Reply to comment on: ‘One-dimensional magnetotelluric inversion using an adaptation of Zohdy's...
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Reply to comment on: ‘One-dimensional magnetotelluric inversion using an adaptation of Zohdy’s resistivity method’ by Chien-chih Chen and Shi-zhe Xu 1 B.A. Hobbs 2 and C.C. Dumitrescu 3 The aim of Zohdy’s resistivity method (Zohdy 1989) is to find the resistivities and depths of a layered model whose calculated apparent resistivities fit, to some measure, an observed resistivity sounding curve. A starting model is first found using natural scale lengths associated with electrode spacings and thereafter a suitable final model is determined through an iterative scheme. In the adaptation of that method to magnetotelluric data, Hobbs and Dumitrescu (1997) determined a starting model by transforming magnetotelluric frequencies to length scales using either the Bostik or Schmucker transformations and then proceeded by iteration in a manner analogous to that of Zohdy (1989). It is good that Chen and Xu (1999) are examining the effectiveness of the starting model and iteration procedure as many variants must be possible. However, the aim of our inversion scheme was to produce a model whose responses fit the observations to some suitable measure – the measure chosen being the x 2 misfit defined eloquently for magnetotelluric data by Weaver and Agarwal (1993). Since there is inherent non-uniqueness in the magnetotelluric inverse problem, there will be an infinity of models satisfying this same criterion. It seems that Chen and Xu (1999) have introduced a further measure in that they particularly wish to satisfy the observations of phase at the four highest frequencies. This has been achieved at the expense of a poorer fit elsewhere. The approach of Chen and Xu (1999), which determines a model satisfying their requirements, differs in two ways from that of Hobbs and Dumitrescu (1997). Firstly, they form a starting model by the Bostik transformation of a ‘smooth’ version of the field data (the method of smoothing is not given, though it appears from the data fit that the apparent resistivities and phases have been smoothed independently). Hobbs and Dumitrescu (1997) improved stability by a joint smoothing of apparent resistivity and phase, taking into account the constraint of Weidelt (1972), so that the subsequent Bostik transformation applied to data that were constrained to be 1D. Secondly, in changing the automatic nature of the Zohdy adaptation by imposing a specific depth refinement factor, Chen and Xu (1999) introduce a form of subjective weighting, q 1999 European Association of Geoscientists & Engineers 607 Geophysical Prospecting, 1999, 47, 607–609 1 Received January 1999, accepted February 1999. 2 Department of Geology and Geophysics, The University of Edinburgh, Grant Institute, West Mains Road, Edinburgh EH9 3JW, UK. 3 Veritas GeoServices Ltd, 2200, 715-5 AvenueSW, Calgary, Alberta, Canada T2P 5A2.
Transcript of Reply to comment on: ‘One-dimensional magnetotelluric inversion using an adaptation of Zohdy's...
Reply to comment on: ‘One-dimensionalmagnetotelluric inversion using an adaptation ofZohdy’s resistivity method’ by Chien-chih Chenand Shi-zhe Xu1
B.A. Hobbs2 and C.C. Dumitrescu3
The aim of Zohdy’s resistivity method (Zohdy 1989) is to find the resistivities anddepths of a layered model whose calculated apparent resistivities fit, to some measure,an observed resistivity sounding curve. A starting model is first found using naturalscale lengths associated with electrode spacings and thereafter a suitable final model isdetermined through an iterative scheme. In the adaptation of that method tomagnetotelluric data, Hobbs and Dumitrescu (1997) determined a starting model bytransforming magnetotelluric frequencies to length scales using either the Bostik orSchmucker transformations and then proceeded by iteration in a manner analogous tothat of Zohdy (1989). It is good that Chen and Xu (1999) are examining theeffectiveness of the starting model and iteration procedure as many variants must bepossible. However, the aim of our inversion scheme was to produce a model whoseresponses fit the observations to some suitable measure – the measure chosen being thex2 misfit defined eloquently for magnetotelluric data by Weaver and Agarwal (1993).Since there is inherent non-uniqueness in the magnetotelluric inverse problem, therewill be an infinity of models satisfying this same criterion. It seems that Chen and Xu(1999) have introduced a further measure in that they particularly wish to satisfy theobservations of phase at the four highest frequencies. This has been achieved at theexpense of a poorer fit elsewhere.
The approach of Chen and Xu (1999), which determines a model satisfying theirrequirements, differs in two ways from that of Hobbs and Dumitrescu (1997). Firstly,they form a starting model by the Bostik transformation of a ‘smooth’ version of thefield data (the method of smoothing is not given, though it appears from the data fitthat the apparent resistivities and phases have been smoothed independently). Hobbsand Dumitrescu (1997) improved stability by a joint smoothing of apparent resistivityand phase, taking into account the constraint of Weidelt (1972), so that the subsequentBostik transformation applied to data that were constrained to be 1D. Secondly, inchanging the automatic nature of the Zohdy adaptation by imposing a specific depthrefinement factor, Chen and Xu (1999) introduce a form of subjective weighting,
q 1999 European Association of Geoscientists & Engineers 607
Geophysical Prospecting, 1999, 47, 607–609
1 Received January 1999, accepted February 1999.2 Department of Geology and Geophysics, The University of Edinburgh, Grant Institute, West Mains Road,
Edinburgh EH9 3JW, UK.3 Veritas GeoServices Ltd, 2200, 715-5 Avenue SW, Calgary, Alberta, Canada T2P 5A2.
enhancing the role of the phase at the four highest frequencies, which they justify bydiscussion. Hobbs and Dumitrescu (1997) imposed no a priori conditions and allowedthe iterations to proceed unhindered.
For comparison, the method described by Hobbs and Dumitrescu (1997) has beenapplied to the COPROD data investigated by Chen and Xu (1999) and the resultingmodel and responses, together with the field data, are shown in Fig. 1. The automaticprocedure applied the depth correction multiplicative factor of 0.9 three times,resulting in a depth-refinement factor of 0.729. Thereafter, after an initial divergence,the iterations converged at iteration number 129 with a x2 misfit of 39.87. SinceE(x2) ¼ 30 (the number of data values), the x2 obtained indicates the data are notrepresentable by 1D solutions, a point well known for the COPROD data set. The fitshown in Fig. 1 appears to conflict with that in Chen and Xu (1999) – for a depthrefinement of 0.729, the response might be expected to lie between their curves labelled‘d-ref 0.6’ and ‘d-ref 1.0’. The model obtained here is similar in form to that ofConstable, Parker and Constable (1987); but the thin conducting layer is nearersurface with a resistivity of 26 Qm between depths 38 and 46 km.
References
Chen C.-c. and Xu S.-z. 1999. Comment on ‘One-dimensional magnetotelluric inversion usingan adaptation of Zohdy’s resistivity method’ by B.A. Hobbs and C.C. Dumitrescu. GeophysicalProspecting 47, 603–606.
608 B.A. Hobbs and C.C. Dumitrescu
q 1999 European Association of Geoscientists & Engineers, Geophysical Prospecting, 47, 607–609
Figure 1. Apparent resistivity and phases of the COPROD data (solid circles and error bars)together with the responses and model derived from the scheme of Hobbs and Dumitrescu(1997).
Constable S.C., Parker R.L. and Constable C.G. 1987. Occam’s inversion: a practical algorithmfor generating smooth models from electromagnetic sounding data. Geophysics 52, 289–300.
Hobbs B.A. and Dumitrescu C.C. 1997. One-dimensional magnetotelluric inversion using anadaptation of Zohdy’s resistivity method. Geophysical Prospecting 45, 1027–1044.
Weaver J.T. and Agarwal A.K. 1993. Automatic 1D inversion of magnetotelluric data by themethod of modelling. Geophysical Journal International 112, 115–123.
Weidelt P. 1972. The inverse problem of geomagnetic induction. Journal of Geophysics 38,257–289.
Zohdy A.A.R. 1989. A new method for the interpretation of Schlumberger and Wennersounding curves. Geophysics 54, 245–253.
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q 1999 European Association of Geoscientists & Engineers, Geophysical Prospecting, 47, 607–609
