An alternate and effective approach to Hilbert transform in geophysical applications N. Sundararajan n , Ali Al-Lazki Department of Earth Science, Sultan Qaboos University, Post Box 36, Postal Code 123, Alkhod, Muscat, Oman article info Article history: Received 4 October 2010 Received in revised form 25 January 2011 Accepted 8 February 2011 Available online 10 March 2011 Keywords: Fourier transform Hartley transform Analytical signal Amplitude Origin Phase Frequency abstract The Hilbert transform defined via the Hartley transform in contrast with the well-known Fourier transform is mathematically illustrated with a couple of geophysical applications. Although, the 1-D Fourier and Hartley transforms are identical in amplitude with a phase difference of 451, the Hilbert transform effectively differs when defined as a function of the Hartley transform in certain geophysical applications. It may be noted here that the Hilbert transform defined through Fourier and Hartley transforms while possessing the same magnitude differs in phase by 2701. It is derived and shown mathematically that the evaluation of depth of subsurface targets is directly equal to the abscissa of the point of intersection of the gravity (magnetic) field and the Hartley–Hilbert transform; however, it is not the case with the Fourier–Hilbert transform. The practical applications are illustrated with the interpretation of gravity anomaly due to an inclined sheet-like structure across the Mobrun ore body, Noranda, Quebec, Canada, and the vertical magnetic anomaly due to a cylindrical structure over a narrow band of quartzite magnetite, Karimnagar, India. The entire process can be automated. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction The theory and applications of the Hilbert transform have been known since the early 1970s in processing and interpretation of potential field data (Nabighian, 1972; Cerevony and Zahradnik, 1975; Stanley and Green, 1976; Sundararajan, 1983). It may be noted here that the vertical and horizontal components of a magnetic field on the surface of the earth form a Hilbert trans- form pair (Cerevony and Zahradnik, 1975). The geophysical applications of the Hilbert transform are multifaceted. In poten- tial fields, this transform is known for its simple and elegant approach in processing and interpretation (Sundararajan, 1983; Sundararajan et al., 1998). Its application to attribute analysis in seismics for direct detection of hydrocarbons is of prime impor- tance (Taner et al., 1979). Another significant application is the amplitude versus offset analysis (AVO) in which the auxiliary functions of the Hilbert transform assume special significance in processing of seismic data for direct delineation of hydrocarbon resources. In potential field applications, the theory of the Hilbert trans- form is closely tied to the theory of Fourier transforms, the first and best known of all integral transforms. The Hilbert transform can also be related through the Hartley transform with the same physical significance; however, it ensures efficiency and economy particularly when we handle large amounts of data due to its inherent nature of being real, unlike its progenitor the Fourier transform, which is complex (Bracewell, 1983; Sundararajan, 1995). For the simple fact that both Fourier and Hartley trans- forms yield the same information at every frequency they may be realized as mathematical twins (Sundararajan, 1997). Thus, the Hilbert transform assumes a special importance when it is defined via the Hartley transform instead of the traditional Fourier transform. However, there exists an interesting feature that the amplitude/magnitude of these two versions of the Hilbert transforms are the same while the phase differs by 2701, which is explained mathematically hereunder. This paves the way for straightforward interpretation of certain geophysical data unlike the Fourier–Hilbert transform as discussed in the text. The interpretation of a gravity anomaly due to an inclined sheet- like structure across the Mobrun ore body, Noranda, Quebec, Canada, and the vertical magnetic anomaly due to a cylindrical structure over a narrow band of quartzite magnetite, Karimnagar, India, illustrate the advantages of the Hartley–Hilbert transform over the Fourier–Hilbert transform. 2. Hartley–Hilbert transform The Hilbert transform of f(x) via the Hartley transform can be defined as (Sundararajan and Srinivas, 2010) hhðxÞ¼ 1 p Z 1 0 ½OðoÞcosðoxÞþ EðoÞsinðoxÞdo, ð1Þ Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/cageo Computers & Geosciences 0098-3004/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cageo.2011.02.003 n Corresponding author. Tel.: þ968 24142285; fax: þ968 24413415. E-mail address: sundararajan_n@yahoo.com (N. Sundararajan). Computers & Geosciences 37 (2011) 1622–1626