time. The differences between the elm and R software ecosystems, which see some communities decline over time, and the more conventional software ecosystems of Python, Java and JavaScript, which do not see many declining communities, are particularly marked.

Spatial averaging of the vertical magnetic field in central‐loop configuration

Geophysics ◽  
1993 ◽  
Vol 58 (10) ◽  
pp. 1507-1510 ◽  
Author(s):  
Wei Qian ◽  
Laust B. Pedersen
Keyword(s):  
Tensor Decomposition ◽  
Data Interpretation ◽  
Impedance Tensor ◽  
Logarithmic Scale ◽  
Resistivity Structure ◽  
One Dimensional ◽  
Transient Electromagnetic ◽  
The Earth ◽  
Band Limited ◽  
Central Loop

Local resistivity heterogeneities can cause static shifts in the magnetotelluric (MT) impedance tensor that severely complicate data interpretation; the apparent resistivity is shifted on a logarithmic scale across the recorded frequency range while the phase has a band‐limited response. Different techniques such as electromagnetic array profiling (EMAP) (Torres‐Verdín and Bostick, 1992) and tensor decomposition (Zhang et al., 1987; Groom and Bailey, 1989; 1991) have been developed in the MT community to recognize and remove static shifts. Sternberg, et al. (1988) and Pellerin and Hohmann (1990) suggest that central‐loop transient electromagnetic (TEM) soundings can obtain an unbiased estimate of the regional resistivity structure of the earth and thereby correct for magnetotelluric static shifts. The regional resistivity structure of the earth must be one‐dimensional (1-D) for this method to work well.

scholarly journals Three-Way Tensor Decompositions: A Generalized Minimum Noise Subspace Based Approach

2018 ◽  
Author(s):  
Le Trung Thanh ◽  
Viet-Dung Nguyen ◽  
Nguyen Linh-Trung ◽  
Karim Abed-Meraim
Keyword(s):  
Computational Complexity ◽  
State Of The Art ◽  
Tensor Decomposition ◽  
Decomposition Algorithms ◽  
Subspace Analysis ◽  
Tensor Decompositions ◽  
Noise Subspace ◽  
Parallel Factor ◽  
Value Decomposition ◽  
Minimum Noise

Tensor decomposition has recently become a popular method of multi-dimensional data analysis in various applications. The main interest in tensor decomposition is for dimensionality reduction, approximation or subspace purposes. However, the emergence of “big data” now gives rise to increased computational complexity for performing tensor decomposition. In this paper, motivated by the advantages of the generalized minimum noise subspace (GMNS) method, recently proposed for array processing, we proposed two algorithms for principal subspace analysis (PSA) and two algorithms for tensor decomposition using parallel factor analysis (PARAFAC) and higher-order singular value decomposition (HOSVD). The proposed decomposition algorithms can preserve several desired properties of PARAFAC and HOSVD while substantially reducing the computational complexity. Performance comparisons of PSA and tensor decomposition of our proposed algorithms against the state-of-the-art ones were studied via numerical experiments. Experimental results indicated that the proposed algorithms are of practical values.

scholarly journals A QBO-signal in mesospheric water vapor measurements at ALOMAR (69.29° N, 16.03° E) and in model calculations by LIMA over a solar cycle

2009 ◽  
Vol 9 (1) ◽  
pp. 883-903 ◽  
Author(s):  
G. R. Sonnemann ◽  
P. Hartogh ◽  
S. Li ◽  
M. Grygalashvyly ◽  
U. Berger
Keyword(s):  
Water Vapor ◽  
Solar Cycle ◽  
Middle Atmosphere ◽  
Transport Processes ◽  
Domain Model ◽  
Time Interval ◽  
High Latitudes ◽  
Model Calculations ◽  
The Real ◽  
Mixing Ratios

Abstract. Microwave water vapor measurements between 40 and 80 km over a solar cycle (1996–2006) were carried out in high latitudes at ALOMAR (69.29° N, 16.03° E), Norway. Three larger interuptions in the winters of 1996/97 and 2005/06, and from spring 2001 to spring 2002, a few smaller interruptions of monitoring occurred during this period. The observed year-to-year variability is not directly related to the solar activity. The analysis of the observations by the Fast Fourier Transform (FFT) method revealed peaks close to two years, particularly in the upper monitoring domain. Model calculations by means of the real date model LIMA, Leibniz-Institute Middle Atmosphere model, reflect essential patterns of the water vapor variation. The FFT-analysis of the calculated water vapor mixing ratios also showed peaks of around two years. The real period of the QBO during the monitoring period ranged quite close to two years within the time interval considered, with the exception of the years 2001/02 when the period was essentially longer. Although the QBO is a phenomenon occurring in the zonal wind of the tropical stratosphere, we suppose an influence of the QBO on the water vapor distribution of the mesosphere of high latitudes controlled by transport processes. A possible link could be given by the planetary wave activity triggered by the QBO.

scholarly journals Three-Dimensional Modelling of EUVE Observations of the Io Plasma Torus

1996 ◽  
Vol 152 ◽  
pp. 457-464
Author(s):  
N. Thomas ◽  
D.E. Innes ◽  
R. Lieu
Keyword(s):  
Three Dimensional ◽  
Emission Lines ◽  
Shift Parameter ◽  
Model Calculations ◽  
Euv Emission ◽  
First Results ◽  
Plasma Torus ◽  
Io Plasma Torus ◽  
Semi Empirical ◽  
D Model

First results from a 3-D model of EUVE observations of the Io Plasma Torus are reported. The semi-empirical model calculations follow a method previously used to describe visible and near-UV emissions. The extension to EUV wavelengths is described. Several EUV emissions have been successfully modelled although some discrepancies remain at this stage. Most EUV emissions peak at a jovicentric distance of ≈ 5.8 RJ. The observed dawn-dusk asymmetry of the torus was well fitted with a shift parameter (ϵ) of 0.03. The modelling also indicates that optical depth effects need to be considered for several EUV emission lines.

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