ARTICLE IN PRESS

Computers & Geosciences 35 (2009) 2095–2099

Contents lists available at ScienceDirect

Computers & Geosciences

0098-30

doi:10.1

$ Cod� Corr

E-m

journal homepage: www.elsevier.com/locate/cageo

Quantification of crustal geotherms along with their error bounds forseismically active regions of India: A Matlab toolbox$

Kirti Srivastava �, Likhita Narain, Swaroopa Rani, V.P. Dimri

National Geophysical Research Institute, Council of Scientific and Industrial Research, Hyderabad 500007, Andhra Pradesh, India

a r t i c l e i n f o

Article history:

Received 11 September 2008

Received in revised form

24 December 2008

Accepted 29 December 2008

Keywords:

Stochastic

Gaussian

Random thermal conductivity

Heat conduction

Matlab

04/$ - see front matter & 2009 Elsevier Ltd. A

016/j.cageo.2008.12.013

e available from server at http://www.iamg.

esponding author.

ail address: kirti_ngri@rediffmail.com (K. Sriv

a b s t r a c t

Determination of the thermo-mechanical structure of the crust for seismically active regions using

available geophysical and geological data is of great importance. The most important feature of the

intraplate earthquakes in the Indian region are that the seismicity occurs within the entire crust. In

Latur region of India an earthquake occurred in the upper crust. In such situations, quantifying the

uncertainties in seismogenic depths becomes very important. The stochastic heat conduction equation

has been solved for different sets of boundary conditions, an exponentially decreasing radiogenic heat

generation and randomness in thermal conductivity. Closed form analytical expressions for mean and

variance in the temperature depth distribution have been used and an automatic formulation has been

developed in Matlab for computing and plotting the thermal structure. The Matlab toolbox presented

allows us to display the controlling thermal parameters on the screen directly, and plot the subsurface

thermal structure along with its error bounds. The software can be used to quantify the thermal

structure for any given region and is applied here to the Latur earthquake region of India.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The thermal structure of the Earth’s crust is influenced by itsgeothermal parameters such as thermal conductivity, radiogenicheat sources and initial and boundary conditions. Two approachesto modeling are commonly used to estimate the subsurfacetemperature field: (1) the deterministic approach and (2) thestochastic approach. In the deterministic approach the subsurfacetemperature field is obtained by assuming that the controllingthermal parameters are known with certainty. However, due tothe inhomogeneous nature of the Earth’s interior there are someuncertainties in estimates of the geothermal parameters. Un-certainties in these parameters may arise from the measurementinaccuracies or a lack of information about the parametersthemselves. In the stochastic approach the uncertainties inparameters are incorporated and an average picture of thethermal field along with its associated error bounds is deter-mined. To assess the properties of the system at a glance we needto obtain the mean value that gives the average picture, alongwith the variance or standard deviation that indicate variability orerrors associated with the system behavior due to errors in thesystem input.

ll rights reserved.

org/CGEditor/index.htm

astava).

Subsurface temperatures are seen to be very sensitive toperturbations in the input thermal parameters and hence severalstudies have been carried out to quantify perturbations in thetemperature and heat flow using stochastic analytical and randomsimulation techniques. Uncertainty in the heat flow was quanti-fied by Vasseur et al. (1985) using a least squares inversiontechnique incorporating uncertainties in the temperature andthermal conductivities. The effect of variations in heat source onthe surface heat flow has also been studied (Vasseur and Singh,1986; Nielson, 1987). In most studies the stochastic heat equationwas solved using the small perturbation method. Srivastava andSingh (1998) used the small perturbation method to analyticallysolve the steady-state heat conduction equation, incorporatingGaussian uncertainties in the heat sources, and obtained theclosed form analytical expressions for the mean temperature fieldand its variance. The thermal structure is obtained numericallyusing the random simulation method or a Monte Carlo simulation.Using a Monte Carlo simulation, several authors have studied thesubsurface thermal field along with its error structure byincorporating randomness in the controlling thermal parameterssuch as thermal conductivity, radiogenic heat production andboundary conditions (Royer and Danis, 1988; Gallagher et al.,1997; Jokinen and Kukkonen, 1999a, b). This numerical modelingis very useful when studying nonlinear problems, but sometimesa simple 1-D analytical solution to the mean behavior and itsassociated error bounds is more useful for quantifying theuncertainty.

ARTICLE IN PRESS

K. Srivastava et al. / Computers & Geosciences 35 (2009) 2095–20992096

Stochastic differential equations in other fields are now beingsolved by yet another approach called the decomposition method(Adomian, 1994; Serrano, 1995). Adomian’s method of decom-position has now been generalized as a general analytic procedureto solve deterministic or stochastic and linear or nonlinearequations. It has been shown to be systematic, robust, andsometimes capable of handling large variances in the controllingparameters. In a recent study, using this new approach thestochastic heat conduction equation was solved by Srivastava andSingh (1999), incorporating uncertainties in the thermal con-ductivity, where the solution for the temperature field wasobtained using a series expansion method. The thermal con-ductivity was considered to be a random parameter with a knownGaussian colored noise correlation structure. Later, Srivastava(2005) extended the study to obtain the analytical expressions formean heat flow and its variance.

Graphical user interface (GUI) packages in Matlab are becom-ing very popular with geoscientific researchers. Witten (2004)developed a sequence of Matlab m-files and two graphical userinterfaces to display raw or processed geophysical data to producethe final graphics. In this paper, we have developed a simplegraphical user interface viewer which is a MATLAB-based soft-ware consisting of m-files. This software is the first package thatcomputes the thermal structure along with its error structure,with the important advantage that it is not a compiled code. Them-file in the package is integrated through a GUI and thecontrolling thermal parameters such as crustal thickness, radio-genic heat production, characteristic depth, surface temperature,surface heat flow, mean thermal conductivity, coefficient ofvariability in thermal conductivity and correlation length scaleare all given on the screen. The program model is intended as auser friendly tool to deal with various aspects of analysis andinterpretation of subsurface geothermal structure for any givenregion.

2. Mathematical formulation

The heat conduction equation with random thermal conduc-tivity can be expressed as

d

dzð �K þ K 0ðzÞÞ

dT

dz

� �¼ �A0e�z=D (1)

where T is the temperature (1C), A(z) the radiogenic heat source(mW/m3), and K(z) the thermal conductivity (W/m1C) expressedas the sum of a deterministic component �K and randomcomponent K0(z). K0(z) has mean zero and a Gaussian colorednoise correlation structure represented by

EðK 0ðzÞÞ ¼ 0 (2)

EðK 0ðz1ÞK0ðz2ÞÞ ¼ s2

K e�rjz1�z2j (3)

where sK2 is the variance in thermal conductivity, r the

correlation decay parameter (or 1/r the correlation length scale)and z1 and z2 the depths.

The analytical expressions derived by Srivastava et al. (2006)for the temperature depth distribution along with its error boundsfor two sets of boundary conditions, i.e. (i) surface temperatureand surface heat flow Qs (mW/m2) and (ii) surface temperatureand basal heat flow Q* (mW/m2), have been used in this study.

The solution to Eq. (1) for the first set of boundary conditions isMean temperature

�T ¼ EðTðzÞÞ ¼ T0 þQs

�Kzþ

A0D2

�K1�

z

D� e�z=D

� �(4)

Variance in the temperature

s2T ¼ c1� Term1þ c2� Term2þ c3� Term3þ c4� Term4 (5)

where CK ¼ sK/K is the coefficient of variability in thermalconductivity and

c1 ¼ C2K A2

0ð1� rDÞ2= �K2

c2 ¼ C2K A0rðrD� 1ÞðQs � A0DÞ= �K

2

c3 ¼ c2

c4 ¼ C2Kr

2ðQs � A0DÞ2= �K2

Term1 ¼1

4ðr� 1=DÞ2fðrD� 1Þð2z2 � 2zD� D2e�2z=D þ D2

Þ

þ4½zðr� 1=DÞ þ 1�

ðrþ 1=DÞ2½�zðrþ 1=DÞ � e�zðrþ1=DÞ þ 1�

þ ½2zDþ D2e�2z=D � D2�g

þ1

4ðrþ 1=DÞ2fðrDþ 1Þð2z2 � 2zD� D2e�2z=D þ D2

Þ

þ4

ðr� 1=DÞ2½�zðr� 1=DÞe�zðrþ1=DÞ þ e�2z=D � e�zðrþ1=DÞ�

� ½2zDþ D2e�2z=D � D2�g

Term2 ¼1

r2f2rðz2D� 2zD2

� 2D3e�z=D þ 2D3Þ

�ð1þ rzÞ

ðrþ 1=DÞ2½zðrþ 1=DÞ þ e�zðrþ1=DÞ � 1�

þe�rz

ðr� 1=DÞ2½�zðr� 1=DÞ þ ezðr�1=DÞ � 1�g

Term3 ¼1

ðr� 1=DÞ2fðr� 1=DÞðz2D� 2zD2

� 2D3e�z=D þ 2D3Þ

�zðr� 1=DÞ þ 1

r2½rzþ e�rz � 1�

þ ½zDþ D2e�z=D � D2�g

þ1

ðrþ 1=DÞ2fðrþ 1=DÞðz2D� 2zD2

� 2D3e�z=D þ 2D3Þ

þe�zðrþ1=DÞ

r2½�rzþ erz � 1�

� ½zDþ D2e�z=D � D2�g

Term4 ¼1

r2

2

3rz3 �

ðrzþ 1Þ

r2ðrzþ e�rz � 1Þ

�

þe�rz

r2½�rzþ erz � 1�g

For the second set of boundary condition i.e. surface tempera-ture and basal heat flow Srivastava (2005) have given the solutionto the mean heat flow and its variance as

Mean temperature

�T ¼ EðTðzÞÞ ¼ T0 þQ�

�Kzþ

A0D2

�K1�

z

De�L=D � e�z=D

� �(6)

Variance in the temperature

s2T ¼ c1� Term1þ c2� Term2þ c3� Term3

þ c4� Term4 (7)

where the constants are different and are given as

c1 ¼ C2K A2

0ð1� rDÞ2= �K2

ARTICLE IN PRESS

K. Srivastava et al. / Computers & Geosciences 35 (2009) 2095–2099 2097

c2 ¼ C2K A0rðrD� 1ÞðQ� � A0De�L=DÞ= �K

2

c3 ¼ c2

c4 ¼ C2Kr

2ðQ� � A0De�L=DÞ2= �K

2

The terms Term1, Term2, Term3 and Term4 are same as givenabove.

3. Numerical examples and discussion

The analytical solutions given by Srivastava et al. (2006) for themean and variance of the temperature depth distribution for twodifferent sets of boundary conditions have been used to quantifythe mean temperature along with its error bounds for theseismically active Latur region of India. The Latur region in theDeccan volcanic province in central India experienced a devastat-ing earthquake of magnitude Mw ¼ 6.2 on 30th September, 1993.The Deccan volcanic province is covered by a thin layer ofcontinental flood basalts. The earthquake occurred in a seismicallyquiet zone, highlighting the unstable nature of the highlyfragmented Indian Shield. The focal depth of the earthquake assuggested by USGS was around 9 km, and the focal mechanismwas a reverse fault with NW–SE-orientated nodal planes. Thisearthquake was studied by several researchers from differentgeophysical angles. Singh et al. (1995) attributed the intraplateseismicity in the Deccan volcanic province of India to the presenceof fluids in the upper and middle crust. Gupta et al. (1996) carriedout various geophysical investigations of the crustal structure in

Fig. 1. Plot of mean temperature and mean 7 1 S.D. for Set 1 when bounda

the region. From magnetotelluric studies they found a fluid-filledconducting zone at 6–10 km depth that would have enhanced theaccumulation of stresses in the crust, causing a mechanicalfailure. Agrawal and Pandey (1999) provided an alternativeexplanation in which the seismic activity stems from a hotunderlying asthenosphere that causes continuous build up oflocalized stresses. Patro et al. (2006) suggested that the seismicityoccurred in a relatively thick and highly resistive upper crust.From a thermal perspective, Roy and Rao (1999) carried out heatflow measurements in four bore holes in the vicinity of therupture zone. One of the bore holes was drilled in the surfacerupture zone and penetrated 338 m in the Deccan basalts and271 m further down into the Archaean granite-gneiss basement.They found that the surface heat flow in the region is about43 mW/m2, which is characteristic of a low-heat-flow regime.Gupta et al. (1990) characterized the Dharwar craton by low-heat-flow values, ranging from 25 to 50 mW/m2, similar to values inother Archaean terrains in the world. In this study, we have usedan exponential model for the heat generation as it has been usedextensively in the literature to quantify the thermal structure. Themean radiogenic heat production has been taken to be about2.6mW/m3, bearing in mind the dominance of granite in the crust.The characteristic depth has been taken to be about 12 km. Fromdeep seismic sounding results Kaila et al. (1979) found an averagecrustal thickness of about 37 km for the Dharwar craton.

Thermal conductivity measurements for the bore hole sampleshave been found to vary from 2.6 to 3.5 W/m 1C (Roy and Rao,1999), so a mean thermal conductivity value of about 3.0 W/m 1Chas been taken in this study. The random component parametersare the coefficient of variability in thermal conductivity which is

ry conditions employed are surface temperature and surface heat flow.

ARTICLE IN PRESS

K. Srivastava et al. / Computers & Geosciences 35 (2009) 2095–20992098

taken to be 0.3 and the correlation length scale which isconsidered to be about 15 km. The basal heat flow estimated forthis region is about 12 mW/m2 and the surface temperature forthe region is considered to be 30 1C.

In the GUI the controlling input parameters can be givendirectly on the screen and both the plot and the results aredisplayed instantaneously. For the Latur region, we used thecontrolling parameters as given above. Set 1 uses the surfacetemperature and surface heat flow as the boundary conditions andSet 2 uses the surface temperature and basal heat flow as itsboundary conditions. The coefficient of variability in thermalconductivity is in the range 0–1. To incorporate a 30% error in thethermal conductivity we need to take the coefficient of variabilityto be 0.3. The correlation length scale should be less than themodel length. In many situations, one third of the model lengthfor the correlation length scale is a good approximation. Theinput-controlling parameters are given in the boxes and theoutput of the temperature depth profile along with the mean 7one standard deviation is plotted; also the results of depth, meantemperature, standard deviation, mean plus one standard devia-tion and mean minus one standard deviation are displayed on thescreen.

Numerical values for Set 1

Fig. 2. Plot of mean temperature

Crustal thickness (L)

37 km

Radiogenic heat production (A)

2.6mW/m3

Characteristic depth (D)

12 km

Surface temperature (T0)

30 1C

Surface heat flow (Qs)

43 mW/m2

and mean 7 1 S.D. for Set 2 when bound

Random thermal conductivity

ary conditions employed are surface tem

Mean thermal conductivity �K 3

.0 W/m 1C

Coefficient of variability Ck 0

.3

Correlation length scale 1/r 1

5 km

The results obtained are plotted in Fig. 1. From the figure, weobserve that the mean temperature at the base of the crust(37 km) is about 295 1C and the standard deviation is about 29 1C.The crustal temperature here varies between 323 and 266 1C.

Numerical values for Set 2

Crustal thickness (L) 3

7 km

Radiogenic heat production (A) 2

.6mW/m3

Characteristic depth (D) 1

2 km

Surface temperature (T0) 3

0 1C

Basal heat flow (Q*) 1

2 mW/m2

Random thermal conductivity

Mean thermal conductivity �K 3

.0 W/m 1C

Coefficient of variability Ck 0

.3

Correlation length scale 1/r 1

5 km

Results obtained are plotted in Fig. 2. From the figure, weobserve that at the base of the crust the mean temperature isabout 276 1C and the standard deviation is about 24 1C. Weobserve that there is a difference in the temperatures between

perature and basal heat flow.

ARTICLE IN PRESS

K. Srivastava et al. / Computers & Geosciences 35 (2009) 2095–2099 2099

Sets 1 and 2. This is due to the different boundary conditionsemployed. Also the error bounds are seen to be highly dependenton the coefficient of variability and the correlation length scale, asseen from the closed form expressions. The mean temperaturesobtained fall well within the estimated deterministic tempe-ratures obtained by earlier workers. Previous studies haveexamined the thermal structure from a deterministic point ofview. In this study, only the thermal conductivity has beenconsidered to be a random parameter. Other parameters such asradiogenic heat sources and boundary conditions can also beconsidered to be random. Obtaining an analytic solution to astochastic differential equation with more than two randomparameters would be difficult. However, using the perturbationmethods or Monte carlo simulation one can obtain the solution tothe problem. Sometimes the solution to a simple stochasticequation with one random parameter is essential as it gives anaverage picture, and the standard deviation (which is thevariability indicator) indicates the errors associated with thesystem behavior due to errors in the system input. Hence, thisstudy has helped with quantifying the errors in the temperaturedepth distribution, and the software package developed can beused for any given region.

4. Conclusions

A package has been developed to compute and plot the errorbounds on subsurface temperatures due to errors in the thermalconductivity for a 1-D steady-state conductive earth model fortwo different sets of boundary conditions and an exponential heatsource. The analytical expressions for error bounds on thetemperature depth distribution as given by Srivastava et al.(2006) have been used and a graphical user interface has beendeveloped in MatLaboratory. This is a very user-friendly packagethat allows the controlling input thermal parameters such ascrustal thickness, radiogenic heat production, characteristic depth,surface temperature, surface heat flow, mean thermal conductiv-ity, coefficient of variability in thermal conductivity and correla-tion length scale to be specified and these can be shown directlyon the screen. A plot of mean temperature along with its errorbounds is also displayed directly on the screen. The developedpackage has been applied to quantify the conductive geothermsalong with their error bounds for the Latur earthquake region ofIndia. The mean temperatures at the base of the crust in theregion are seen to be between 260 and 325 1C indicating that theregion is characteristic of a cold type of crust. The package can beapplied to quantify the thermal state of the crust, along with itserror bounds, for any given region.

Acknowledgements

The authors wish to thank the students of the Department ofMathematics, Osmania University for their help. The comments

and suggestions made by two anonymous reviewers have helpedus in improving the paper.

Appendix A. Supporting Information

Supplementary data associated with this article can be foundin the online version at doi:10.1016/j.cageo.2008.12.013.

References

Adomian, G., 1994. Solving Frontier Problems in Physics—The DecompositionMethod. Kluwer Academic, Boston, 372pp.

Agrawal, P.K., Pandey, O.P., 1999. Relevance of hot underlying aesthenosphere tothe occurrence of Latur earthquake and Indian peninsular shield seismicity.Geodynamics 28, 303–316.

Gallagher, K., Ramsdale, M., Lonergan, L., Marrow, D., 1997. The role thermalconductivities measurements in modeling the thermal histories in sedimen-tary basins. Marine and Petroleum Geology 14, 201–214.

Gupta, M.L., Sunder, A., Sarma, S.R., 1990. Heat flow and heat generation inDharwar cratons and implications for the southern Indian shield geotherm andlithospheric thickness. Tectonophysics 194, 107–122.

Gupta, H.K., Sarma, S.V.S., Harinarayana, T., Virupakshi, G., 1996. Fluids below thehypocentral region of Latur earthquake India, geophysical indicators. Geophy-sical. Research Letters 23, 1569–1572.

Jokinen, J., Kukkonen, I.T., 1999a. Random modeling of lithospheric thermalregime: Forward simulation applied in uncertainty analysis. Tectonophysics306, 277–292.

Jokinen, J., Kukkonen, I.T., 1999b. Inverse simulation of lithospheric thermal regimeusing the Monte Carlo method. Tectonophysics 306, 293–310.

Kaila, K.L., Roy Choudhury, K., Reddy, P.R., Krishna, V.G., Narain, H., Subbotin, S.I.,Sollogub, V.B., Chekunov, A.V., Kharetchko, G.E., Lazarenko, M.A., Ilchenko, T.V.,1979. Crustal structure along Kavali–Udipi profile in the Indian peninsularshield from deep seismic sounding. Journal Geological Society of India 20,307–333.

Nielson, S.B., 1987. Steady state heat flow in a random medium and linear heat flowheat production relationship. Geophysical Research Letters 14, 318–321.

Patro, B.P.K., Nagarajan, N., Sarma, S.V.S., 2006. Crustal geoelectric structure andthe focal depths of major stable continental region earthquakes in India.Current Science 90 (1), 107–113.

Roy, S., Rao, R.U.M., 1999. Geothermal investigations in the 19993 Latur earthquakearea, Deccan Volcanic Province, India. Tectonophysics 306, 237–252.

Royer, J.J., Danis, M., 1988. Steady state geothermal model of the crust andproblems of boundary conditions: Application to a rift system, the southernRhinegraben. Tectonophysics 156, 239–255.

Serrano, S.E., 1995. Forecasting scale dependent dispersion from spills inheterogeneous aquifers. Journal of Hydrology 169, 151–169.

Singh, R.P., Sato, T., Nyland, E., 1995. The geodynamic context of the Latur (India)earthquake, 30, September 1993. Physics of the Earth and Planetary Interiors91, 245–251.

Srivastava, K., 2005. Modeling the variability of heat flow due to random thermalconductivity of the crust. Geophysical Journal International 160 (2), 776–782.

Srivastava, K., Singh, R.N., 1998. A model for temperature variation in sedimentarybasins due to random radiogenic heat sources. Geophysical Journal Interna-tional 135, 727–730.

Srivastava, K., Singh, R.N., 1999. A stochastic model to quantify the steady statecrustal geotherms subject to uncertainty in thermal conductivity. GeophysicalJournal International 138, 895–899.

Vasseur, G., Lucazeau, F., Bayer, R., 1985. The problem of heat flow densitydetermination from inaccurate data. Tectonophysics 121, 23–34.

Vasseur, G., Singh, R.N., 1986. Effect of random horizontal variation in radiogenicheat source distribution on its relationship with heat flow. Journal GeophysicalResearch 91, 10397–10404.

Witten, A., 2004. A Matlab based three dimensional viewer. Computers &Geosciences 30, 693–703.