See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/257139474 Upward And Downward Continuation Chapter · January 2007 DOI: 10.1007/978-1-4020-4423-6_311 CITATIONS READS 0 3,690 1 author: Dhananjay Ravat University of Kentucky 127 PUBLICATIONS 2,209 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: An informal gravity and magnetic community effort to advance data bases, modeling, and analysis efforts View project All content following this page was uploaded by Dhananjay Ravat on 06 May 2014. The user has requested enhancement of the downloaded file. Ravat, D., 2007, Upward and Downward Continuation, Encyclopedia of Geomagnetism and Paleomagnetism, D. Gubbins and E. Herrero-Bervera (eds.), Springer, 974-976. 974 UPWARD AND DOWNWARD CONTINUATION Table U2 Dimensions and units of physical quantities used in geomagnetism Physical quantity Dimension SI Emu Conversion factor Length Mass Time Charge L M T Q m kg s coulomb (C) cm G s coulomb 102 103 1 1 Electric current Potential difference Electric field Resistance Resistivity Conductivity QT1 L2 MT2 Q LMT2 Q L2 MT1 Q2 L3 MT1 Q2 L2 M1 TQ2 ampere (A) volt (V) V m1 ohm ohmm siemensm1 abamp emu emu emu emu emu 10 108 106 109 1011 1011 Magnetic flux Magnetic induction B Magnetic field intensity Inductance Permeability Magnetic moment density Magnetic polarization Susceptibility L2 MT1 Q1 MT1 Q1 L1 T1 Q L2 MQ2 LMQ2 L1 T1 Q MT1 Q1 Dimensionless weber (W) tesla (T) A m1 henry (H) H m1 A m1 T wSI maxwell gauss oersted ðMT1 Q1 Þ emu Dimensionless emu ðMT1 Q1 Þ gauss wemu 108 104 103 =4p 109 4p 107 103 =4p 104 4p LMTQ denote length, mass, time, and charge. The conversion factor in the right column should be used to multiply a value in emu to yield the SI value. Note the difference in definition for H, M, and w between the two systems. The siemen is sometimes called the mho. volume of a material M. Furthermore, the definition of M differs by a numerical factor of 4p between the two systems, which has the undesirable effect that the dimensionless susceptibility w differs. The magnetic polarization P (usually denoted as J in paleomagnetism but this is used in MHD (q.v.) exclusively for electric current density), has the same dimensions as B in both systems. Confusion propagates because of sloppy terminology: it is standard practice in geomagnetism and paleomagnetism to refer to B as the magnetic field rather than magnetic induction, and magnetization is used to mean either M or P (Table U2). David Gubbins Bibliography Blakely, R.J., 1996. Potential Theory in Gravity and Magnetic Applications. Cambridge: Cambridge University Press. Butler, R.F., 1992. Paleomagnetism: Magnetic Domains to Geologic Terranes. Boston: Blackwell Scientific. Jackson, J.D., 1999. Classical Electrodynamics, 3rd edition. New York: Wiley. Cross-references Magnetohydrodynamics UPWARD AND DOWNWARD CONTINUATION Potential fields known at a set of points can be expressed at neighboring higher or lower spatial locations in a source free region using the continuation integral that results from one of Green’s theorems (see, e.g., Blakely, 1995). The principal uses of this concept are to adjust altitude of observations to a datum as an aid to the interpretation of a survey (see Crustal magnetic field), reduce short-wavelength data noise by continuing the field upward, and increasing the horizontal resolution of anomalies and their sources by continuing the field downward. It is possible to continue the field upward or downward in a number of different ways depending on the application at hand; for example, designing continuation operators in spatial or wavenumber space (Henderson and Zietz, 1949; Dean, 1958), using harmonic functions (Courtillot et al., 1978; Shure et al., 1982; Fedi et al., 1999), and deriving physical property variations of sources causing the fields (Dampney, 1969; Emilia, 1973; Langel and Hinze, 1998). Applications also vary widely: from environmental and exploration applications involving short-wavelength anomaly fields over small height differences (a few meters to kilometers) to global distribution of anomalies measured by satellites in which anomalies are downward continued from satellite altitudes (300–700km) to Earth’s surface and also downward continuing the core part of the Earth’s field all the way to the top of the core to decipher features of core circulation over time. The effect of upward/downward continuation process on the fields can be understood by examining the continuation operator in the wavenumber domain. The operator has the form ejkjz , where jkj is the wavenumber (jkj ¼ 2l where l is the full wavelength) and z is the continuation level (Dean, 1958). The negative sign in the exponent indicates upward continuation (away from the sources of the field) and the positive sign implies the downward continuation (toward the sources of the field). The response of the continuation operator with respect to wavelength is illustrated in Figure U5, which shows that shorter wavelengths are attenuated and smoothed in the process of upward continuation, whereas in downward continuation the shorter wavelengths are amplified and sharpened. Both operations are susceptible to errors in the data and their results can be rendered invalid or at least severely compromised due to the quality of data. For example, if measurement errors are primarily short-wavelength, then the nature of downward continuation operator which amplifies primarily the shortwavelength components of the data can severely distort the downward continued result. On the other hand, if the long-wavelength portion of the field is contaminated, for example, by inaccurate compilation of different surveys having different base levels, then the retention of the corrupt long wavelengths in the process of upward continuation can render the result unusable (Ravat et al., 2002). The most straightforward upward/downward continuation of a field is performed from one level surface to another level surface (Henderson UPWARD AND DOWNWARD CONTINUATION 975 Figure U5 Amplitude response of upward and downward continuation operators with respect to wavelength for certain heights (z ) of continuation. and Zietz, 1949; Henderson, 1970). This is often useful for interpretation and joining two adjacent surveys carried out at different altitudes. As aid in interpretation, upward continuation allows one to assess the effect of deeper sources because in this process the effect of shallower, short-wavelength features is attenuated. Preferential upward and downward continuation operators have been designed that can help attenuate only the shallow, short-wavelength part of the spectrum, leaving the deeper, long-wavelength part unaltered or, alternatively, preferentially amplify only the deeper part of the spectrum without the deleterious effects of amplifying short-wavelength noise (Pawlowski, 1995). Thus, under certain situations, it is possible to isolate a magnetic anomaly signal from different depth layers of the crust. Downward continuation into the region of sources leads the continuation integral to diverge even in the case of noise-free data; in the case of high data density noise-free data the depths at which the continuation integral blows up (data begin to vary wildly) can be used to infer the depth to the top of the shallow magnetic sources in the region. When airborne magnetic surveys (see Aeromagnetic surveying) are conducted in rugged terrain made up of magnetic formations, it is not advisable to view the data at constant altitude because effects of topographic variations can lead to anomaly artifacts. In such situations, one might prefer to “continue” level survey data at some constant distance away from topography (on a constant terrain clearance or “draped” surface). Challenges of maintaining the constant terrain clearance of aircrafts in a rugged topography may require one to adjust the data further until the survey is accurately draping. Conversely, flying conditions can lead to unintentional altitude variations in surveys originally intended to be flown at constant barometric altitude (level survey), and such surveys need datum corrections as well. Two types of procedures have been commonly used in accomplishing datum transformations from level-to-drape and drape-to-level surfaces: Taylor’s series approximation and equivalent source concept. Taylor’s series allows extrapolation of a function to nearby points and, given vertical derivatives of the field and certain approximations regarding behavior of the field, the series yields adequate values of levelto-drape transformation. Similarly, an iterative Taylor’s series can be used for drape-to-level transformation (Cordell and Grauch, 1985). The equivalent source method (Dampney, 1969) employs Green’s equivalent layer concept and uses a set of sources with arbitrary magnetization (often induced dipoles because of their simplicity; Emilia, 1973) to approximate the field. This process is equivalent to finding the potential that satisfies the observed field. The inverted magnetization of the sources is then used to predict the field in the neighborhood of observations. Use of local harmonic functions (Fedi et al., 1999) can also be useful for these purposes. Dhananjay Ravat Bibliography Blakely, R.J., 1995. Potential Theory in Gravity and Magnetic Applications. Cambridge:Cambridge University Press. Cordell, L., and Grauch, V.J.S., 1985. Mapping basement magnetization zones from aeromagnetic data in the San Juan basin, New Mexico. In Hinze, W.J. (ed.) The Utility of Regional Gravity and Magnetic Anomaly Maps. Tulsa: Society of Exploration Geophysicists, pp. 181–197. Courtillot, V., Ducruix, J., and Le Moüel, J.L., 1978. Inverse methods applied to continuation problems in geophysics. In Sabatier, P.C. (ed.) Applied Inverse Problems. Berlin: Springer-Verlag, pp. 48–82. Dampney, C.N.G., 1969. The equivalent source technique. Geophysics, 34: 39–53. 976 UPWARD AND DOWNWARD CONTINUATION Dean, W.C., 1958. Frequency analysis for gravity and magnetic interpretation. Geophysics, 23: 97–127. Emilia, D.A., 1973. Equivalent sources used as an analytic base for processing total magnetic field profiles. Geophysics, 38: 339–348. Fedi, M., Rapolla, A., and Russo, G., 1999. Upward continuation of scattered potential field data. Geophysics, 64: 443–451. Henderson, R.G., 1970. On the validity of the use of the upward continuation integral for total magnetic intensity data. Geophysics, 35: 916–919. Henderson, R.G., and Zietz, I., 1949. The upward continuation of anomalies in total magnetic intensity fields. Geophysics, 14: 517–534. Langel, R.A., and Hinze, W.J., 1998. The Magnetic Field of the Earth’s Lithosphere: The Satellite Perspective. Cambridge: Cambridge University Press. View publication stats Pawlowski, R.S., 1995. Preferential continuation for potential-field anomaly enhancement. Geophysics, 60: 390–398. Ravat, D., Whaler, K.A., Pilkington, M., Purucker, M., and Sabaka, T., 2002. Compatibility of high-altitude aeromagnetic and satellitealtitude magnetic anomalies over Canada. Geophysics, 67: 546–554. Shure, L., Parker, R.L., and Backus, G.E., 1982. Harmonic splines for geomagnetic modeling. Physics of the Earth and Planetary Interiors, 28: 215–229. Cross-references Aeromagnetic Surveying Crustal Magnetic Field
