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A Procedure to Determine the Optimal Sensor Positions for Locating AE Sources in Rock Samples

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Abstract

Within a research work aimed to better understand frost weathering mechanisms of rocks, laboratory tests have been designed to specifically assess a theoretical model of crack propagation due to ice segregation process in water-saturated and thermally microcracked cubic samples of Arolla gneiss. As the formation and growth of microcracks during freezing tests on rock material is accompanied by a sudden release of stored elastic energy, the propagation of elastic waves can be detected, at the laboratory scale, by acoustic emission (AE) sensors. The AE receiver array geometry is a sensitive factor influencing source location errors, for it can greatly amplify the effect of small measurement errors. Despite the large literature on the AE source location, little attention, to our knowledge, has been paid to the description of the experimental design phase. As a consequence, the criteria for sensor positioning are often not declared and not related to location accuracy. In the present paper, a tool for the identification of the optimal sensor position on a cubic shape rock specimen is presented. The optimal receiver configuration is chosen by studying the condition numbers of each of the kernel matrices, used for inverting the arrival time and finding the source location, and obtained for properly selected combinations between sensors and sources positions.

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Abbreviations

i, j :

Indexes counting the ith and jth AE transducers

t i :

Arrival time to the ith transducer

t j :

Arrival time to the jth transducer

t 0 :

Source origin time

(x 0, y 0, z 0):

Source spatial coordinates

(x i , y i , z i ):

ith Transducer spatial coordinates

(x j , y j , z j ):

jth Transducer spatial coordinates

v :

Velocity

a i,j , b i,j , c i,j , d i,j , p i,j :

Auxiliary parameters that depend on known values related to transducers i and j

p :

Vector containing theoretical model quantities related to transducers i and j

\({\mathbf{p}}^{ * }\) :

Vector containing actual measured quantities related to transducers i and j

q :

Vector of unknown parameters

H :

Kernel matrix of physical and mathematical information for the selected problem

e :

Vector of the measurement departures from the theoretical model

H + :

Pseudo-inverse of the matrix H

k(H):

Condition number of the matrix H

||H||:

L2-norm of the matrix H

λ max :

Maximum singular values of H

λ n :

Smallest non-zero singular value of H

δ p :

Small deviation in p vector

δ q :

Small deviation in q vector

||δ p|| :

L2-norm of δ p

||δ q|| :

L2-norm of δ q

U :

Matrix of the left singular vectors of the matrix H

V :

Matrix of the right singular vectors of matrix H

S :

Matrix of the singular values λ n of the matrix H

I :

Identity matrix

Δx :

Grid node spacing

\(\overline{V}\) :

Estimated or hypothesized average wave velocity

Δt :

Accuracy of the first arrival time picking

K :

Matrix of condition numbers

k :

Condition number

k opt :

Condition number of the optimal transducer configuration

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Authors and Affiliations

  1. Department of Structural, Building and Geotechnical Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Turin, Italy

    S. Duca & C. Occhiena

  2. Department of Environment, Land and Infrastructure Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Turin, Italy

    L. Sambuelli

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  1. S. Duca
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  2. C. Occhiena
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  3. L. Sambuelli
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Correspondence to L. Sambuelli.

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Duca, S., Occhiena, C. & Sambuelli, L. A Procedure to Determine the Optimal Sensor Positions for Locating AE Sources in Rock Samples. Rock Mech Rock Eng 48, 481–493 (2015). https://doi.org/10.1007/s00603-014-0582-0

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  • DOI: https://doi.org/10.1007/s00603-014-0582-0

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