E F F E C T S OF NUMBER OF EVENTS AND R E L A Y POINT DENSITY ON
A C C U R A C Y OF THREE-DIMENSIONAL AE-TOMOGRAPHY
Y . K o b a y a s h i T . Shiotani ^ and K . Oda '
' Department o f C i v i l Engineering, College o f Science and Technology, N i h o n University, T o k y o , Japan / Faculty o f C i v i l Engineering and Geosciences,
D e l f t University o f Technology, D e l f t , The Netherlands. Email: kobayashi.yoshilcazu@nihon-u.ac.jp ^ Graduate School o f Engineering / Graduate School of Business,
K y o t o University, K y o t o , Japan.
Department o f C i v i l Engineering, College o f Science and Technology, N i h o n University, T o k y o , Japan.
A B S T R A C T
This paper introduces results o f numerical investigations on accuracy o f elastic wave velocity distribution in Three-dimensional AE-Tomography. A series o f numerical analyses is conducted by changing number o f events and density o f relay points f o r the investigation. AE-Tomography is an identification problem and its number o f observation equations is in proportion to the number o f events, and further, the density o f the relay points is immediately correlated w i t h the resolution o f the source location that is significant f o r calculation o f travel time f r o m the source location to receivers. The investigation is carried out on a model o f tetrahedron that is meshed by using tetrahedral cells, and 8 receivers are settled at apexes and middle o f the edges of surfaces o f the model. The results o f the investigation demonstrate that the accuracy o f the source location and wave velocity distribution is strongly conelated w i t h the number o f events. I t was also demonstrated that the density o f the relay points affects to the estimated elastic wave velocity distribution, and better estimation o f the elastic wave velocity distribution is confirmed i n case o f dense installation o f relay points. However, these tendencies are confirmed qualitatively and suggest that more investigation is required to f i g u r e out the effects o f them to the estimated wave velocity distribution.
K E Y W O R D S
AE-Tomography, Elastic wave tomography, system identification, relay points, source location, ray-trace. I N T R O D U C T I O N
Elastic wave tomography has been applied f o r soundness evaluation o f concrete structures because o f its capability that enables to f i g u r e out elastic wave velocity distribution. Generally, the correlation between the elastic wave velocity and deterioration o f the material is w e l l k n o w n , and the extent o f the deterioration can be f i g u r e out f r o m the elastic wave velocity distribution. I t is note that source information such as source location and stimulated time are required f o r the elastic wave velocity tomography and it raises the cost o f investigation. On the other hand, as N D T by the elastic wave, A E Testing has also been adopted f o r the soundness evaluation. This technique assesses the deterioration o f the concrete f r o m the distribution o f A E events since it is a natural that A E events are concentrates i n the v i c i n i t y o f the defects such as cracks. However, the elastic wave velocity distribution must be assumed or given f o r computing the source locations in this technique. Thus, since each technique requires the information that is obtained by another technique, these techniques f o r m circular references each other. AE-Tomography was proposed by Schubert (2006) to unite the advantages o f the both techniques without the circular references. AE-Tomography figures out the source location o f A E events and elastic wave velocity distribution simultaneously, although the estimation o f the source location and elastic wave velocity distribution has been conventionally conducted separately as A E - T e s t i n g and Elastic wave velocity tomography. This algorithm o f AE-Tomography assumes straight ray-path on cross section o f interest to s i m p l i f y the computation. However, i t w o u l d cause error o f estimated elastic wave velocity distribution since the ray-path is not straight i n actual cases because the ray-path cannot be defined as a straight line on heterogeneous elastic wave velocity f i e l d as well known, and the heterogeneity generally exists i n the concrete structures. Hence, algorithms were proposed by the authors f o r A E - T o m o g r a p h y to take the heterogeneity o f the wave velocity distribution and reflection and d i f f r a c t i o n o f t h e elastic wave into consideration i n t w o - and three-dimensional manner (e.g. Kobayashi and Shiotani, 2012). A l t h o u g h it was demonstrated that these algorithms
estimate the source locations o f A E events and elastic wave velocity distribution adequately, the effects o f observation and model condition f o r the result of AE-Tomography have not been discussed yet. Therefore i n this paper, a series o f numerical analyses is carried out to investigate the characteristic o f the effect o f observation and model condition on the accuracy of estimated elastic wave velocity tomogram.
T H R E E - D I M E N S I O N A L A E - T O M O G R A P H Y
Ray-Trace technique
Conventionally, straight ray-path has been assumed to s i m p l i f y the computational procedure i n the source location and elastic wave velocity tomography. However, it is well k n o w n that the ray-path is not straight on heterogeneous wave velocity distribution, and it w o u l d affect results o f them since the heterogeneity o f the wave velocity distribution is always observed i n actual cases such as severely deteriorated concrete structures. The authors have proposed a ray-trace technique (Kobayashi, 2011) that considers the heterogeneity o f the elastic wave velocity. This method carries out the ray-trace on f i n i t e element mesh, and relay points are introduced to reduce its mesh dependency o f the ray-trace as illustrated in Figure 1. I t is assumed that elastic wave velocity is constant in individual cells and ray-path is defined as a polyline. This polyline is f o r m e d by connecting segments that are defined as a straight line that connects arbitrary combination o f two nodal or relay points. Then, travel time is computed in all o f ray-paths f r o m a source to a receiver, and the ray-path that gives m i n i m u m travel time is adopted as the ray-path f r o m the source to the receiver. Finally, the travel time on the ray-path is obtained as f o l l o w s .
U j ^ T - T i j (1) This algorithm is used f o r both o f estimation o f the source location and wave velocity distribution. This
algorithm has been extended f o r three-dimensional problem, and hexahedral and tetrahedral cells are available in present. r - O - O - O - p O - O - O - pO-CHO-| - o - o - o - r - O - O - O - i i N o o o ( O O < > (. (J O V ,) i.) O O ( ) O O O ( 1 O i> C / / i cy>o o ( 0 O < ^ O O O ( ) rj O O (' ) n ó O (• ^ O O O t i O OSD c O O < > () O () <: > . ) ! ) 1 1 ) O O O < |i O O O (' i—O-O-O—J j (1 .1 i-> V • n < -. , ( ) f,1 O ! ) 1 ) O O O c 1 (.1 r i r; < ') O i"i ( ' :] ,[) O O O c O O ^ ) (1 O O i' ) O O O < ) O r i ( ^ O O O c ' i ft n o ( i' -) O n n ( ) O O ! ) <^ ,'.)(': C, ( > V ) (' O C ) O n o < r O O <j ) o ^ 8 - a J 2_o O O i > , w w . < ) (; u O <: j) o u n t ,) ) O (.1 !. ) O O O <: ) (; O O ( ;) O O O <|: i O O O c O O < . • Cl i ) U l ) ( ) t, ï ' ^ ^ S O c S O O O c ;> :> u 11 i. ) '' 11 (.1 i; 1 : n l> <[ ) O O O ^ ) (•/ () (1 ( p a ^ o c 3 O O O <^ - ; ' O O f i O O O ( > O O O c ) ; . ) 11 i 1 (, p O o^ i D O O O < ') :, ; O O (; } . . r, : j ) O O O ) O O O E
O Relay point
< Represented ray-path
Figure 1 Conceptual sketch o f ray-trace w i t h relay points Source location and wave velocity distribution
The authors proposed two-dimensional algorithm f o r AE-Tomography (Kobayashi and Shiotani, 2012). I n this algorithm, the source locations and occurrence times o f A E events are computed by using arrival time at receivers and given elastic wave velocity distribution on first stage, and then theoretical travel times are calculated by carrying out ray-trace f r o m the source location to the receivers. Finally, the elastic wave velocity distribution is updated by m i n i m i z i n g difference between the observed arrival times and theoretical arrival times i n all events. This procedure is iteratively conducted until adequate criteria are achieved. I n this section, the algorithm is briefly introduced.
The source location technique is a significant part o f the algorithm o f AE-Tomography. Conventionally, the source location has been estimated by assuming homogeneous elastic wave velocity distribution and straight ray-path f r o m the source to the receiver on a cross section o f interest. A l t h o u g h these assumptions w o u l d be useful i n term o f reduction o f computational cost, it w o u l d be a cause o f inaccurate result because the ray-path is not straight in actual case especially i f the wave velocity shows strong heterogeneity as well as the ray-trace. Thus, the raytrace algorithm that was introduced i n previous section has been adopted f o r the algorithm o f A E -Tomography. I n this algorithm, the ray-trace is carried out f r o m a receiver on a given elastic wave velocity distribution on a primary stage. This procedure gives travel time Ty f r o m the receiver to all o f the nodal and relay points as illustrated in Figure 1. I n which, / is a number o f nodal or relay point at the receiver and j is a
number that is assigned to a nodal or relay point o f destination. Then, the possible occunence time o f the A E event can be estimated as f o l l o w s at the nodal or relay point o f destination j.
t i j ^ T i - T j (2)
in w h i c h , T; is the a n i v a l time at receiver /. This procedure is conducted f r o m all o f the receivers, then each nodal or relay point has plural possible occun-ence time o f A E events. The number o f the possible occurrence time is identical to the number o f receivers, and these times must be the same i f the ray-path and elastic wave velocity are exactly represented. However, it is impossible in almost case because the resolution o f the representations depends on the mesh. Therefore, variance the possible occurrence time at the nodal or relay point j is computed at all o f the nodal and relay point at first, then a nodal or relay point that gives m i n i m u m variance o f the possible occurrence time is chosen as the source location since the possible occurrence times w o u l d be close each other near the source location and the variance w o u l d be small i n consequence. A n d furthermore, the occurrence time is estimated as average o f the possible occurrence time as f o l l o w s .
f, = (3) in which, N is number o f receivers.
Because the source location and occunence time o f the A E events, the estimation o f the elastic wave velocity can be estimated by conventional technique o f elastic wave velocity tomography. I n this stage, the theoretical arrival time at receivers can be obtained by using the estimated occurrence time and the travel time. Since the travel time f r o m all o f the receivers to the source location are already obtained as Ty, the theoretical arrival time
T'i is given as
T'i = tj + Tj (4)
Finally, the elastic wave velocity distribution is updated by using Simultaneous Iterative Reconstruction Technique (SIRT). S I R T is simple and robust method, and updates the elastic wave velocity distribution as f o l l o w s .
V (T - Tpiui / v , A^. = 2 . Ti / Z ' ' ' '
-i ' i
in w h i c h , AS,^ is variation o f slowness i n cell k, l^i is a length o f the segment o f ray-path i i n cell k and is a total length o f the ray-path /. Consequently, the slowness o f cell k is updated by the variation ASj^ as f o l l o w s .
S',^S,+ AS, (5)
where Sj^ is a updated slowness o f cell k. This procedure is iteratively carried out until the adequate criteria are achieved. It is noteworthy that the source location is updated in every iterative procedure as w e l l as the elastic wave velocity distribution.
N U M E R I C A L V E R I F I C A T I O N
A series o f numerical analyses is carried out to figure out the characteristic o f the algorithm f o r threedimensional A E T o m o g r a p h y on its accuracy i n a tetrahedral model that is illustrated i n Figure 2. For the A E -Tomography, the number o f events and resolution o f the source location are significant parameter on its accuracy under the same model because AE-Tomography is an identification problem and the number o f events and resolution o f the source location correlates the number o f observation equations and accuracy o f the observations, respectively. Thus, the number o f events and the resolution o f the source location are adopted as parameters f o r the investigation. I t is note that the resolution o f the source location can be controlled by changing the number o f the relay points. The conditions o f cases on the investigation aie .shown i n Table ? For the arrival time at the receivers, firstly virtual source locations are randomly generated i n the model as illustrated i n Figure 2 as black spheres, the arrival times are obtained by carrying out the ray-trace f r o m the virtual source location. I f the number o f events is less than the number o f generated sources, adequate number o f events is chosen f r o m the generated source. 8 receivers are installed as illustrated in Figure 2 as green sphere. Homogeneous wave velocity distribution is assumed in the analyses as the initial wave velocity distribution f o r AE-Tomography, and it is set to 4000 m/s.
R E S U L T S A N D D I S C U S S I O N S
Figures 3 to 6 show the estimated wave velocity distribution in each case. I n these figures, it is shown that the computed elastic wave velocity distribution i n the cases o f that number o f events is 100 are closer to the target
elastic wave velocity distribution than the cases o f that number o f events is 12. I t is natural that the better elastic wave velocity distribution is obtained in cases o f large number o f events because number o f observation equations is larger i f the number o f events is larger.
z W a v o V o l D C i t y [ m / s ] 4000 3000 3B00 3700 3600 3500 3400 3300 3200 3100 3000
Figure 2 Configuration o f target model Table 1 Parameters f o r Cases
N Num. o f Events Case 1 3 12 Case 2 3 100 Case 3 5 12 Case 4 5 100 V[m/s] 4000 3900 saoo 3700 3600 3500 3400 3300 — 3200
•
3100•
3000Figure 3 Wave velocity distribution i n Case
V[iii/s] 4000 - 3900 h 3800 3700 3600 3500 3700 3600 3500 3400 3400 — 3300 3200 — 3100 3000
Figure 4 Wave velocity distribution in Case 2
The number o f variables that is identified in these analyses is 71 because number o f cells is 71 and i t is assumed that the wave velocity is constant in individual cell i n this model. Since the number o f receiver is 8 i n all cases, the number o f observation equation is 96 i f the number o f events is 12. Thus, the number o f observation equations to the number o f variables ratio is 1.35. This ratio is very l o w in application o f the elastic wave tomography, and i t w o u l d be the reason o f the estimated elastic wave velocity distribution. Even i n cases 1 and 3, the slow velocity area is not detected at lower left area o f the figures although it should be identified as w e l l as the target elastic wave velocity distribution. On the other hand, the estimated elastic wave velocity I t is caused because insufficient number o f events. Generally, the l o w wave velocity area is detected i n case 2 and 4 although the velocity is higher than the target wave velocity because the number o f observation functions is
1200 and the ratio is 16.90. However, although this ratio is extremely higher than the case 1 and 3, the estimated wave velocity and target wave velocity are not quantitatively consisted w i t h each other. This may be possible that the resolution o f the estimated wave velocity is also depend on the configuration o f the receiver installation. Thus, the optimized receiver configuration would be required to study. Furthermore, it w o u l d be necessary to consider about the possibility o f the shadow zone problem during the identification procedure. On the effects o f the density o f the relay points, the weak correlation is found in the cases o f N = 3 and N = 5. I n contrast o f case 1 and case 3, w h i l e the l o w velocity area is found i n case 1, l o w velocity area was more averagely identified in case 3. In the case o f case 2 and case 4, the same tendency is observed. The reason o f that the l o w velocity area is locally estimated in case 1 and 2 w o u l d be f r o m the distribution o f the events since the events are randomly generated in the model and consequently the distribution is biased. Thus, the effect o f N should be more deeply considered by carrying out more cases o f analyses with various N and changing the distribution o f the events to figure out the cause o f this result.
Figure 5 Wave velocity distribution in Case 3 Figure 6 Wave velocity distribution i n Case 4 C O N C L U S I O N S
This paper presented the results o f numerical investigation on three-dimensional A E - T o m o g r a p h y . Consequently the f o l l o w i n g conclusions were drawn.
1. The quality o f the estimated elastic wave velocity distribution is affected by the number o f events that is directly conelates with the number o f observation equations.
2. The slow velocity area was correctly detected in case that the number o f events is sufficient.
3. The estimated elastic velocity distribution was not quantitatively consistent w i t h the target elastic wave velocity distribution even in the case o f sufficient number o f events. The cause o f this result should be more deeply considered.
4. The effect N was not clearly figure out by the series o f analyses. This should be studied more deeply by using more wide range o f N .
A C K N O W L E D G M E N T S
The authors gratefully acknowledge the financial support provided by Tobishima Corporation. R E F E R E N C E S
Schubert, F. (2006). ' T o m o g r a p h y Techniques f o r Acoustic Emission M o n i t o r i n g " , 9th European NDT
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Kobayashi, Y . and T o m o k i , S. (2012). "Seismic Tomography w i t h Estimation o f Source Location f o r Concrete Structures", Structural Faults and Repair-2012. Edinburgh
Kobayashi, Y . (2011). "Mesh-independent ray-trace algorithm f o r concrete structures". Proceedings o f the f i f t h conference on emerging technologies in N D T , pp. 103-109
