Received: 15 September 2011 – Accepted: 30 September 2011 – Published: 4 January 2012
Abstract. In this paper we suggest the use of diffusion-neural-networks, (neural networks with intrinsic fuzzy logicabilities) to assess the relationship between isoseismal areaand earthquake magnitude for the region of Greece. It isof particular importance to study historical earthquakes forwhich we often have macroseismic information in the formof isoseisms but it is statistically incomplete to assess mag-nitudes from an isoseismal area or to train conventional ar-tificial neural networks for magnitude estimation. Fuzzy re-lationships are developed and used to train a feed forwardneural network with a back propagation algorithm to obtainthe final relationships. Seismic intensity data from 24 earth-quakes in Greece have been used. Special attention is be-ing paid to the incompleteness and contradictory patterns inscanty historical earthquake records. The results show thatthe proposed processing model is very effective, better thanapplying classical artificial neural networks since the mag-nitude macroseismic intensity target function has a strongnonlinearity and in most cases the macroseismic datasets arevery small.
1 Introduction
A significant stage of modern seismic hazard techniques isthe assessment of the magnitudes (M) of the most severehistoric earthquakes. The difficulty in identifying large seis-mogenic faults corresponding to historical events and assess-ing their magnitudes increases the need to derive methodolo-gies to estimate the magnitudes of historic earthquakes fromthe macroseismic information depicted in the historic records(i.e. isoseisms).
Obviously, there is no established consensus on how to getan objective estimate of the magnitude of an earthquake frommacroseismic data alone. In the absence of a simple physical
link betweenM and macroseismic intensity, the only alterna-tive is to investigate the validity of an empirical relationshipusing statistical methodologies.
During the past years, the relationship between macroseis-mic information and earthquake magnitude has been inves-tigated by many researchers either by providing some func-tional form, (e.g. Gupta and Nuttli, 1976; Vassileva, 2001;Tselentis and Danciu, 2008), between seismic intensity, (herewhen we refer to seismic intensity we mean the ModifiedMercalli intensity MMI), andM throughout regressive anal-ysis. Destructive earthquakes are infrequent with very smallprobability of occurrence. Thus, observations used in esti-mating seismic magnitudes from intensity data are incom-plete and form a small sample.
Recently, many researchers have established regressionanalysis models based on powerful statistical tools to developvarious nonlinear relationships between MMI,M, and vari-ous seismic engineering parameters (e.g. Tselentis and Dan-ciu, 2008; Tselentis and Vladutu, 2010; Tselentis, 2011).Furthermore, the empirical relationships derived from thistypes of analyses are normally characterized by large scatter-ing of the data due to the inherent uncertainty of the intensityparameter and by the uncertainty on the hypocentral depths.It is well known that for the same magnitude earthquake, thedeeper the hypocenter the smaller the effects on the Earth’ssurface, thus the magnitude of shallow earthquakes is over-estimated while the magnitude of deeper events is underesti-mated.
To overcome this depth uncertainty, Galanopoulos (1961)proposed the use of the area or average radius of individualisoseisms, or the felt area (the area where the earthquake wasactually felt by people). This methodology gives a better esti-mate ofM than a single MMI value. Even with this method-ology, however, the hypocentral depth plays a significant rolebecause it controls the decay of intensity with distance.
Published by Copernicus Publications on behalf of the European Geosciences Union.
38 G-A. Tselentis and E. Sokos: Relationship between isoseismal area and magnitude of historical earthquakes
Another approach which has also been used to estimateseismic magnitudes from macroseismic information (iso-seisms) was to synthesize isoseisms numerically correspond-ing to various seismogenic faults (e.g. Suhadolc et al., 1988;Zahradnic, 1989; Sirovich, 1996) and compare it with ob-served data throughout a trial and error process. Recently,for non-linear modelling, neural networks and neuro-fuzzymodelling approaches have received a great deal of attention(Huang and Leung, 1999; Tselentis and Vladutu, 2010; Tse-lentis, 2011). Neuro-fuzzy modelling is concerned with theextraction of models from numerical data representing thebehaviour of a system. The models in this case are rule-basedand use the formalism of fuzzy logic, i.e. they consists of setsof fuzzy “if-then” rules with possibly several premises (Mor-aga, 2000). These models, known as hybrid neural fuzzy,combine the stochastic and artificial intelligence approachesand they are particularly suited for data sets with very weakstatistical properties.
In the present paper, we investigate the efficiency of hybridfuzzy-neural-network models in correlating seismic magni-tudes with the isoseismal area using a data set of 24 earth-quakes from Greece (Table 1), for which we have well knownmacroseismic information in the form of isoseisms.
2 Data
Most of the intensity information was available through theEuropean Strong Motion Database (Ambraseys et al., 2004)and completed by the macroseismic database developed byKalogeras et al. (2004). The macroseismic database coversmost of the strong earthquakes occurring in Greece and foreach event, MMI values are assigned to every recording sta-tion.
The isoseisms corresponding to each earthquake were con-structed using the kriging methodology. Kriging is a sta-tistical technique that estimates unknown values at specificpoints in space using data values from known locations. Themain assumption when using kriging is that the data anal-ysed are samples of a regionalized variable, as is assumedto be the case with intensity data. A regionalized variablevaries continuously in such a manner that points near eachother have a certain degree of spatial correlation, but pointsthat are widely separated are statistically independent.
The kriging estimator applied in the macroseismic datasetconsidered in the present paper is given by
Ij =
n∑i=1
wij MMI i (1)
whereIj is the predicted intensity value at any grid node, n isthe number of points used to interpolate at each node, MMIi
is the intensity value at thei-th point andwij is the weightassociated with thei-th data value when estimatingIj . Theweights are solutions of a system of linear equations which
Fig. 1. Example of Isoseisms constructed by the kriging method-ology and corresponding macroseismic data for 3 earthquakes inAttica.
are obtained by assuming thatI is a sample-path of a randomprocess and that the error of prediction is minimal.
The kriging algorithm assigns weights to each point basedon the distance between the point to be interpolated and thedata location (h), as well as the inter-data spacing. Other pa-rameters, such as length scale, repeatability, and direction de-pendence of data are also considered for assigning weights.These parameters are entered into the algorithm via the var-iogramγ (h), which is an analytical tool that quantifies thedegree of spatial autocorrelation of data.
In the present investigation, the isoseismals that more ac-curately represented the observed intensity data field werechosen by modelling a simple linear variogram based on thekriging options of Surfer Package from Golden Software. Adetailed explanation of the kriging algorithm and the vari-ogram parameters can be found in De Rubeis et al. (2005).Figure 1 depicts some examples of the estimated isoseismsfor the case of 3 earthquakes in Attica (Central Greece) andthe corresponding macroseismic data.
Since the relation between isoseismal area and seismic mag-nitude is strongly nonlinear, artificial neural networks (ANN)are particularly suited for treating macroseismic data (Tunget al., 1994; Davenport, 2004; Tselentis 2011).
An ANN is an information processing paradigm that isinspired by the way biological nervous systems, such asthe brain, process information. ANN can be understood asa mappingf : Rn → Rm, defined byy = f (x) = g(W.x),wherex is the input vector,y is the output vector,W is theweight matrix andg is a nonlinear activation function. Themappingf can be decomposed into a series of mappings re-sulting in a multi-layer network (Fig. 2):
Rn → Rp → Rq → ... → Rm
The algorithm for computingW is the training algorithm.The most popular ANN (this kind of ANN will be used in thepresent investigation) are the multi layer back propagationnetworks (Rumelhart and McClelland, 1986), whose trainingalgorithm is the well-known gradient descendent method. Inthe learning phase of training such a network, we present thepatternxp = {ipi} as an input and ask that the network adjustthe set of weights in all the connecting links and also all thethresholds in the nodes such that the desired outputsyp =
{tpk} are obtained at the output nodes. When this adjustment
FIG.1
FIG.2
FIG.3
Fig. 2. General topology of a feed-forward ANN with one hiddenlayer(b).
40 G-A. Tselentis and E. Sokos: Relationship between isoseismal area and magnitude of historical earthquakes
is completed, we present to the network another pairxp,yp
and ask that the network also learns this association. In otherwords, we ask that the network find a single set of weightsW and biases that will satisfy all the input-output pairs whichare presented to it.
In general, the output{opk} of the network will not be thesame as the target values[tpk}. For each pattern, the squareof the error is
EP =
∑k
(tpk −opk
)2 (2)
And the average system error is given by
E =1
p
∑p
∑k
(tpk −opk)2 (3)
whereP is the sample size. A true gradient search forminimum system error should be based on the minimizationof Eq. (3).
Thus, an ANN is a learning machine whose function de-pends on the training examples, it does not recognize anyfunctional relation between the input data but it determinesa numerical relation among the state parameters. Accord-ing to the principle of information diffusion, (Huang, 2002),we can increase the certainty of the determined relation ifwe increase the number of the training examples with thehelp of an appropriate information scattering function. ANNtrained in this manner are called diffusion neural networks(e.g. Huang and Moraga, 2004).
However, neural information processing models generallyassume that the patterns used for training an ANN are com-patible. If the patterns are contradictory, the neural networkdoes not converge because the adjustments of the weightsand thresholds do not know where to turn.
In theory, neural networks, and fuzzy systems are equiva-lent in that they are convertible, yet in practice each has itsown advantages and disadvantages. For neural networks, theknowledge is automatically acquired by the back propaga-tion algorithm, but the learning process is relatively slow andanalysis of the trained network is difficult (black box). Nei-ther is it possible to extract structural knowledge (rules) fromthe trained neural network, nor can we integrate special in-formation about the problem into the neural network in orderto simplify the learning procedure.
Fuzzy systems are more favourable in that their behaviourcan be explained based on fuzzy rules and thus their perfor-mance can be adjusted by tuning the rules. But since, in gen-eral, knowledge acquisition is difficult and also the universeof discourse of each input variable needs to be divided intoseveral intervals, applications of fuzzy systems are restrictedto the fields where the number of input variables is small.
Various researchers (e.g. Monostori and Egresits, 1994;Hernandez et al., 1995; Radeva and Radev, 2002; Radeva,2002) have developed fuzzy neural networks with strongernonlinear mapping abilities than the conventional ANN. This
kind of ANN have promising application prospects in nonlin-ear modeling, fuzzy identification and self-organizing fuzzycontrol for complex systems such macroseismic data sets(Huang and Liu, 1985).
Hybrid-fuzzy-neural-networks are expected to be very ef-fective in estimating the relationship between isoseismal areaand earthquake magnitude although data are usually scanty,incomplete and contradictory. The basics of information dif-fusion and fuzzy theory will be presented in the next section.
4 Information diffusion
If there are only few data available in the examination of aphenomenon, we can assign these to some already existingstatistical distribution (e.g. the Bayes method), and the struc-tured sample will have an informational value. The questionarises: what to do in the case when we do not know a prioristatistical distribution? From a small data sample, any clas-sical ANN cannot recognize a nonlinear function. This is thecase we mostly encounter when we deal with macroseismicdatasets.
In such a case, the theory offuzzy setscan be applied witha very good efficiency. This theory enables the processingof uncertain information, to be more precise, it writes downthe fuzzy logical assertions in an exact mathematical form(Zadeh, 1974).
Suppose that we haven observationsX, of magnitudesmi and isoseismal areassi obtained fromn historical earth-quakes. Our task is to find some sort of relationR betweenthem. Letf be an operator (mathematical procedure) of theobservations which is employed to estimateR. Let R(f,X)
denote the estimation ofR by f .If n is large, thenf can be a probabilistic or statistical op-
erator andR(f,X) is a statistical relationship. In the casethatn is small (inadequate data sample), we generally cannotobtain a statistical or physical relation fromX. Under thissituation we need to employ fuzzy relationship based onX.Let us suppose that we are given a sampleX of n real val-ued observations,xi (i = 1,n), which have two components,earthquake magnitudemi and isoseismal areaAi , whose un-derlying relationship is to be estimated
X = {x1,x2,...,xn} = {(A1,m1),(A2,m2),...,(An,mn)} (4)
To reduce scattering it is preferable to use the logarithm ofthe isoseismal area
s = log10A (5)
and in this case our sample becomes
X = {x1,x2,...,xn} = {(s1,m1),(s2,m2),...,(sn,mn)}
Let S= {s1,s2,...,sn} be a sample taken from the universeof discourseU, any mapping fromSxU to [0,1]
G-A. Tselentis and E. Sokos: Relationship between isoseismal area and magnitude of historical earthquakes 41
FIG.1
FIG.2
FIG.3 Fig. 3. The information diffusion function is used to diffuse infor-mationsi to all the elements of the universe of discourseU .
is called an information diffusion ofS on U if it satisfiesthe following condition for everysi in S and for everyu′,u′′
in U if |u′−s| ≤ |u′′
−s|
thenµ(s,u′) ≥ µ(s,u′′) (7)
µ(si,u) is called an information diffusion function ofSonU. If U is discrete,µ can be written asµ(si,ui). As we candeduce from Eq. (7),µ is a convex function aboutU.
In other wordsµ defines a fuzzy subset on the universe ofdiscourseSxU. Obviously, an available fuzzy relation wouldapproximately reveal the information structure implied bythe observationsX.
The trivial diffusion function is defined as1, if u = s
µ(s,u)= sεS,uεU (8)
0, otherwiseThe simplest diffusion function is the linear distribution
function with respect to a discrete monitoring spaceU. WhenS= {si |i = 1,n} and the monitoring spaceU = {uj |j = 1,m}
has steps of equal length1 we get the 1-dimensional linearinformation distribution which can be written as
1−|s −u|/1, if |s −u| ≤1
µ(s,u)= sεS,uεU (9)
0, otherwisewhere1 = uj+1−uj .The principle of information diffusion (Huang and Shi,
2002) alerts that, when we use an incomplete data set to es-timate a relationship, there must exist reasonable diffusionmeans to change observations into fuzzy sets to partly fill thegap caused by incompleteness and improve the original esti-mate.
Obviously, if our macroseismic observations (data set)Xis incomplete, this implies that the patterns are insufficient.In other words, we need more patterns to train the BP net-work for obtaining a more accurate estimate of input-outputrelation. The simplest model to derive the required patternsis based on the similarities of information and molecules.
Fig. 4. Information diffusion function ofsi on a continuous universeof discourseU .
Taking into consideration the molecular diffusion theory, weconsider the following normal diffusion function
µ(sj ,u)= e−(u−sj )2
2h2 sεS,uεU (10)
h is the normal diffusion coefficient which can simply be cal-culated (Huang, 1997) by
h =
1.6987(b−a)/(n−1) for 1< n≤ 51.4456(b−a)/(n−1) for 6≤ n ≤ 71.4230(b−a)/(n−1) for 8≤ n ≤ 91.4208(b−a)/(n−1) for 10≤ n
(11)
Whereb = max{si}, 1≤ i ≤ n; a = min{si}, 1≤ i ≤ n; andn the number of pairs (si,mi).
Thus, it is obvious that with the help of the normal diffu-sion function, we can transform any one input-output obser-vation (si,mi) into two fuzzy subsets
Ai =
∫U
µ(si,u)
u
Bi =
∫V
µ(mi,v)
v(12)
Obviously, for a particular pair of observations (si,mi) wehaveAi → Bi .
Information diffusion can be represented schematically inFig. 3, where observationsi is diffused to every point ofUwith different values. In the case of a continuous universe ofdiscourse, information diffusion ofsi can be represented bya fuzzy membership functionµ(si,u) as presented in Fig. 4.
In order to preserve more information we use the corre-lation product enconding (Kosko, 1992), to produce the re-quired fuzzy relationships, therefore we can write
µRi(u,v) = µAi(u)µBi(v),uεU,vεV (13)
with Ri depicting the corresponding rule (relation betweenu,v). Thus, we getn fuzzy relationships fromn earthquakeobservations.
42 G-A. Tselentis and E. Sokos: Relationship between isoseismal area and magnitude of historical earthquakes
For the case that we haven observations(s1,m1),(s2,m2),...,(sn,mn) then, using the informa-tion distribution approach, we can obtain n fuzzy IF-THENrules (Ri) : A1 → B1,A2 → B2,...,An → Bn (as it is shownin Fig. 5).
Let us suppose thatso is a known crisp input value and thatwe want to find a way to estimate the corresponding mag-nitudemo from so andRi . Considering that the universe ofdiscourseU where allsi belong (the monitoring space) is dis-crete. Its elementsuj are the controlling points. Thus,so isnot just equal to some valueuj in universeU. We can applythe information distribution formula (9) to get the followingfuzzy subset (information diffusion ofso)
1−|so −uj |/1 if |so −uj | ≤1
µso(uj ) =
0 if |so −uj | ≥1 (14)
where1 = uj+1−uj
Then, a fuzzy consequent∼ mo from ∼ so andRi can bewritten as
µmo(v) =
∑v
µso(v)µRi(u,v) (15)
Changing the intensity componentµBi(v) in Eq. (13) intothe following fuzzy subsets:
1 if v = mi
µBi(v) =
0 if v 6= mi (16)
And
µAi(v) if v = mi
µRi(u,v) =
0 if v 6= mi (17)
Equation (15) can be written as
if v = mi
µmo(v) =
0 if v 6= mi (18)
Let
Wi =
∑v
µSo(v)µAi(v) (19)
Wi , can be considered as the possibility that componentmo may bemi .
Then integrating all results coming fromR1, R2,...,Rn therequired valuemo can be derived from the following (gravitycenter)
mo =
n∑i=1
wimi
n∑i=1
Wi
(20)
FIG.4
FIG.5
Fig. 5. Fuzzy rule “IFu = A THEN v = B” through observationsi ,mi using information diffusion.
This result means that we have estimated the requiredvaluemo, by fuzzy influence based on the diffusion methodof the known valueso. The above procedure is summarizedin the flow chart depicted in Fig. 6.
Thus, throughout the information diffusion approxima-tion reasoning technique, we can transform the originalX = {x1,x2,...,xn} = {(s1,m1),(s2,m2),...,(sn,mn)} sampleof observations to∼ X = {(s1,∼ m1),(s2,∼ m2),...,(sn,∼
mn)}, which is finaly used to train the BP ANN (Fig. 7).
5 Results
After trying a various number of hidden layers of the BPANN, we found that the optimum network which resulted inthe least errors is that consisting of 7 hidden layers (Table 2,Fig. 8).
For supervised training of the ANN, a subset of two thirdsof the total data was used. The individual sites assigned to thetraining set were selected at random from the complete set ofrecords. The other third of the data was used for testing theANN after it had been trained.
Next, we used Eq. (20) to calculate the new magnitudes byapplying the information diffusion technique and used themagain to train the ANN. Figure 9 depicts the three resultsof our model corresponding to conventional ANN, linear re-gression and the hybrid-neural-fuzzy and the neural network.
In order to compare the mean square errorsE of the threeestimators, the linear regression (LR), the hybrid fuzzy neu-ral network (HFN), and the conventional neural network es-timator (NN) are computed as follows:
Fig. 6. Flowchart depicting the information diffusion method.
FIG.7
(a) (b)
FIG.8
Fig. 7. System architecture of the hybrid neural-fuzzy networkadopted in the present investigation.
EHFN =1
24
24∑i=1
(yi −yHFNi)2= 0.16
ELR =1
24
24∑i=1
(yi −yNNi)2= 0.23
Obviously, the HFN estimator is better than the linear regres-sion estimator and the conventional neural network estimatorsince it is more precise, nearer to real value, and more stablethan the conventional neural estimator.
44 G-A. Tselentis and E. Sokos: Relationship between isoseismal area and magnitude of historical earthquakes
FIG.7
(a) (b)
FIG.8 Fig. 8. The selected neural network consisting of 7 hidden layers(a) and the corresponding convergence curve(b).
FIG.9 Fig. 9. Relationship between logarithmic isoseismal area and earth-quake magnitude estimated by the back propagation NN (dashedcurve) and the proposed hybrid fuzzy-neural methodology (solidcurve). The lines corresponds to the list square fit of the data.
6 Conclusions
Neural information processing models largely assume thatthe samples for training a neural network are sufficient. Oth-erwise, there exists a non-negligible error between the realfunction and estimated function from a trained network. Toreduce the error in this paper, we suggest a hybrid fuzzy neu-ral network to learn from a small sample.
The obtained error is less than the error of the conventionalANN. The results show that the hybrid fuzzy neural model isvery effective in the case of treating macroseismic M,MMIdatasets of historic earthquakes where the target function hasa strong nonlinearity and the given sample is very small.
Acknowledgements.This paper was motivated by the pioneeringwork of C. Huang on hybrid fuzzy neural theory. We also thankN. Chingsao for helping us to understand the intricacies of fuzzysystems. All ANN calculations were performed using AlyudaNeurointelligence algorithm.
Edited by: M. E. ContadakisReviewed by: two anonymous referees
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