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B - Appendix BGlossary of symbols
- Leland Timothy Long, Georgia Institute of Technology, Ronald Douglas Kaufmann
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1 - Gravitational attraction
- Leland Timothy Long, Georgia Institute of Technology, Ronald Douglas Kaufmann
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Preface
- Leland Timothy Long, Georgia Institute of Technology, Ronald Douglas Kaufmann
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Summary
Gravity data acquisition and analysis is most often presented in outline form as one of the smaller chapters in books on general geophysical exploration methods. This limited description means that the details of field techniques and data analysis are lost or greatly abbreviated and left to the individual to learn through experience. The objective of this book is to offer a detailed presentation of gravity data acquisition and analysis in a single package. The examples are taken from geophysical engineering problems as well as the analysis of regional and global data.
The objective is to completely cover the information needed for a novice to understand how and why gravity data are acquired and analyzed. A student completing a course using this text could easily acquire gravity data and would be prepared to initiate independent research on the analysis of potential data. A consulting geophysicist will find a base of both theoretical- and application-oriented information in this text, while a geologist or engineer can use this book to better understand the advantages and limitations of the gravity method. The general approach of this text has evolved over the past 20 years through experience gained from the acquisition of more than 40,000 values of gravity and in teaching courses in potential methods.
The text is intended for a wide range of users. It is written so that the basic applications are easily understood by those with limited training in mathematics. At the same time, the text occasionally introduces more advanced topics from potential theory for those with greater skills in mathematics. The text does not present extensive equations for the many possible specific models. Some simple shapes lead to complex equations that are computationally intense and generally of little practical use. Instead, the text presents the simpler models as a means of illustrating concepts or as a method of approximating structures. Sufficient background is presented in the equations and analysis techniques for those wishing to create their own more detailed models. In general, methods that allow automatic modeling of the gravity fields using approximations will be emphasized. Inversion methods are presented for the geophysicists needing more advanced analysis techniques for larger datasets.
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Acquisition and Analysis of Terrestrial Gravity Data
- Leland Timothy Long, Ronald Douglas Kaufmann
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- 05 February 2013
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- 17 January 2013
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Gravity surveys have a huge range of applications, indicating density variations in the subsurface and identifying man-made structures, local changes of rock type or even deep-seated structures at the crust/mantle boundary. This important one-stop book combines an introductory manual of practical procedures with a full explanation of analysis techniques, enabling students, geophysicists, geologists and engineers to understand the methodology, applications and limitations of a gravity survey. Filled with examples from a wide variety of acquisition problems, the book instructs students in avoiding common mistakes and misconceptions. It explores the increasing near-surface geophysical applications being opened up by improvements in instrumentation and provides more advance-level material as a useful introduction to potential theory. This is a key text for graduate students of geophysics and for professionals using gravity surveys, from civil engineers and archaeologists to oil and mineral prospectors and geophysicists seeking to learn more about the Earth's deep interior.
Frontmatter
- Leland Timothy Long, Georgia Institute of Technology, Ronald Douglas Kaufmann
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- Acquisition and Analysis of Terrestrial Gravity Data
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A - Appendix ACommon definitions and equations in potential theory
- Leland Timothy Long, Georgia Institute of Technology, Ronald Douglas Kaufmann
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- Acquisition and Analysis of Terrestrial Gravity Data
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References
- Leland Timothy Long, Georgia Institute of Technology, Ronald Douglas Kaufmann
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2 - Instruments and data reduction
- Leland Timothy Long, Georgia Institute of Technology, Ronald Douglas Kaufmann
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- Acquisition and Analysis of Terrestrial Gravity Data
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- 17 January 2013, pp 10-40
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6 - Interpretation of density structure
- Leland Timothy Long, Georgia Institute of Technology, Ronald Douglas Kaufmann
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- Acquisition and Analysis of Terrestrial Gravity Data
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Summary
Introduction
The density distribution required to generate a gravity anomaly is non-unique. While a density distribution generates a unique anomaly, there are many density distributions that can generate that same anomaly. Hence, gravity anomalies alone cannot determine the distribution of densities. However, because the gravity field is generated by a distribution of single poles of attraction, it is generally simpler to interpret than dipole fields like the magnetic field. The non-uniqueness of the potential field prevents one from obtaining an unconstrained solution for the density structure. The only parameter that can be defined from noise-free data is the maximum depth to some part of the structure, and that useful information is itself based on the assumption that part of the anomaly is equivalent to a point mass. Also, if the gravity anomaly is well defined, the excess or missing mass can be computed directly from the gravity data, although the distribution of that mass in the subsurface cannot be defined. The non-uniqueness of the potential field can only be overcome by assuming that the structure fits some pre-conceived understanding concerning its density, shape, and position.
In finding a density model for a gravity anomaly the details of the assumptions will influence and often determine the solution. In effect, any solution that satisfies the restriction concerning the maximum depth to the top of the structure will be determined principally by the assumptions concerning the nature and shape of the density structure. The most common assumptions are based on well-understood shapes for geologic structures. Assumptions appropriate for a flat and shallow sedimentary basin differ greatly from those appropriate for the thin vertical plane of an intrusive dike, and the gravity anomalies would also have significantly different shapes. On the other hand, a sphere at depth might duplicate the approximate shape of an anomaly from a symmetrical shallow basin, but such a model would be an unrealistic solution where the surface geology clearly shows the existence of a basin. An obvious constraint on any model is that the modeled structure should not extend above the surface, a contradiction often created with the application of models that are too simple.
5 - Manipulation of the gravity field
- Leland Timothy Long, Georgia Institute of Technology, Ronald Douglas Kaufmann
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Summary
Objective of gravity field manipulation
The gravity field may be displayed as discrete points corresponding to points of observation, as an extrapolation of these points to a regularly spaced grid, and as an interpretation in the form of a contour map. Each representation has intrinsic advantages and deficiencies. Although the distribution of observations in the field is rarely in a form that is convenient for an efficient application of computer-based analysis techniques, discrete point locations offer the most precise and honest representation of the data. Also, discrete point locations minimize the introduction of artifacts often associated with extrapolations in graphical presentations of gravity data. The extrapolation of gravity values to a regularly spaced grid simplifies the application of computer programs for analysis; but, the precision of the interpolated value at individual grid point locations can vary widely over the map area. The precision will be low, for example, in areas where the data density is low and where the data distribution is poorly suited for the chosen extrapolation technique. The precision will be high in areas where multiple observations define the value at the grid point location. A contoured presentation displays the data in a form that is easy to visualize. However, a contoured presentation always adds some element of interpretation to the field observations. Contouring programs require assumptions concerning the smoothness of the field in order to extrapolate values to areas without field observations. If the contouring technique, for example, fits a plane to the closest points, then small errors in a tight cluster of data could result in a steep slope to the plane. Extrapolating this plane to adjacent areas without supporting data could lead to large deviations from the actual field value, a common problem with some older contouring programs. Also, contouring programs rely on spatial gridding algorithms such as minimum curvature and kriging, each with their own benefits and limitations of how accurately they honor the data and how they interpolate values. The manual contouring of many older maps by skilled draftsmen often provides a more pleasing and realistic representation of the gravity field.
Index
- Leland Timothy Long, Georgia Institute of Technology, Ronald Douglas Kaufmann
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7 - The inversion of gravity data
- Leland Timothy Long, Georgia Institute of Technology, Ronald Douglas Kaufmann
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Summary
Introduction
The inverse problem in potential theory is an exercise in reducing an infinite distribution of possible solutions down to one that is physically realistic and acceptable to an experienced interpreter. This chapter formalizes the determination of distributions of densities given assumptions concerning the properties of the density anomalies and the causative structures. Although the densities of materials surrounding a gravity observation uniquely determine the gravity field, the gravity data alone cannot determine a unique distribution of densities. The density can vary smoothly within a geologic structure, or it can be discontinuous, such as at boundaries of distinct geologic units. In either smooth or discontinuous density distributions there exists an infinite variety in the densities that can be found to fit any set of gravity observations. The only restriction is that the anomalous densities fall above some maximum depth. In general, density is neither a smooth nor uniform function of position. Density is discontinuous across boundaries at all scales, from mineral grain boundaries to the edges of major geologic structures. The details of the density structure are usually so complex that a complete definition would be neither practical nor advisable. In most cases, average or bulk densities for geologic units may be constrained to fall within narrow limits and these limits may be used to limit the density models that generate the gravity anomaly. In effect, the limitations placed on density by the composition and physical condition of geologic units to a large extent define acceptable density structures. On the other hand, the non-uniqueness of gravity data cannot be escaped. There exists an infinitely large set of density models that will satisfy a given gravity anomaly and most of these are not physically realistic. The task of gravity data inversion is to automatically find geologically acceptable density structures that also satisfy the observed gravity anomalies. Hence, reasonable densities for the lithology and acceptable shapes for structures are used to constrain the search for the density structure in gravity data inversion. In the inverse problem, the assumptions are formulated as constraint equations, sometimes referred to as the cost functions. The role of the cost functions or constraints is to add sufficient information to allow a unique solution for a density model consistent with the observed anomalies and expressed constraints.
3 - Field acquisition of gravity data
- Leland Timothy Long, Georgia Institute of Technology, Ronald Douglas Kaufmann
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Summary
Introduction
Gravity data acquisition is an exercise in the documentation of detail and common sense. The practicality of developing detailed survey plans beforehand depends on the type of survey. A survey over a small area where the locations of data points can be determined in advance could benefit from a site visit and detailed preplanning. However, for more widely spaced data, such as in regional surveys for crustal-scale structures, the observation points most likely have not been visited before the survey and their exact location and condition may not be determined until the sites are visited. For these surveys, planning in detail for each site is rarely practical because unanticipated legal and physical barriers to access of survey locations may change with time. In addition, the preservation of safe working conditions related to traffic or weather in the field often requires on-the-spot changes in survey plans. With or without pre-planning, detailed notes on the survey should be taken at the time of data acquisition. Complete and detailed notes can significantly reduce data reduction errors and are a necessary component of a quality-assurance program. Without such notes, correction of anomalous points would require a repeat occupation of the field site. Guidelines for documenting field acquisition, either in a notebook or in a computer will help to maintain complete and consistent notes and will speed data reduction.
The field survey techniques must accommodate the size and terrain of the survey area and the available instruments. Each survey will be unique and could require some modifications to survey technique. However, the data documentation should be consistent to encourage a systematic reduction process and record of quality control. Documentation starts with familiarizing the survey crew with the area through a site visit, perhaps when establishing base stations, or by examining maps and aerial photographs. Interactive web sites provide a useful tool for examination of the acquisition area. The importance of documentation and its contribution to efficiency is fully realized when a survey is interrupted or when problems in meter drift or locations are discovered after the survey team has left the field. Although GPS systems and meters equipped with computers reduce the need for hard copies, backups should be designed to prevent loss of data in power failures or computer crashes. Ultimately, a backup of essential data should be printed in a format easily interpreted by optical character recognition.
8 - Experimental isostasy
- Leland Timothy Long, Georgia Institute of Technology, Ronald Douglas Kaufmann
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Summary
The isostatic reduction
The complete Bouguer anomaly has the pronounced effect of removing the attraction of the mass between the point of observation and mean sea level or another convenient datum for reduction. In mountainous regions with an average elevation of 600 m the mass removed introduces Bouguer anomalies on the order of −67 mGal. According to the principle of isostasy, these negative Bouguer anomalies are attributed to the zones of anomalous negative mass below the datum that compensate and support the surface load. The thicker mountainous crust is in effect floating on the denser asthenosphere. In the isostatic reduction the surface mass is not removed as it is in the Bouguer anomaly. Instead, the surface mass is moved to below the datum of reduction. The free air reduction is equivalent to moving the mass into a thin layer at the surface, making the free air reduction a kind of isostatic reduction. In another sense, the isostatic reduction may be viewed as the removal of the topography and the removal of the compensating mass according to a particular model for the compensation. In general, the distribution of compensating mass is unknown. In this chapter we discuss the character of the relation between topography and the Bouguer gravity anomaly and the determination of the distribution of compensating mass.
The isostatic response function
The isostatic response function is an abstract quantity that cannot be measured directly. Conceptually, it is the Bouguer gravity anomaly resulting from the load of an unrealistic very narrow and tall mountain in isostatic equilibrium. Consequently, the isostatic response function can be a complicated consequence of the elastic strength and composition of the affected rocks. Also, over time, viscoelastic deformation of the underlying rocks, inelastic deformation like faulting, and elastic deformation under stress make the isostatic response function a complex function of time and tectonic history. The Pratt–Hayford and Airy–Heiskanen compensation models (see Chapter 2 and Heiskanen and Moritz, 1967) for computation of the isostatic anomaly illustrate classical models for the expected character of the isostatic response function. In Figure 8.1 a point load corresponding to these models of isostatic compensation is approximated by a 1.0-km high topographic feature with a width of 20 km. In Figure 8.2 the Bouguer anomaly, that is the approximated isostatic response function, for these two models are similar, supporting the observation that the isostatic anomaly computed using either of these two models is very similar.
4 - Graphical representation of the anomalous field
- Leland Timothy Long, Georgia Institute of Technology, Ronald Douglas Kaufmann
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Summary
Map scale and implied accuracy
Gravity data are usually presented in a graphical mode as a contour or relief map. The contour map may be color coded to indicate the magnitude of the anomaly. Also, the contours may be shaded to simulate a low angle of illumination or displayed as a surface to provide a three-dimensional visualization of the data (Figure 4.1). Such presentations facilitate visual perception of the shape and trends of the potential field. Color coding and shading have the advantage of minimizing the mental effort needed to convert contour lines and point plots of data to a visual image of the surface. The similarity of this image to the familiar presentation of topography facilitates interpretation and identification of anomalies. With an understanding of how density anomalies contribute to the anomalous field, observed anomalies can be translated into estimates of structures that ultimately lead to more quantitative interpretations of structure.
Often, access to an area is limited and in order to obtain sufficient detail for interpretation, the data are obtained along lines and projected onto straight-line profiles that are approximately parallel to the line. These slices through the gravity field of a study area can be analyzed in more detail and provide interpreted information more quickly than from the larger data set required to cover a two-dimensional area. Whether it is a line of closely spaced data or a section drawn through a two-dimensional map, these one-dimensional profiles generally require that the trend in the anomalies be known before an analysis is started. Applications of two-dimensional models require that the slices be projected onto a line perpendicular to the trend of the structure.
Contents
- Leland Timothy Long, Georgia Institute of Technology, Ronald Douglas Kaufmann
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- Acquisition and Analysis of Terrestrial Gravity Data
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Contributors
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- By Rose Teteki Abbey, K. C. Abraham, David Tuesday Adamo, LeRoy H. Aden, Efrain Agosto, Victor Aguilan, Gillian T. W. Ahlgren, Charanjit Kaur AjitSingh, Dorothy B E A Akoto, Giuseppe Alberigo, Daniel E. Albrecht, Ruth Albrecht, Daniel O. Aleshire, Urs Altermatt, Anand Amaladass, Michael Amaladoss, James N. Amanze, Lesley G. Anderson, Thomas C. Anderson, Victor Anderson, Hope S. Antone, María Pilar Aquino, Paula Arai, Victorio Araya Guillén, S. Wesley Ariarajah, Ellen T. Armour, Brett Gregory Armstrong, Atsuhiro Asano, Naim Stifan Ateek, Mahmoud Ayoub, John Alembillah Azumah, Mercedes L. García Bachmann, Irena Backus, J. Wayne Baker, Mieke Bal, Lewis V. Baldwin, William Barbieri, António Barbosa da Silva, David Basinger, Bolaji Olukemi Bateye, Oswald Bayer, Daniel H. Bays, Rosalie Beck, Nancy Elizabeth Bedford, Guy-Thomas Bedouelle, Chorbishop Seely Beggiani, Wolfgang Behringer, Christopher M. Bellitto, Byard Bennett, Harold V. Bennett, Teresa Berger, Miguel A. Bernad, Henley Bernard, Alan E. Bernstein, Jon L. Berquist, Johannes Beutler, Ana María Bidegain, Matthew P. Binkewicz, Jennifer Bird, Joseph Blenkinsopp, Dmytro Bondarenko, Paulo Bonfatti, Riet en Pim Bons-Storm, Jessica A. Boon, Marcus J. Borg, Mark Bosco, Peter C. Bouteneff, François Bovon, William D. Bowman, Paul S. Boyer, David Brakke, Richard E. Brantley, Marcus Braybrooke, Ian Breward, Ênio José da Costa Brito, Jewel Spears Brooker, Johannes Brosseder, Nicholas Canfield Read Brown, Robert F. Brown, Pamela K. Brubaker, Walter Brueggemann, Bishop Colin O. Buchanan, Stanley M. Burgess, Amy Nelson Burnett, J. Patout Burns, David B. Burrell, David Buttrick, James P. Byrd, Lavinia Byrne, Gerado Caetano, Marcos Caldas, Alkiviadis Calivas, William J. Callahan, Salvatore Calomino, Euan K. Cameron, William S. Campbell, Marcelo Ayres Camurça, Daniel F. Caner, Paul E. Capetz, Carlos F. Cardoza-Orlandi, Patrick W. Carey, Barbara Carvill, Hal Cauthron, Subhadra Mitra Channa, Mark D. Chapman, James H. Charlesworth, Kenneth R. Chase, Chen Zemin, Luciano Chianeque, Philip Chia Phin Yin, Francisca H. Chimhanda, Daniel Chiquete, John T. Chirban, Soobin Choi, Robert Choquette, Mita Choudhury, Gerald Christianson, John Chryssavgis, Sejong Chun, Esther Chung-Kim, Charles M. A. Clark, Elizabeth A. Clark, Sathianathan Clarke, Fred Cloud, John B. Cobb, W. Owen Cole, John A Coleman, John J. Collins, Sylvia Collins-Mayo, Paul K. Conkin, Beth A. Conklin, Sean Connolly, Demetrios J. Constantelos, Michael A. Conway, Paula M. Cooey, Austin Cooper, Michael L. Cooper-White, Pamela Cooper-White, L. William Countryman, Sérgio Coutinho, Pamela Couture, Shannon Craigo-Snell, James L. Crenshaw, David Crowner, Humberto Horacio Cucchetti, Lawrence S. Cunningham, Elizabeth Mason Currier, Emmanuel Cutrone, Mary L. Daniel, David D. Daniels, Robert Darden, Rolf Darge, Isaiah Dau, Jeffry C. Davis, Jane Dawson, Valentin Dedji, John W. de Gruchy, Paul DeHart, Wendy J. Deichmann Edwards, Miguel A. De La Torre, George E. Demacopoulos, Thomas de Mayo, Leah DeVun, Beatriz de Vasconcellos Dias, Dennis C. Dickerson, John M. Dillon, Luis Miguel Donatello, Igor Dorfmann-Lazarev, Susanna Drake, Jonathan A. Draper, N. Dreher Martin, Otto Dreydoppel, Angelyn Dries, A. J. Droge, Francis X. D'Sa, Marilyn Dunn, Nicole Wilkinson Duran, Rifaat Ebied, Mark J. Edwards, William H. Edwards, Leonard H. Ehrlich, Nancy L. Eiesland, Martin Elbel, J. Harold Ellens, Stephen Ellingson, Marvin M. Ellison, Robert Ellsberg, Jean Bethke Elshtain, Eldon Jay Epp, Peter C. Erb, Tassilo Erhardt, Maria Erling, Noel Leo Erskine, Gillian R. Evans, Virginia Fabella, Michael A. Fahey, Edward Farley, Margaret A. Farley, Wendy Farley, Robert Fastiggi, Seena Fazel, Duncan S. Ferguson, Helwar Figueroa, Paul Corby Finney, Kyriaki Karidoyanes FitzGerald, Thomas E. FitzGerald, John R. Fitzmier, Marie Therese Flanagan, Sabina Flanagan, Claude Flipo, Ronald B. Flowers, Carole Fontaine, David Ford, Mary Ford, Stephanie A. Ford, Jim Forest, William Franke, Robert M. Franklin, Ruth Franzén, Edward H. Friedman, Samuel Frouisou, Lorelei F. Fuchs, Jojo M. Fung, Inger Furseth, Richard R. Gaillardetz, Brandon Gallaher, China Galland, Mark Galli, Ismael García, Tharscisse Gatwa, Jean-Marie Gaudeul, Luis María Gavilanes del Castillo, Pavel L. Gavrilyuk, Volney P. Gay, Metropolitan Athanasios Geevargis, Kondothra M. George, Mary Gerhart, Simon Gikandi, Maurice Gilbert, Michael J. Gillgannon, Verónica Giménez Beliveau, Terryl Givens, Beth Glazier-McDonald, Philip Gleason, Menghun Goh, Brian Golding, Bishop Hilario M. Gomez, Michelle A. Gonzalez, Donald K. Gorrell, Roy Gottfried, Tamara Grdzelidze, Joel B. Green, Niels Henrik Gregersen, Cristina Grenholm, Herbert Griffiths, Eric W. Gritsch, Erich S. Gruen, Christoffer H. Grundmann, Paul H. Gundani, Jon P. Gunnemann, Petre Guran, Vidar L. Haanes, Jeremiah M. Hackett, Getatchew Haile, Douglas John Hall, Nicholas Hammond, Daphne Hampson, Jehu J. Hanciles, Barry Hankins, Jennifer Haraguchi, Stanley S. Harakas, Anthony John Harding, Conrad L. Harkins, J. William Harmless, Marjory Harper, Amir Harrak, Joel F. Harrington, Mark W. Harris, Susan Ashbrook Harvey, Van A. Harvey, R. Chris Hassel, Jione Havea, Daniel Hawk, Diana L. Hayes, Leslie Hayes, Priscilla Hayner, S. Mark Heim, Simo Heininen, Richard P. Heitzenrater, Eila Helander, David Hempton, Scott H. Hendrix, Jan-Olav Henriksen, Gina Hens-Piazza, Carter Heyward, Nicholas J. Higham, David Hilliard, Norman A. Hjelm, Peter C. Hodgson, Arthur Holder, M. Jan Holton, Dwight N. Hopkins, Ronnie Po-chia Hsia, Po-Ho Huang, James Hudnut-Beumler, Jennifer S. Hughes, Leonard M. Hummel, Mary E. Hunt, Laennec Hurbon, Mark Hutchinson, Susan E. Hylen, Mary Beth Ingham, H. Larry Ingle, Dale T. Irvin, Jon Isaak, Paul John Isaak, Ada María Isasi-Díaz, Hans Raun Iversen, Margaret C. Jacob, Arthur James, Maria Jansdotter-Samuelsson, David Jasper, Werner G. Jeanrond, Renée Jeffery, David Lyle Jeffrey, Theodore W. Jennings, David H. Jensen, Robin Margaret Jensen, David Jobling, Dale A. Johnson, Elizabeth A. Johnson, Maxwell E. Johnson, Sarah Johnson, Mark D. Johnston, F. Stanley Jones, James William Jones, John R. Jones, Alissa Jones Nelson, Inge Jonsson, Jan Joosten, Elizabeth Judd, Mulambya Peggy Kabonde, Robert Kaggwa, Sylvester Kahakwa, Isaac Kalimi, Ogbu U. Kalu, Eunice Kamaara, Wayne C. Kannaday, Musimbi Kanyoro, Veli-Matti Kärkkäinen, Frank Kaufmann, Léon Nguapitshi Kayongo, Richard Kearney, Alice A. Keefe, Ralph Keen, Catherine Keller, Anthony J. Kelly, Karen Kennelly, Kathi Lynn Kern, Fergus Kerr, Edward Kessler, George Kilcourse, Heup Young Kim, Kim Sung-Hae, Kim Yong-Bock, Kim Yung Suk, Richard King, Thomas M. King, Robert M. Kingdon, Ross Kinsler, Hans G. Kippenberg, Cheryl A. Kirk-Duggan, Clifton Kirkpatrick, Leonid Kishkovsky, Nadieszda Kizenko, Jeffrey Klaiber, Hans-Josef Klauck, Sidney Knight, Samuel Kobia, Robert Kolb, Karla Ann Koll, Heikki Kotila, Donald Kraybill, Philip D. W. Krey, Yves Krumenacker, Jeffrey Kah-Jin Kuan, Simanga R. Kumalo, Peter Kuzmic, Simon Shui-Man Kwan, Kwok Pui-lan, André LaCocque, Stephen E. Lahey, John Tsz Pang Lai, Emiel Lamberts, Armando Lampe, Craig Lampe, Beverly J. Lanzetta, Eve LaPlante, Lizette Larson-Miller, Ariel Bybee Laughton, Leonard Lawlor, Bentley Layton, Robin A. Leaver, Karen Lebacqz, Archie Chi Chung Lee, Marilyn J. Legge, Hervé LeGrand, D. L. LeMahieu, Raymond Lemieux, Bill J. Leonard, Ellen M. Leonard, Outi Leppä, Jean Lesaulnier, Nantawan Boonprasat Lewis, Henrietta Leyser, Alexei Lidov, Bernard Lightman, Paul Chang-Ha Lim, Carter Lindberg, Mark R. Lindsay, James R. Linville, James C. Livingston, Ann Loades, David Loades, Jean-Claude Loba-Mkole, Lo Lung Kwong, Wati Longchar, Eleazar López, David W. Lotz, Andrew Louth, Robin W. Lovin, William Luis, Frank D. Macchia, Diarmaid N. J. MacCulloch, Kirk R. MacGregor, Marjory A. MacLean, Donald MacLeod, Tomas S. Maddela, Inge Mager, Laurenti Magesa, David G. Maillu, Fortunato Mallimaci, Philip Mamalakis, Kä Mana, Ukachukwu Chris Manus, Herbert Robinson Marbury, Reuel Norman Marigza, Jacqueline Mariña, Antti Marjanen, Luiz C. L. Marques, Madipoane Masenya (ngwan'a Mphahlele), Caleb J. D. Maskell, Steve Mason, Thomas Massaro, Fernando Matamoros Ponce, András Máté-Tóth, Odair Pedroso Mateus, Dinis Matsolo, Fumitaka Matsuoka, John D'Arcy May, Yelena Mazour-Matusevich, Theodore Mbazumutima, John S. McClure, Christian McConnell, Lee Martin McDonald, Gary B. McGee, Thomas McGowan, Alister E. McGrath, Richard J. McGregor, John A. McGuckin, Maud Burnett McInerney, Elsie Anne McKee, Mary B. McKinley, James F. McMillan, Ernan McMullin, Kathleen E. McVey, M. Douglas Meeks, Monica Jyotsna Melanchthon, Ilie Melniciuc-Puica, Everett Mendoza, Raymond A. Mentzer, William W. Menzies, Ina Merdjanova, Franziska Metzger, Constant J. Mews, Marvin Meyer, Carol Meyers, Vasile Mihoc, Gunner Bjerg Mikkelsen, Maria Inêz de Castro Millen, Clyde Lee Miller, Bonnie J. Miller-McLemore, Alexander Mirkovic, Paul Misner, Nozomu Miyahira, R. W. L. Moberly, Gerald Moede, Aloo Osotsi Mojola, Sunanda Mongia, Rebeca Montemayor, James Moore, Roger E. Moore, Craig E. Morrison O.Carm, Jeffry H. Morrison, Keith Morrison, Wilson J. Moses, Tefetso Henry Mothibe, Mokgethi Motlhabi, Fulata Moyo, Henry Mugabe, Jesse Ndwiga Kanyua Mugambi, Peggy Mulambya-Kabonde, Robert Bruce Mullin, Pamela Mullins Reaves, Saskia Murk Jansen, Heleen L. Murre-Van den Berg, Augustine Musopole, Isaac M. T. Mwase, Philomena Mwaura, Cecilia Nahnfeldt, Anne Nasimiyu Wasike, Carmiña Navia Velasco, Thulani Ndlazi, Alexander Negrov, James B. Nelson, David G. Newcombe, Carol Newsom, Helen J. Nicholson, George W. E. Nickelsburg, Tatyana Nikolskaya, Damayanthi M. A. Niles, Bertil Nilsson, Nyambura Njoroge, Fidelis Nkomazana, Mary Beth Norton, Christian Nottmeier, Sonene Nyawo, Anthère Nzabatsinda, Edward T. Oakes, Gerald O'Collins, Daniel O'Connell, David W. Odell-Scott, Mercy Amba Oduyoye, Kathleen O'Grady, Oyeronke Olajubu, Thomas O'Loughlin, Dennis T. Olson, J. Steven O'Malley, Cephas N. Omenyo, Muriel Orevillo-Montenegro, César Augusto Ornellas Ramos, Agbonkhianmeghe E. Orobator, Kenan B. Osborne, Carolyn Osiek, Javier Otaola Montagne, Douglas F. Ottati, Anna May Say Pa, Irina Paert, Jerry G. Pankhurst, Aristotle Papanikolaou, Samuele F. Pardini, Stefano Parenti, Peter Paris, Sung Bae Park, Cristián G. Parker, Raquel Pastor, Joseph Pathrapankal, Daniel Patte, W. Brown Patterson, Clive Pearson, Keith F. Pecklers, Nancy Cardoso Pereira, David Horace Perkins, Pheme Perkins, Edward N. Peters, Rebecca Todd Peters, Bishop Yeznik Petrossian, Raymond Pfister, Peter C. Phan, Isabel Apawo Phiri, William S. F. Pickering, Derrick G. Pitard, William Elvis Plata, Zlatko Plese, John Plummer, James Newton Poling, Ronald Popivchak, Andrew Porter, Ute Possekel, James M. Powell, Enos Das Pradhan, Devadasan Premnath, Jaime Adrían Prieto Valladares, Anne Primavesi, Randall Prior, María Alicia Puente Lutteroth, Eduardo Guzmão Quadros, Albert Rabil, Laurent William Ramambason, Apolonio M. Ranche, Vololona Randriamanantena Andriamitandrina, Lawrence R. Rast, Paul L. Redditt, Adele Reinhartz, Rolf Rendtorff, Pål Repstad, James N. Rhodes, John K. Riches, Joerg Rieger, Sharon H. Ringe, Sandra Rios, Tyler Roberts, David M. Robinson, James M. Robinson, Joanne Maguire Robinson, Richard A. H. Robinson, Roy R. Robson, Jack B. Rogers, Maria Roginska, Sidney Rooy, Rev. Garnett Roper, Maria José Fontelas Rosado-Nunes, Andrew C. Ross, Stefan Rossbach, François Rossier, John D. Roth, John K. Roth, Phillip Rothwell, Richard E. Rubenstein, Rosemary Radford Ruether, Markku Ruotsila, John E. Rybolt, Risto Saarinen, John Saillant, Juan Sanchez, Wagner Lopes Sanchez, Hugo N. Santos, Gerhard Sauter, Gloria L. Schaab, Sandra M. Schneiders, Quentin J. Schultze, Fernando F. Segovia, Turid Karlsen Seim, Carsten Selch Jensen, Alan P. F. Sell, Frank C. Senn, Kent Davis Sensenig, Damían Setton, Bal Krishna Sharma, Carolyn J. Sharp, Thomas Sheehan, N. Gerald Shenk, Christian Sheppard, Charles Sherlock, Tabona Shoko, Walter B. Shurden, Marguerite Shuster, B. Mark Sietsema, Batara Sihombing, Neil Silberman, Clodomiro Siller, Samuel Silva-Gotay, Heikki Silvet, John K. Simmons, Hagith Sivan, James C. Skedros, Abraham Smith, Ashley A. Smith, Ted A. Smith, Daud Soesilo, Pia Søltoft, Choan-Seng (C. S.) Song, Kathryn Spink, Bryan Spinks, Eric O. Springsted, Nicolas Standaert, Brian Stanley, Glen H. Stassen, Karel Steenbrink, Stephen J. Stein, Andrea Sterk, Gregory E. Sterling, Columba Stewart, Jacques Stewart, Robert B. Stewart, Cynthia Stokes Brown, Ken Stone, Anne Stott, Elizabeth Stuart, Monya Stubbs, Marjorie Hewitt Suchocki, David Kwang-sun Suh, Scott W. Sunquist, Keith Suter, Douglas Sweeney, Charles H. Talbert, Shawqi N. Talia, Elsa Tamez, Joseph B. Tamney, Jonathan Y. Tan, Yak-Hwee Tan, Kathryn Tanner, Feiya Tao, Elizabeth S. Tapia, Aquiline Tarimo, Claire Taylor, Mark Lewis Taylor, Bishop Abba Samuel Wolde Tekestebirhan, Eugene TeSelle, M. Thomas Thangaraj, David R. Thomas, Andrew Thornley, Scott Thumma, Marcelo Timotheo da Costa, George E. “Tink” Tinker, Ola Tjørhom, Karen Jo Torjesen, Iain R. Torrance, Fernando Torres-Londoño, Archbishop Demetrios [Trakatellis], Marit Trelstad, Christine Trevett, Phyllis Trible, Johannes Tromp, Paul Turner, Robert G. Tuttle, Archbishop Desmond Tutu, Peter Tyler, Anders Tyrberg, Justin Ukpong, Javier Ulloa, Camillus Umoh, Kristi Upson-Saia, Martina Urban, Monica Uribe, Elochukwu Eugene Uzukwu, Richard Vaggione, Gabriel Vahanian, Paul Valliere, T. J. Van Bavel, Steven Vanderputten, Peter Van der Veer, Huub Van de Sandt, Louis Van Tongeren, Luke A. Veronis, Noel Villalba, Ramón Vinke, Tim Vivian, David Voas, Elena Volkova, Katharina von Kellenbach, Elina Vuola, Timothy Wadkins, Elaine M. Wainwright, Randi Jones Walker, Dewey D. Wallace, Jerry Walls, Michael J. Walsh, Philip Walters, Janet Walton, Jonathan L. Walton, Wang Xiaochao, Patricia A. Ward, David Harrington Watt, Herold D. Weiss, Laurence L. Welborn, Sharon D. Welch, Timothy Wengert, Traci C. West, Merold Westphal, David Wetherell, Barbara Wheeler, Carolinne White, Jean-Paul Wiest, Frans Wijsen, Terry L. Wilder, Felix Wilfred, Rebecca Wilkin, Daniel H. Williams, D. Newell Williams, Michael A. Williams, Vincent L. Wimbush, Gabriele Winkler, Anders Winroth, Lauri Emílio Wirth, James A. Wiseman, Ebba Witt-Brattström, Teofil Wojciechowski, John Wolffe, Kenman L. Wong, Wong Wai Ching, Linda Woodhead, Wendy M. Wright, Rose Wu, Keith E. Yandell, Gale A. Yee, Viktor Yelensky, Yeo Khiok-Khng, Gustav K. K. Yeung, Angela Yiu, Amos Yong, Yong Ting Jin, You Bin, Youhanna Nessim Youssef, Eliana Yunes, Robert Michael Zaller, Valarie H. Ziegler, Barbara Brown Zikmund, Joyce Ann Zimmerman, Aurora Zlotnik, Zhuo Xinping
- Edited by Daniel Patte, Vanderbilt University, Tennessee
-
- Book:
- The Cambridge Dictionary of Christianity
- Published online:
- 05 August 2012
- Print publication:
- 20 September 2010, pp xi-xliv
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- Chapter
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