Int. J. Sensor Networks, Vol. x, No. x, xxxx 1

Distributed Tomography with Adaptive Mesh Refinementin Sensor Networks

Goutham Kamath*Department of Computer Science,Georgia State University,Atlanta, GA, 30303, USAEmail: gkamath1@student.gsu.edu*Corresponding author

Lei ShiDepartment of Computer Science,Georgia State University,Atlanta, GA, 30303, USAEmail:lshi1@student.gsu.edu

Edmond ChowCollege of Computing,Georgia Institute of Technology,Atlanta, GA, 30332, USAEmail:echow@cc.gatech.edu

Wen-Zhan SongDepartment of Computer Science,Georgia State University,Atlanta, GA, 30303, USAEmail:wsong@gsu.edu

Abstract: Existing seismic instrumentation systems do not yet have the capability to recover thephysical dynamics with sufficient resolution in real time. Currently, seismologists use centralizedtomography inversion algorithm for which the data is gathered either manually from each station orby using limited number of expensive broadband stations. This scheme can take months to generatetomography and also lack the resolution due to limited number of sensors. It also introduces abottleneck in computation and increases the risk of data loss in case of node failures, especially thebase station. To address these issues a distributed approach is required which can avoid costly datacollection from large number of sensors and perform in-network imaging to obtain high resolutionreal-time tomography. In this paper, we present a distributed adaptive mesh refinement solution toinvert seismic tomography over large dense network, which avoids centralized computation andexpensive data collection. Our approach first discretizes the high fidility data and later filters themusing adaptive mesh to make it well-conditioned. We show that this filtered well conditioned systemhas lower dimension and improved convergence rate than the original system, thereby decreasing thecommunication overhead over the network. The system is implemented and evaluated using a COREemulator and the results show that our method is able to obtain high-resolution images in real-timeby distributing the computation load over the network.

Keywords: Distributed Sensing, Adaptive Mesh, Seismic Tomography, Sensor Network, In-networkComputing

Reference to this paper should be made as follows: Kamath, G.,Shi, L., Chow, E. and Song W.(2015) ‘Distributed Tomography with Adaptive Mesh Refinement in Sensor Networks’, Int. J. SensorNetworks, Vol. x, No. x, pp.xxx–xxx.

Biographical notes: Goutham Kamath received his B.E. degree from India in 2009 and M.S inElectrical Engineering from University of Wyoming in 2012. He is currently pursuing his PhD in theDepartment of Computer Science, Georgia State University. His research interests include wirelesssensor networks, distributed systems and mobile ad-hoc networks.

Copyright © 2015 Inderscience Enterprises Ltd.

2 G. Kamath et al

Lei Shi received his BSc degree in Software Engineering from Tongji University in 2007, and theMSc degree in Computer Science from Shanghai Jiao Tong University in 2010. He is currentlypursuing the PhD degree in the Department of Computer Science of Georgia State University. Hisresearch interests include wireless sensor networks, in-network processing and distributed systems.

Edmond Chow did his B.A.Sc. from University of Waterloo in 1993 and PhD from University ofMinnesota, 1997. After graduating he worked in LLNL until 2005 and later worked at D.E. Shawresearch until 2010. He is currently an associate professor at Georgia Tech. His research interestincludes developing and applying numerical methods and high-performance computing to solvelarge-scale scientific problems.

Wen-Zhan Song is a professor in Georgia State University. His research mainly focuses on sensorweb, smart grid and smart environment where sensing, computing, communication and control playa critical role and need a transformative study. His research has received 6 million+ research fundingfrom NSF, NASA, USGS, Boeing and etc since 2005, and resulted in 80+ journal articles, conferencearticles and book chapters in this area.

1 Introduction

Current volcano data collection and monitoring systems lackthe capability of obtaining real time information and alsorecovering the physical dynamics of seismic activity withsufficient resolution. At present, the seismic tomographyprocess involves aggregating raw data from seismic sensorsinto a centralized server for post-processing and analysis.The raw seismic samples are typically in the range of 16−24 bit at 50− 200Hz. These high precision sampling fromeach node makes it extremely difficult to collect raw, real-time data from a large-scale dense sensor network dueto severe limitations on energy and bandwidth. Due tothese restrictions many of the threatening active volcanoesworldwide use fewer than 20 nodes (Song et al., 2009),limiting our ability to understand dynamics and physicalprocesses of volcanoes in real-time. The centralized solutionintroduces a bottleneck in computation and also increases therisk of data loss in case of node failures. The advancementin current wireless sensor technology makes it possible todeploy and maintain a large-scale network for environmentalmonitoring and surveillance. However, the currently usedtomography algorithms cannot be easily implemented underthis distributed scenario as it relies on centralized processing.Thus, real-time volcano tomography requires a practicalapproach which is distributed, scalable, and efficient withrespect to tomography computation.

Tomography can be defined as the science of computingreconstructions in 2D and 3D from projections, i.e., solving thesystem obtained by integrations along the rays that penetratea domain Ω, typically a rectangle in 2D, and a box in 3D. Letus assume that there are P number of sensors deployed forvolcano monitoring. The discretized version of this problemtakes the form

Ax = B (1)

where,

A =

A1

A2

...Ap

;B =

B1

B2

...Bp

;Ai ∈ Rmi×n;Bi ∈ Rmi

where, Bi is a data or measurement vector, x ∈ Rn is an-dimensional model parameter vector and Ai ∈ Rmi×n is adata kernel which acts as a linear operator that establishesthe relationship between data and model parameters in i-thsensor. Further details regarding the formulation are providedin section 3.

To recover the magma/slowness modelx, geophysicists areinterested in solving (1) and at present these inversion methodsrely on centralized data gathering scheme (sendingAi, Bi toa base station) and has been implemented on volcanoes such asMount St. Helens (Lees, 1992), Mount Rainier (Moran et al.,1999) as well as many others. Currently, seismic communityfaces two major hurdles in-order to obtain both real-time andhigh-resolution tomography. First, the data logged at eachstations are of high fidelity and due to bandwidth limitations,information from only a small number of sensors can beobtained for real time imaging. Now, with limited sensorsthey cannot achieve high resolution tomography. On the otherhand, large arrays of sensors can be deployed to obtain high-resolution tomography. However, limited bandwidth restrictsthe transfer of large volume of data from these sensors in whichcase they resort for manual data gathering which takes monthsfor post processing and imaging. So to obtain high resolutionand real time imaging we need large number of sensor stationswhich has the capability of performing in-network computingand also avoid costly data collection. The authors in (Liu et al.,2013) have discussed the use of low-cost sensor for P-phasedetection of earthquake. The earthquake hypo-center detectionforms the basic step for seismic tomography and we extendthis further to obtain in-network imaging. Here, we assumethat the sensors used are low-cost and has low computationalpower eg: Raspberry Pi/Beagle-Bone/Android.

In this paper we present a distributed approach tosolve tomography problem using quadtree/octree basedadaptive mesh refinement (AMR). Quadtree/octree basedparametrization reduces the data overhead in the model spacewhile maintaining enough information that are needed forimage reconstruction. In this method, we simultaneouslydiscretize the high fidelity data on each node and later filterthem using adaptive mesh to make it well-conditioned. Thesefiltered well conditioned systems have lower dimension thanthe original problem and we prove that it has improved

Distributed Tomography with Adaptive Mesh Refinement in Sensor Networks 3

convergence rate. This method is implemented and evaluatedusing a CORE emulator and the results show that we are ableto obtain high-resolution images in real-time by distributingthe computation load over the network. We also comparethis method with (Kamath et al., 2013) and show that this issuperior in-terms of communication cost and also messageoverhead.

The rest of the paper is organized as follows. Section 2presents related work on adaptive mesh and iterative methodsfor reconstruction. In section 3 we provide backgroundon seismic tomography inversion and present the problemformulation. In section 4 we first discuss the advantages ofAMR and explain how AMR is achieved using quadtree/octreedecomposition. In section 5 we introduce the algorithmdistributed tomography using adaptive mesh refinement (DT-AMR) that solves seismic tomography problem in sensornetworks. Simulation results are shown in section 7. Finallywe conclude the paper in section 8.

2 Related Work

The tomography inversion process mainly involve solvinglarge sparse system of linear equations (1). Due to theproperty of sparsity and its sheer volume, iterative methodsbecame almost mandatory as they are more efficient in-termsof memory and computational requirements compared todirect methods (Saad, 2003). Several parallel and distributediterative methods have been developed and are currentlyused to solve a large variety of problems (Heath et al.,1991; Bertsekas and Tsitsiklis, 1991). All these methods aredeveloped mainly for GPU computing, where communicationbandwidth does not cause significant bottleneck. However inour case, the sensors which are deployed in harsh environmentcommunicate through limited bandwidth and has resourceconstraints such as power.

A popular iterative method for solving overdeterminedsystems was proposed by Kaczmarz (KACZ) (Kaczmarz,1937) which is an alternating projection method. This methodis also known under the name Algebraic ReconstructionTechnique (ART) in computer tomography (Herman, 1980).This algorithm does not require the full matrix to be inmemory at one time and can incorporate new information (raypaths), on the fly. The vectors of unknowns are updated afterprocessing each equation of the system and this cycle repeatsuntil convergence. These iterative algorithms are distributedby averaging the boundary information, e.g., ComponentAveraging (CAV) (Censor et al., 2001) and ComponentAveraged Row Projections (CARP) (Gordon and Gordon,2005). A survey paper comparing various block parallelmethods based on their performance on GPU’s is (Elble et al.,2010).

(Kamath et al., 2013) proposed an algorithm calledcomponent average distributed multi-resolution evolvingtomography (CA-DMET) which involved modification ofcomponent average type algorithms such as CAV and CARPfor seismic tomography. This was the first algorithm whichwas designed to run distributedly to solve tomographyproblem. Although they were able to obtain the image it

failed to deliver images with sufficient resolution and theconvergence stalled after certain iteration. Moreover, theiralgorithm was developed using regular grids i.e. the partialdifferential equation is discretized over a regular cell of samedimensions. In this paper, our main goal is to show that regulargrid partition is not suitable for distributed tomography and wedevelop irregular grid method which outperforms CA-DMETin terms of convergence and also communication overhead.

Adaptive mesh refinement (AMR) has been studied widelyand has been used as discretization tool for partial differentialequation as early as 1980 (Berger and Oliger, 1984). However,only until early 90’s it was used by seismology communityto solve inverse problem on small set of data (Michelini,1995). (Vesnaver, 1996) used SVD to interactively changethe boundaries while, (Curtis and Snieder, 1997) usedgenetic algorithm to optimize the parametrization. Thesealgorithms were suitable for small size data sets and requiredhigh computational power to run efficiently. (Spakman andBijwaard, 2001) came up with a less computation intensivesolution to parametrize the coefficient and this algorithm couldrun efficiently even for large matrices. However, this algorithmis only suitable for centralized architecture and is not feasibleto be implemented in a distributed scenario. Adaptive mesh haslots of advantages over regular mesh in terms of tomographyinversion and we will discuss this in section 4. To performadaptive mesh in a distributed case, we had to come up withsome novel method which was computationally light and alsosatisfy the requirements such as faster convergence. To thebest of our knowledge, our work is the first attempt to computeseismic tomography using adaptive mesh over a distributedsensor networks.

3 Problem Formulation

In this section we will first provide an insight on seismictomography and later explain the formulation of discretizedinverse problem which we intend to solve. For the sakeof simplicity we assume 2D model in order to derive theequations and can be easily extended to 3D case. Let us assumethere are P seismic sensors deployed on a square domainΩ = [0, 1]× [0, 1] (in arbitrary units), and also we are givena unknown function (slowness or velocity parameter) f(t) =f(t1, t2) that we wish to reconstruct. This function createsdamping of the signal (p-wave) caused due to earthquakewhich penetrates an infinitesimally small part dτ of a rayat position t such that it is proportional to the f(t)(d)τ .The data in tomography problem consists of measurementsof the damping of signals following ray tracing through thedomain Ω. See (Iyer and Dawson, 1993) for more details onmathematical formulation.

Let us first view this problem in terms of ith seismic node.The eth measurement be, for e = 1, · · · ,mi represents thetotal damping of signal e that penetrates Ω as shown inFigure 1. Here, mi denotes number of earthquake eventsdetected by i-th sensor node. Now, all the points te on rayecan be obtained by,

te(τ) = te,0 + τde, τ ∈ R (2)

4 G. Kamath et al

ray e

a1 a2

a3

a4 a5 a6 a7 a8

a21

a12

a57 a58 a59 a60a61 a62 a63 a64

a22

a30 a31

a40

Figure 1 Example of discretized tomography with N = 8 insensor i. The e-th ray intersect a total of 7 pixels, andthus the e-th row of matrix Ai has 7 non-zero elements(in columns 3, 12, 21, 22, 30, 31, and 40)

where te,0 is an arbitrary point on the ray, and de is a(unit) vector along the direction of the ray. Due to the aboveassumption, the damping associated with the e-th ray is givenby,

be =

∫ ∞−∞

f(te(τ))dτ e = 1, · · · ,mi (3)

where dτ denotes the integration along the ray.Next we will show how the continuous function f(t) can

be discretized by dividing Ω into grids of pixels. Thesegrids can be regular or irregular, each one having its ownadvantages and disadvantages. This papers core idea is to showhow discretization based on irregular grid can improve theconvergence rate thereby decreasing the communication cost.We also show that irregular mesh reduces the packet overheadwhile achieving better result compared to regular grids on adistributed network. But before that we shall explain how theproblem is formulated on a regular N ×N grid and later inthe next section we will show how it can be modified basedon irregular grid using adaptive mesh refinement technique.

When (3) is discretized onN ×N array of pixels assumingf(t) takes a constant value fkl over the pixels (k,l)

f(t) = fkl for t1 ∈ Ik and t1 ∈ Ilwhere,

Ik = [(k − 1)/N, k/N ], k = 1, · · · , NIl = [(l − 1)/N, l/N ], l = 1, · · · , N

Substituting (3) into (3)

be =∑

(k,l)∈raye fkl∆Lekl

where,∆Lekl = length of raye in pixel (k,l)

We can re-write the above equation in a more simplifiedform by numbering pixel (k,l) with j i.e. xj = fkl, where j =(l − 1)N + k , we get

be =

n∑j=1

aejxj , e = 1, · · · ,mi; n = N2

In the matrix form the equation formed at the ith sensornode is given by,

Aixi = Bi Ai ∈ Rmi×n, x ∈ Rn (4)

where,

Bi = [b1, · · · , bmi]T

aej =

∆Lekl (k,l) ∈ raye0 otherwise

Now for P number of sensors i.e. i ∈ 1, · · · , Pwe have

Ax = B (5)

where,

A =

A1

A2

...AP

;B =

B1

B2

...BP

;Ai ∈ Rmi×n;Bi ∈ Rmi

Current algorithms that solves (5) assumes a centralizedsetup that involve collection of data Ai, Bi from all thesensor nodes (mostly manually or sometimes using expensivebroadband station) to a centralized server. Although thecentralized techniques are well established and prove to bemore efficient in terms of final solution the process involved indata gathering turns out to be extremely slow (several months).With the increasing demand for high resolution and real-timeprediction there is a need for new distributed approach whichis scalable and efficient in terms of computation. In this paper,we present a technique that solves (4) locally using AMRinstead of (5) to obtain slowness/velocity model x with lowerdimension. In our approach since we solve (4) locally, weavoid collectingAi, Bi to a centralized server. In this method,computation is completely distributed however it involvesexchanging of partial solution xwhich is of lower volume andsparse. Equation (4) and (5) falls under the category of inverseproblems and the matrices arising from these equations aregenerally ill-posed. These ill-posed problem exhibit slowerconvergence rate which in our case increased communicationover the networks. In the next section we will exploit thegeometric features of the seismic tomography using AMRthrough which we can decrease the communication cost bymaking the problem well-posed.

4 Adaptive Mesh Refinement

Many geophysical inverse problems are ill-conditioned i.e.model space contains more details than it can be resolvedusing available data space (Bertero et al., 1985). Model spacematrixA typically has large null space and because of this, few

Distributed Tomography with Adaptive Mesh Refinement in Sensor Networks 5

portions of the solution cannot be resolved leading to its non-uniqueness. Geophysicists commonly use two approachesto overcome this problem: firstly, by making the problemwell conditioned using some a-priori information such assmoothness constraints, regularization etc (Herman, 1980).However, obtaining reliable prior information is hard andalso sometimes the smoothness constraints can be unrealistic.The second approach is by identifying the eigenvalue andeigenvector corresponding to the null space of model and laterremoving them explicitly. Although this method reduces theeffective information content of the data set in the model spacein a nontrivial manner, it can be used to obtain maximumamount of information that can be resolved from the currentdata set which will then reduce the amount of additional a-priori information to be included in the solution.

Removal of null space from the model data is equivalent toparameter reduction and here we will explain this with a simpleexample. Consider the path geometry of the ray produced bysource (circle) and receiver (square) as shown in Figure 2.Let us suppose that we have an error-free measurement ofaverage velocity and each of the four cells in Figure 2(a)have exactly equal ray path coverage. From measurementsalong the two left-hand paths, the quantity v1 + v2 can bedetermined exactly. The quantity v1− v2, however, remainsentirely unresolved by the data, and hence velocities v1 andv2 cannot be determined. Similarly, the combination v3 + v4may be determined exactly, but not v3− v4 and hence v3 andv4 cannot be determined.

Let v = [v1, · · · , v2]T , then well-determinedcombinations are e1.v and e2.v. Whereas, the undeterminedcombinations are e3.v and e4.v, where

e1 =

1100

; e2 =

0011

; e3 =

1−100

; e4 =

001−1

Figure 2 Cell geometries for the event (circle) station (square)paths shown. In (a) the cells bisect each path whereas in(b) each cell contains exactly one path length. Both cellgeometries have exactly the same homogeneous pathcoverage within each cell

Vectors e1 to e4 are the eigenvectors of the inverse problemand v1 to v4 the velocities we are interested. Parametercombinations parallel to eigenvectors e1 and e2 are completelydetermined if and only if (iff) the corresponding eigenvaluesare large, however combinations parallel to eigenvectors e3and e4 are undetermined iff they correspond to small or zeroeigenvalues.

Remark 1: Removal of parameters that creates null spaceinvolve modification of the grid structure based on raycoverage which also changes the resolution and informationcontent.

4.1 Adaptive mesh in seismic tomography

Now we will provide the motivation for using adaptive meshrefinement in seismic tomography with the help of real data.The distribution of earthquake events and the ray coverageover a region of interest are typically non-uniform in nature. InFigure 3, we plot some data collected from the seismic sensorsplaced over Mt. St. Helens,WA,USA. This data set is obtainedfrom 78 active stations spread across hundreds of km and thedata life time is over a span of 10 years. Figure 3(a) showsaround 1140 earthquake events distributed over a region. Itis observed that these events are non-uniformly distributedhaving dense population near crater region compared to therest. Figure 3(b) shows the histogram plot of number of eventsdetected by each station. From this we infer that, few stationsreceive upto 1000 good events where as others fail to receiveeven a single event. This in turn corresponds to an unevendistribution of rays among each sensor nodes. Figure 3(c) and(d) plots the ray path traced by a station named YEL placedover top of volcano with coordinates 46N12.58122W11.27.The entire region is viewed as a large cube discretized intoregular sized voxels. From Figure 3(c) and (d), it is discernedthat a large portion of the area is devoid of any penetration bythese rays. These regular grid spacing creates ill-conditionedsystem as model space contains more detail than it can beresolved using available data space (Bertero et al., 1985) andthis situation increases in case of distributed tomography.From analyzing data from two different sensor placed farapart, we saw that the ray coverage pattern in these nodesare different and it is often advantages to use different meshrefinement based on the local data to discretize rather thanuniform mesh on all the nodes.

60 70 80 90

3020

100

−10

X

Dept

h

(a) Earthquake Locations

0 10 20 30 40 50 60 70 800

200

400

600

800

1000

No.

of E

arth

quak

s

Station ID

(b) Event detection distribution

Node - YEL (46N12.58 122W11.27)

(c) Ray Tracing Side View

Node - YEL (46N12.58 122W11.27)

(d) Ray Tracing Top View

Figure 3 Non-uniform distribution of rays and events at Mt StHelens.

6 G. Kamath et al

Several methods like genetic algorithms, Singular ValueDecomposition (SVD) and other direct methods have beenused to reduce the null space. These methods are very effectivein conditioning the problem, however these methods are notfeasible to be implemented on sensors with low computationalpower and memory. For this reason our research directionfocused on reducing the null space by studying the geometryof the problem which involves ray coverage. (Vesnaver, 1996)and (Spakman and Bijwaard, 2001) studied this problem ofirregular grid extensively based on ray geometry and theypointed out the two situations that give rise to null space intomography.

1. pixels/grids not crossed by any ray.

2. groups of two or more ray that are linearly dependent.

By using above results, we tried to remove null space byfirst performing the ray tracing on the regular grid of finestresolution and formulated the problem as shown in (4). Thefinest level of discretization of ray coverage will be of the order128× 128 for 2D and 128× 128× 128 in case of 3D. Directmethods like SVD or genetic algorithm mentioned above willfail to compute null space considering the memory size andcomputational power of our sensors. So, we used a simplertechnique of measuring hit count (rays passing in each grid) toidentify the regions that are not crossed by the rays. Later, thiscriteria was used to merge the grids that did not have enoughrays or had rays which where linearly dependent. Below weshow the mathematical formulation.

Let us assume that we have a regular grid of finestresolution with non-overlapping cells which are defined witha constant functions rj(s), j = 1, 2, · · · , Nr such that:

rj(s) =

R− 1

2j s inside cell j with volume Rj

0 otherwise

The rj cells which are made of fine resolution becomethe building blocks in the construction of irregular cell. Wewish to construct the irregular cells ck(s), k = 1, · · · , Ncwith Nc Nr. The only restriction we have is the irregularconstructed cells are also non overlapping. The relationbetween the ck and the rj is given by

ck(s) =

Nb∑j=1

∆kjrj(s), (6)

∆kj = 1 when rs is a building block of cell ck and 0 otherwise.Calculating ∆ which forms the building blocks for

merging the finest resolution grid to form irregularmesh is computationally expensive. In our method theseare calculated using quadtree/octree based technique andperformed simultaneously on all nodes as it requires onlythe local ray information. The choice of quadtree or octreebased refinement to compute ∆ was mainly for three reason 1)computationally less intensive 2) can be stored efficiently ondevices with low-memory 3) forms structured cell partitioning.The structured cell partitioning plays a key role duringdistributed computing where the solution from different gridsneeds to be merged. In section 5 we show how this will simplify

the problem and reduce the communication overhead. Next,we will explain the adaptive mesh refinement method usingquadtree on 2D tomography problem. The method can beeasily extended to 3D case using octree.

4.2 Adaptive mesh using quadtree/octree

To perform quadtree based AMR on seismic tomography firstwe need to generate a density matrix I which is checkedfor homogeneity criteria and later split. Matrix I should bein Rn×n where n ∈ 2k, k > 0 and this restricts us to havethe finest resolution to be power of 2. We use hit count togenerate I i.e. Iij = number of rays passing through ij-thgrid of finest resolution. There are other ways to choose Isuch as region of interest and other hybrid methods (Spakmanand Bijwaard, 2001) and this requires domain knowledgeand will not be discussed in this paper. Now with only hitcount, I can be generated by performing columnSum(A) andthen reshaping it to n× n. To generate quadtree S we useQTDecomp function which first initializesS to size of I whichbecomes the root. Later this S is divided into four equal sizedsquares if the corresponding blocks in I satisfy the criterionof homogeneity. This process is further continued recursivelyand stops if either the block does not meet the criteria or depthof the tree is k i.e. until finest resolution. After obtaining thequadtree S we can easily generate ∆ by using algorithm [1].In this paper, we do not discuss the efficient way to implementquadtree data structures and further details on this can be foundin (Gargantini, 1982).

Algorithm 1 ∆← TransMat(S)1: for i← root until all nodes do2: if(i ==leafNode(S))3: idx = getIndex(i)4: ∆(i,idx) = 15: endif6: endfor

Next we will show how we can apply AMR to our problemand derive equations that satisfy the travel time setup explainedin section 3. Let A denote the data kernel formed by irregularmesh and the coefficients in Aek denotes the length of e-th rayin k cell. Computing Aek solely by using ray tracing can becomputationally expensive especially because of the irregulargeometry of the mesh. Therefore, to avoid that we use (6) toobtain the relationship between ray lengths on irregular andregular grids, and it is given by,

Aek =

N∑j=1

∆kjAej (7)

Figure 4 shows the relation between rays on two gridsthat is given by (7). In this, a2 is obtained by summinga3, a4, a11, a12 whereas, a7 to a10 maintains the finestresolution as regular grid. This criteria of splitting is decidedby ∆ as discussed earlier and now the equation can be re-written in matrix form as,

A = A∆T (8)

Distributed Tomography with Adaptive Mesh Refinement in Sensor Networks 7

ray e

a1 a2

a3

a4 a5 a6 a7 a8

a21

a12

a57 a58 a59 a60a61 a62 a63 a64

a22a30 a31

a40

(a) Regular Grid

ã1

ray eã2

ã3 ã4

ã5 ã6

ã7 ã8 ã11

ã12

ã13 ã14

ã15 ã16

ã9ã10

(b) Adaptive Grid

Figure 4 Relation between ray tracing in different grids

Substituting (7) and (8) in (4) equation at each nodebecomes,

Bi = Ai∆Ti y

Bi = Aiy

where, Bi and ∆i is the travel time vector andtransformation matrix formed at node i respectively. y is thenew model vector on irregular grid and can be transformed tooriginal grid by x = ∆iy

We summarize the entire process of adaptive meshrefinement using quadtree in Figure 5.

Regular Cell Model Highest Resolution

A

à = A ΔT

b = Ã y

Analyzex = Δy

Δ = TransMat(S)

(Cheap)

(Cheap)

(Solve for y)

Density Matrix I based on Hitcount

S = QTDecomp(I,Threshold)

Figure 5 Flowchart of the mesh refinement process

0 500 1000 150010

−30

10−20

10−10

100

Parameter Size

Sin

gula

r V

alue

s

Before−AMRAfter−AMR

(a) Singular values of A and A

0

5

10

15

0

5

10

15

0

5

10

15

(b) Octree based AMR

Figure 6 Effect of AMR on singular values and Octree basedAMR

We applied adaptive mesh refinement on seismictomography problem and we observed that it can improve thecondition number of matrix A by removing the small singularvalues as shown in Figure 6(a). Figure 6(b) shows the adaptivemesh refinement on 3D problem using octree decompositionand have finer grids at the center of magma and becomescourser towards outside.

Remark 2: Adaptive mesh refinement can be viewed as anon-trivial way of adaptive pre-conditioning as it decreasesthe condition number of the data kernel.

The technique of selecting pre-conditioner using geometryof the problem and simple parametrization technique(quadtree/octree) is computationally less intensive. However,it should be noted that AMR relies on the threshold we chooseand this requires domain knowledge. Also, if thresholds arenot selected carefully it may result in removal of good singularvalues thereby leading us to different or bad solution. We willshow the analysis of threshold sensitivity in the evaluationsection. Until now we have seen the working of AMR forseismic tomography and we have shown how quadtree/octreebased approach is suitable to run on sensor nodes. In the nextsection we will discuss how these transformed system of linearequation can be solved distributedly over a sensor network.

5 Distributed Tomography Inversion usingAdaptive Mesh

In the previous section we have seen how to transform thesystem of linear equation formed over regular grid to amore well-condition system using irregular partitioning. Nowapplying (8) and (4.2) to (5) we get,

A∆T y = B

Ay = B

where,

A =

A1∆T

1

A2∆T2

...Ap∆

Tp

=

A1

A2

...Ap

;B =

B1

B2

...Bp

; Ai ∈ Rmi×ni

From the above equation we see that at each sensor i ∈1, · · · , P Ai ∈ Rmi×n gets transformed to Ai ∈ Rmi×ni .Note that during this transformation the number of rays i.e.mi and right hand side Bi are unchanged while the number ofgrids are changed from n to ni where ni ≤ n. At each sensorstation the new linear subsystem formed after AMR is givenby Aiyi = Bi. The rows of this local subsystem contains raysand can be solved locally using row projection method likeART algorithm (2) due to the advantage it offers as mentionedin section 2.

The asymptotic convergence of this method requires 0 <λ < 2, where λ is called relaxation parameter. Recently the

8 G. Kamath et al

Algorithm 2 xk ←ART(λ,A, b, xk−1)1: xk,0 = xk−1,2: xk,i = xk,i−1 + λ

(bi−〈aTi xk,i−1〉)‖ai‖22

ai, i = 1, · · · ,m3: xk = xk,m

rate of convergence of ART was shown to be governed bycond(A) (Strohmer and Vershynin, 2009) where cond(A) =κ(A) = σmax(A)

σmin(A) denotes the condition number of A whereσmax and σminare its maximum and the minimum singularvalues. The partial solution obtained from all the nodes usingalgorithm (2) has to be combined to form next iterate, butbefore we go into that detail we will show how the transformedsystem at each node can obtain partial solution with fewersteps than the original system.

Theorem 1: Let x = ∆y be the solution to equation Ay =b, where A = A∆T Then algorithm (2) has E‖xk − x?‖22 ≤E‖xk − x?‖22 where, x? is the true solution of original systemAx = b

Proof 1: Let the system of linear equation on regular gridarising due to tomography beAx = b. The transformed systemon irregular grid be Ay = b where A = A∆T ; x = ∆y From(Strohmer and Vershynin, 2009) the convergence rate of ARTon regular grid is given by,

E‖xk − x?‖22 ≤[1− 1

κ(A)

]k‖x0 − x?‖22 (9)

similarly for irregular grid we have,

E‖xk − x?‖22 ≤[1− 1

κ ˜(A)

]k‖x0 − x?‖22 (10)

We have shown that AMR reduces the null space of the systemi.e 0 ≤ σmin(A) σmin(A)

Since κ(A) = σmax(A)σmin(A) and σmax(A) ≈ σmax(A)

We have, κ(A) κ(A), i.e the system becomes well-conditioned after AMR

Now we have[1− 1

κ ˜(A)

]k[1− 1

κ(A)

]kMultiplying both sides by, ‖x0 − x?‖22 we get,[1− 1

κ ˜(A)

]k‖x0 − x?‖22

[1− 1

κ(A)

]k‖x0 − x?‖22 (11)

Using (11) in (9) and (10) we get,

E‖xk − x?‖22 ≤ E‖xk − x?‖22 (12)

Now after taking fewer iteration to solve Aiyi = Bi wewill show how to combine the partial solution to form thenext iterate. We borrow the idea from (Gordon, 2006) whoproposed parallel-ART which is an extension of Cimmino’smethod to a generalized case. (Gordon and Gordon, 2005)and (Gordon, 2006) proved that the intermediate projectionC1, C2, · · · , CP can be averaged to obtain next iterate xk+1.Algorithm (3) describes the parallel-ART.

The line 4 in algorithm (3) performs averaging of all thepartial solution to form the next iterate xk+1 which will be

Algorithm 3 Parallel-ARTINPUT1: A = A1, · · · , AP ; B = B1, · · · , BP OUTPUT1: x← minx ‖Ax− B‖REPEAT1: fork ← 0 until convergence or maximum number of iteration2: do parallel in all blocks i ∈ 1, · · · , P3: xki = ART(λ,Ai, Bi, x

k)4: xk+1 = 1

P

∑Pi=1 x

ki

5: end

used as a initial input to ART in the next iteration. This processcontinue until convergence is met or for fixed maximumnumber of iterations. The partial solution in (Gordon, 2006;Gordon and Gordon, 2005) have same size i.e. xi ∈ Rn.However, in our case due to AMR the partial solution xi ∈Rni and we show how we can perform averaging over partialsolution of different dimension using generalized averaginglemma .

Lemma 1: (Generalized Averaging Lemma)Let xi ∈ Rni be the projection obtaining after performingART(λ, Ai, Bi, xk−1) on ith sensor node with irregular grid.The next iterate takes the form xk+1 ← (D · Xk)/P where,D = (∆1, · · · ,∆P ) and Xk = (xk1 , · · · , xkP )T

Proof 2: Performing ART simultaneously on all the P nodeshaving irregular mesh givesxki ← ART(λ, Ai, Bi, xk−1), Ai ∈ Rmi×ni , Bi ∈Rmi , xki ∈ Rni

From (4.2) we have, ∆i : Rni → Rn where,n is the dimensionof the regular mesh. Therefore, to form the generalizedaverage we needxk+1 = ((∆1x

k1), · · · , (∆P x

kP ))/P

xk+1 = ((∆1, · · · ,∆P )(xk1 , · · · , xkP )T )/Pxk+1 = (D · Xk)

The complete DT-AMR is given in algorithm (4). In thisalgorithm the initialization, ray tracing and adaptive mesh isperformed only once and done simultaneously on each node.The steps that involves communication is highlighted in bold.Sending ∆i to SINK in step 4 is done only once and can bedone cheaply as ∆i can be encoded efficiently as mentionedin section 4. The actual communication in the network occursin the line 5 and 7 of distributed tomography which involvesaggregation of partial solution of size ni from all the nodesand then broadcast averaged result back. The worst casecommunication cost for this given by P

∑i dim(xi). Since

this needs to be broadcasted back and algorithm convergesafter k iterations, the worst case communication cost is2kP

∑i dim(xi). Although this cost may seem to be high,

we will see in the next section that it is infact lesser thancentralized methods where each row of Ai, Bi needs totransferred. Moreover, the averaging over the networks can bedone by communicating only with the neighbors and we leavethis to the future work.

Distributed Tomography with Adaptive Mesh Refinement in Sensor Networks 9

Algorithm 4 Distributed Tomography using Adaptive MeshRefinement (DT-AMR)Initialize1: Number of seismic sensors P and AMR threshold T2: Finest resolution dimension Q = d× d or Q = d× d× d3: Initialize a SINK node for aggregation.Ray Tracing at each node i1: Upon the detection of events2: Perform ray tracing on each node simultaneously to obtainAi andBi

Adaptive Mesh at each node i1: Obtain Density matrix I from Ai

2: S = QTDecomp(I,T) ; ∆i = TransMat(S)3: Ai = Ai∆

Ti

4: Send( ∆i)→ SINKDistributed Tomography1: k ← 0, xk ← 02: while not converged do3: In Every node i for 1 ≤ i ≤ P do in parallel4: xi ← ART(λ, Ai, Bi, x

k,Iteration)5: Send( xi)→ SINK to average6: x(k+1) =

1P

∑Pi=1 ∆ixi

7: Broadcast x(k+1) to all the node P8: k ← k + 19: end while

10: Update slowness model: x = xk−1

11: TERMINATE

6 Communication Architecture

In this section, we present the communication architecture toevaluate DT-AMR algorithm on large scale sensor network.The protocol is tested on CORE2 and EMANE3 networkemulators Ahrenholz et al. (2011). The advantage of emulationon CORE is that the code developed here can be transplanted toa Linux-based device, e.g., BeagleBone Black board, virtuallywithout any modifications.

DT-AMR is designed to compute the tomography in awireless mesh network and requires both unicast and broadcastcommunication according to the system architecture and thealgorithm requirements. On most remote deployment sites, itis hard to rely on the pre-existing infrastructures (e.g. cellularinfrastructure). Therefore, we need to utilize the wireless meshnetworking which creates its own infrastructure by multi hoprelays. However, such systems may experience erratic linkqualities and intermittent disconnections among nodes. Thesecharacteristics, combined with unpredictable environmentalconditions, make it difficult to maintain efficient and reliableend-to-end connectivity that spans many hops. For example,the traditional end-to-end protocol like TCP is not suitable fora wireless mesh network in challenging environment becausethe packet lost ratio is higher than a wired network. Forexample in a multi hop transmission using TCP, the sourcenode needs to retransmit the packets through all hops if thepacket gets lost on the path. This decreases the data rate afterseveral hops due to packet loss and congestion control.

To address the challenges in wireless mesh networking,we adopt Disruption-Toleration Networks (DTN) techniquesto maintain efficient and reliable end-to-end connectivity that

TCP UDT NORM

UnicastTransport Layer

Bundle Layer

Application Layer

Broadcast

Convergence Sub-Layer

Cache Management Sub-Layer

TCP Adapter

UDTAdapter

Queue in Mem 2nd Queue in Disk

loadpersist

send2nexthop recv

sendBndl recvBndl

Bundle

ACK

ACKACK

Figure 7 Bundle Layer Architecture

spans many hops for data delivery. In our design, the datais buffered in a bundle and then transferred hop by hop in astore-and-forward manner until it arrives at the destination.Our implementation of DTN technique does not make anychanges to underlying network services, it uses TCP for one-hop reliable bundle transfer, and uses routing table to indicatethe next hop. Figure 7 shows the application interfaces on eachnode for the integration of DTN and routing protocol. For DT-AMR emulation, we implemented a naive data aggregationand dissemination protocol on top of bundle layer. In this setup,we specified a SINK node to which all the other nodes sent itspartial solution. Next, the SINK node computed the (k + 1)-thiterate and disseminated the solution back to all nodes.

Figure 8 shows that the Bundle Layer outperforms TCPwith routing protocol. The test is done using CORE andEMANE for 100 nodes multi-hop network settings. Besidesunicast, we implemented a delay-tolerant broadcasting servicebased on the NACK-Oriented Reliable Multicast (NORM)protocol. Using NORM interface, one node can push a bundlereliably to its one-hop neighbors. Our cache component canreceive and store this broadcast bundle and rebroadcast it againwith NORM to the nodes that are two hops away and so onso forth. A redundancy check module is developed in thecache component which guarantees that each node receivesthe same bundle at most once. The implementation of all thealgorithms of CA-DMET is in ANSI C. The event locationand tomography inversion related code are cross-compiled torun on BeagleBone Black board. All other code can be directlyported to embedded system such as ARM-based CPU or MCU.

7 Evaluation and Validation

In this section, we evaluate the DT-AMT algorithm andpresent the simulation results. First we give the descriptionof the synthetic model used for the simulation along with theexperimental setup and later we compare DT-AMR with theexisting distributed algorithms such as, CARP (Gordon andGordon, 2005) and CAV (Censor et al., 2001). Typically, totest tomography inversion algorithm a synthetic model is used.This serves two purpose: a) the real data set such as from Mt. StHelens do not have a ground truth and it is still uncertain which

10 G. Kamath et al

10 20 30 40 500

100

200

300

400

500

600

Latency (ms)

Thro

ughp

ut (k

bps)

TCP ThroughputBundle Throughput

(a) B.A.T.M.A.N. routing

10 20 30 40 500

100

200

300

400

500

600

Latency (ms)Th

roug

hput

(kbp

s)

TCP ThroughputBundle Throughput

(b) OLSR routing

Figure 8 Performance of Bundle layer vs TCP.

model is reliable. b) The simulation using synthetic modelenables us to investigate individually various phenomenawhich cannot be separated physically. For example, p-wavedata always contain noise due to measurement and scattering,but simulation can indicate the specific effect separately. Forthis reason, we adopt a synthetic data of a fault model from(Hansen and Saxild-Hansen, 2012) which has been widelyused for cross bore-hole tomography (Curtis and Snieder,1997). This fault model is created with velocities of 0.75Vfor the right fault and 1.0V for the left fault as shown in theFigure 9(a).

We evaluated the communication cost of DT-AMRalgorithm using CORE network emulator (Ahrenholz et al.,2011). We select CORE as the development and evaluationplatform because the sensors that will be deployed on the realvolcano will be some-low powered linux based devices suchas android, beagle-bone or raspberry pi. Code developed inCORE emulator can be transferred to these devices withoutany modification. A network of 32 nodes are deployed whichdetects the event and traces the ray as shown in the Figure 9(b).We add Gaussian noise to the obtained travel time to modelthe receiver error. The finest resolution of dimension 32× 32is used as a regular grid to construct adaptive mesh usingquadtree. The threshold for the hitcount is chosen to be 20. Forthe iterative methods, the selection of relaxation parameter ρare critical and in all of our experiments this remains constantthroughout the iterations, i.e., ρk = ρ = 0.25 for all k ≥ 0.Rate of convergence of different algorithms are comparedusing relative updates, φ = |x(k+1) − x(k)|/|x(k)|, residualsχ = ‖Axk − b‖ and absolute error ε = ‖x∗ − xk‖ where x∗

is the ground truth.

(a) Magma/Fault Model

Receiver Source

(b) Ray Traversal

Figure 9 Synthetic Model

7.1 Centralized AMR

Adaptive mesh has been applied on seismic tomography ina centralized setup earlier and has proven to perform better(Vesnaver, 1996)(Spakman and Bijwaard, 2001). However,quadtree based adaptive mesh for seismic tomography hasbeen applied for the first time and we validate our approachusing similar steps. Quadtree based AMR has been developedspecifically to work in distributed environment and we donot expect it to perform better than centralized algorithmsmentioned in (Vesnaver, 1996)(Spakman and Bijwaard, 2001).We perform quadtree based AMR on synthetic model andrun ART which is a common centralized algorithm used forcomputing tomography. We see the advantage of AMR inFigure 10(a) and (b), where the relative and absolute errordecreases significantly when AMR is used. AMR makes thesystem well conditioned i.e reduce the zero eigenvalue andwhen the system is well-conditioned the solution obtained willbe closer to the ground truth. From these tests we can concludethat ART with AMR is better and we can obtain better solution.

0 10 20 3010

−5

100

Iteration

Rel

ativ

e E

rror

AMR−ARTART

(a) Relative Error in ART

0 10 20 30

10−0.9

10−0.6

10−0.3

Iteration

Err

or

AMR−ARTART

(b) Absolute Error in ART

Figure 10 Comparing effect of AMR in Centralized ART

7.2 Distributed AMR

After validating the performance of our quadtree based AMRin centralized setup, we now perform series of experiment ondistributed network. We compare our DT-AMR with standarddistributed algorithms such as CARP and CAV. We useresiduals and absolute error as the parameter for comparisonand results are shown in Figure 11. These plots demonstratethat there is a difference in the initial convergence behaviorin these algorithms both in-terms of residuals and absoluteerror. The final solution obtained from DT-AMR is alsomore closer to the ground truth and can be obtained withfewer iteration. The iteration on the x-axis represents thetotal number exchange of partial solution required during theintermediate step.

DT-AMR takes the advantage of partitioning the systemof equation based on its resolving power at each node makingit more well conditioned. The local computation on thewell conditioned system on single node will accelerate theconvergence and this is shown in Figure 12, where we analyzethe performance on node 1 and 4. We observe that even ateach node the partial solution obtained from DT-AMR issignificantly better than CARP (Theorem 1) and this is themain reason for the improved performance of DT-AMR indistributed environment.

Distributed Tomography with Adaptive Mesh Refinement in Sensor Networks 11

0 10 20 30 4010

0

101

102

103

Iteration

Res

idua

ls

DT−AMRCARPCAV

(a) Residuals

0 10 20 30 4010

0

101

102

Iteration

Err

or

DT−AMRCARPCAV

(b) Absolute Error

Figure 11 Comparing effect of DT-AMR with CARP and CAV

0 10 20 30 40

10−0.015

10−0.012

10−0.009

10−0.006

Iteration

Err

or

DT−AMRCARP

(a) Absolute Error in Node 1

0 10 20 30 4010

−0.011

10−0.009

10−0.007

Iteration

Err

or

DT−AMRCARP

(b) Absolute Error in Node 4

Figure 12 Comparing effect of DT-AMR in single Node

The DT-AMR is designed to run both in clustered and alsoin fully distributed setup. To test this, we group the nodesinto clusters each having cluster heads which collects Ai andbi from its neighbors and run DT-AMR. The simulation wasdone with total 32 nodes and number of clusters varying from8 to 32 (each cluster has one node i.e., fully distributed).From Figure 13 we observe that as cluster number increasesthe convergence rate decreases. This is one of the drawbacksof distributed iterative methods and the reason behind thisis the way these algorithm projects and combine the partialresults. Centralized methods are always better in-terms ofcomputation and final solution especially for inverse problems.

0 10 20 3010

0

101

102

Iteration

Err

or

P = 8P = 16P = 32

(a) Absolute Error in DT-AMR

0 10 20 3010

0

101

102

103

Iteration

Res

idua

ls

P = 8P = 16P = 32

(b) Residual in DT-AMR

Figure 13 DT-AMR with different partitioning

7.3 Communication Cost and Robustness

In section 5 we have seen the summation of partial solutionon the network and dissemination of the computed resultconstitutes the major part of communication in DT-AMR.In this setup, we specified a SINK node to which all theother nodes send its partial solution. Next, the SINK nodecomputed the (k + 1)-th iterate and disseminated the solutionback to all nodes. For fair comparison we implemented similardata aggregation protocol (without the dissemination part) for

centralized scheme where all nodes sent its ray informationto the SINK placed 1) at the corner CENT(C) and 2) in themiddle CENT(M).

To compare the communication cost we increased thenumber of receivers from 32 to 49 which formed 7× 7 gridof mesh network. From Figure 14(a) and (b), we can see thatcommunication cost in a centralized setup is high near thebase station as all the ray information is transferred over thenetwork and its volume depend on the number of earthquakeevents.

In case of DT-AMR we used data aggregation anddissemination as mentioned above with aggregation SINKnode in the middle. Figure 14(c) shows the communicationpattern and from this we can see that the communicationcost for DT-AMR is less than centralized scheme. Thisis mainly because distributed method depends on numberof iteration and typically with semi-convergent property ofiterative method the number of iteration is much less comparedto number of earthquake events. The advantage of DT-AMRmethod is that the communication depends only on theaveraging over the network and this can be done using gossipmethods which involves information only with neighbors andcan provide flat balanced communication pattern. This isbeyond the scope of this paper and will be investigated infuture. AMR decreased the size of partial solution xi fromn→ ni. We validate this effect by comparing volume(bytes)transferred by DT-AMR to that of without AMR. Figure 14(d)shows that in both cases SINK(M) and SINK(C) the volumeof bytes transferred is less in case of DT-AMR.

Next we performed experiments to validate the robustnessof DT-AMR with same data set but with two different packetloss ratios of 20% and 40% at each node. Figure 15 shows thefinal tomography obtained and from this we can see that dueto the lost partial solution at each iteration the solution is notvery accurate compared to one with no lost packet. However,with Figure 15(b) and (c) we are still able to identify the regionof fault although it is not very accurate.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) No Packet Loss

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(b) 20% Packet Loss

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(c) 40% Packet Loss

0 10 20 30

10−0.7

10−0.5

10−0.3

10−0.1

Iteration

Err

or

HitCount−20HitCount−40HitCount−60

(d) DT-AMR with varying hit-count

Figure 15 Loss tolerance and threshold sensitivity of DT-AMR

12 G. Kamath et al

24

6

24

6

0

2

4

6

x 104

XY

Num

ber

of M

essa

ge

(a) Centralize(M)

24

6 24

6

0

2

4

6

8

x 104

YX

Num

ber

of M

essa

ge

(b) Centralize(C)

24

6

24

6

0

2

4

6

x 104

XY

Num

ber

of M

essa

ge

(c) DT-AMR(M)

0

5e+07

1e+08

1.5e+08

2e+08

2.5e+08

3e+08

3.5e+08

4e+08

4.5e+08

SINK(M) SINK(C)

Com

mun

icat

ion

Vol

ume

(byt

es)

DT-AMRNo-AMR

(d) Communication Volume

Figure 14 Communication Cost

Finally, we evaluate the performance of DT-AMR byvarying the threshold of the hit-count from 20 through 60.From Figure 15(d) we can see that the convergence speed andthe final result depends on the selection of hit-count thresholdas mentioned in section 4. To select this threshold we needdomain knowledge and if not selected carefully may result inremoval of good singular values that leads to different or badsolution.

8 Conclusion

In this paper, we presented a distributed approach basedon adaptive mesh refinement that performed tomographyinversion over sensor networks. This algorithm first convertedthe problem to a well-conditioned system by eliminating thenull spaces and retaining the information based on the nodesresolving power. Later, a distributed method was presentedwhich solved the inverse problem over sensor networks byexchanging on the partial solution to form next iterate. Theexperimental evaluation showed that our proposed methodconverges faster than CARP and CAV also obtained betterslowness closer to ground truth. Centralized methods aremore computationally stable and distributed method cannotperform better in terms of getting solution, however interms of communication and message exchanged DT-AMRalgorithm performed better than centralized algorithm. DT-AMR is designed to run on devices with low memory andcomputational power. Further enhancement of this algorithmcan be done by studying its performance on real dataand applying different constraints to form adaptive mesh.Our future research will focus on developing a distributedaveraging operator that can balance the communication cost.

Acknowledgements

Our research is partially supported by NSF-CNS-1066391,NSF-CNS-0914371, NSF-CPS-1135814 and NSF-CDI-1125165.

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