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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112, B06414, doi:10.1029/2006JB004663, 2007
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Continental thermal isostasy: 1. Methods and sensitivity Derrick Hasterok1 and David S. Chapman1 Received 28 July 2006; revised 27 January 2007; accepted 22 February 2007; published 23 June 2007.
[1] Continental elevations result from a combination of compositional and thermal
buoyancy and geodynamic forces. Thermal effects are often masked by compositional variations to crustal density and thickness that produce equal or greater relief. We have developed a method by which compositional variations in the crust may be removed, thereby isolating the thermal contribution to elevation. This isostatic correction normalizes the composition of a region to a standard crustal column 39 km thick having an average density of 2850 kg m3. The crustal thickness and density are computed using one-dimensional seismic VP models and an empirical velocity to density conversion. Continental regions adjusted for compositional effects show that thermal isostasy can produce nearly 3 km of relief between cold shield platforms and hot rift zones, comparable to observed relief between young (hot) and old (cold) regions of oceanic lithosphere. A Monte Carlo analysis is used to estimate the uncertainties in the isostatic correction. Uncertainties in the seismic parameters, surface heat flow, and regression coefficients of the velocity-density relationship are all incorporated into the analysis. The Wyoming Craton is used as a case study to demonstrate the effectiveness of the elevation adjustment. Analyses of seismic VP models yield a crustal thickness of 49.5 ± 4.9 km and density of 2945 ± 13 kg m3. The computed compositional correction to elevation for the Wyoming Craton is 131 ± 180 m, shifting the raw elevation of 1069 m to an adjusted elevation of 938 m. Citation: Hasterok, D., and D. S. Chapman (2007), Continental thermal isostasy: 1. Methods and sensitivity, J. Geophys. Res., 112, B06414, doi:10.1029/2006JB004663.
1. Introduction [2] The contribution of variations in crustal thickness and density to changes in elevation are well known and have been appreciated for more than a century [Pratt, 1855; Airy, 1855]. Less appreciated are the effects of the lithospheric thermal state on continental elevation. Thermal isostasy is the geodynamic process whereby regional variations in the lithospheric thermal regime cause changes in elevation. Elevation changes result from variations in rock density in response to thermal expansion. Thermal isostasy has been invoked principally to explain the bathymetry of the oceans as a function of age [McKenzie, 1967; Sclater and Francheteau, 1970; Parker and Oldenburg, 1973; Davis and Lister, 1974; Crough, 1975; Parsons and Sclater, 1977; Sclater et al., 1980, 1981; Stein and Stein, 1992, 1993]. [3] On continents, thermal isostasy has been used successfully to examine the evolution of regions which mimic oceanic spreading, such as continental rifts and elements of provinces with extensive volcanism or back-arc regions [McKenzie, 1978; Jarvis and McKenzie, 1980; Brott et al., 1981; Lachenbruch and Morgan, 1990; Hyndman et al., 1 Department of Geology and Geophysics, University of Utah, Salt Lake City, Utah, USA.
Copyright 2007 by the American Geophysical Union. 0148-0227/07/2006JB004663$09.00
2005]. However, direct thermal effects on continental elevation are difficult to discern because of the potential masking effect of compositional variations in lithospheric density and thickness. [4] One approach to thermal isostasy examines elevation as a function of heat flow [Han and Chapman, 1995; Nagihara et al., 1996]. Han and Chapman [1995] used this approach in a pilot study to examine continental elevation of several tectonic provinces around the globe, but did not make an assessment of the uncertainties related to removing compositional isostatic effects. [5] Making a detailed assessment of thermal isostatic contributions to elevation is important for understanding the observed elevation and placing constraints on geodynamic models of rifting and mountain building. For example, regions such as the hot Basin and Range, which has thin crust, sits at an elevation significantly above the thicker crust of the North American craton. This study then seeks to extend the earlier pilot study of Han and Chapman [1995] with a much more detailed analysis that leads to a greater understanding of the variations in elevation related to thermal isostasy. [6] We present this work in two parts. In part 1, we (1) develop a method by which crustal contributions to elevation may be normalized and (2) discuss the sensitivity of normalization to surface heat flow and thermal models used in this process. We illustrate the crustal normalization method by a single example using the Wyoming Craton. In
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Figure 1. Thermal isostasy of the oceans. (a) Set of oceanic geotherms based on parameters from GDH1, a plate cooling model by Stein and Stein [1992]. Geotherms are computed at 0.5 Ma and 25 – 150 Ma in steps of 25 Ma. (b) and (c) Bathymetry and heat flow of the seafloor as a function of age. Data in grey are excluded from the analysis for GDH1 parameters (solid curves). (d) Bathymetry as a function of heat flow. The theoretical thermal isostastic curve in Figure 1d is derived by removing age from the GDH1 bathymetry-age and heat flow – age relations. The group of points most disturbed by shallow hydrothermal circulation is outlined by the grey dashed line. part 2 [Hasterok and Chapman, 2007], we (1) demonstrate the effectiveness of this compositional normalization to tectonic provinces of North America, (2) seek to define a general thermal isostatic model, and (3) explore explanations of additional processes and properties giving rise to regions with anomalous elevation. The full sense of our exploring continental thermal isostasy will be realized by reading parts 1 and 2 together.
2. Thermal Isostasy [7] Temperature differences between two different lithospheric thermal states, represented by a regional geotherm, T(z), and a reference geotherm, Tref(z), integrated over depth predict an elevation change, DeT, given by Z
zmax
½T ð zÞ Tref ð zÞdz;
DeT ¼ aV
ð1Þ
0
where the elevation change is scaled by the coefficient of thermal expansion, aV. We assume both lithospheric regions have the same adiabat. The maximum depth of integration, zmax, is the depth at which the colder geotherm converges to a mantle adiabat. 2.1. Oceanic Thermal Isostasy [8] To understand the influence of thermal variations on continental elevation, it is useful to look first at the simpler case of oceanic lithosphere where lateral compositional variations are small. Thermal isostasy within the oceans is well established [McKenzie, 1967; Sclater and Francheteau, 1970; Parker and Oldenburg, 1973; Davis and Lister, 1974; Crough, 1975; Parsons and Sclater, 1977; Sclater et al., 1980, 1981; Stein and Stein, 1992, 1993]. As oceanic
lithosphere spreads away from a ridge the lithosphere cools, thickens, and contracts, causing it to subside. Figure 1a illustrates the time evolution of an oceanic geotherm. The ocean lithosphere initially cools rapidly as shown by the large difference between the 0.5 Ma geotherm and the 25 and 50 Ma geotherms. At longer times, the geotherms reach near steady state as shown by the 100 to 150 Ma geotherms. [9] Because the seafloor also conveniently records age, the effect of thermal isostasy can be observed when bathymetry and heat flow are plotted with respect to age (Figures 1b and 1c). Bathymetry is proportional to the integrated thermal state of the oceanic lithosphere (equation (1)) and is explained very well by a simple one-dimensional (1-D) model of lithospheric cooling. Likewise, the lithospheric cooling model predicts surface heat flow, which agrees well with observations for ages greater than about 60 Ma (Figure 1c). However, heat flow is systematically low in younger seafloor compared to the model. This heat flow deficit can be explained by the mining of heat through vigorous, shallow hydrothermal circulation in young seafloor [Harris and Chapman, 2004, and references therein]. Heat mining affects surface heat flow measurements but leaves the deeper thermal regime of the lithosphere largely unaffected. Hence bathymetry is a more robust measure of thermal isostasy in the oceans than heat flow. [10] Would one have been able to discover thermal isostasy acting within the oceans if the age of seafloor were not expressed by marine magnetic anomalies? Figure 1d shows oceanic bathymetry plotted as a function of heat flow and a model prediction derived by eliminating time from the equations governing bathymetry and heat flow as a function of age. The coincident heat flow and bathymetry data used to compute the GDH1 model (bathymetry from 0 – 165 Ma
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Figure 2. Thermal isostasy on continents. (a) A family of continental geotherms parametric in surface heat flow values of 40 to 100 mW m2 in increments of 10 mW m2. The geotherm family is used to compute the theoretical thermal isostatic curve shown in Figure 2b using equation (1). (b) Observed elevations of individual tectonic provinces of North America are plotted as a function of the observed heat flow. Wyoming Craton is labeled WYC. The zero-elevation heat flow used to compute the thermal isostatic curve (black) is 40 mW m2 (dashed lines). and heat flow from 50 – 165 Ma) cluster tightly, only defining well a small portion of the thermal isostatic model. Comparison of model fits with data in Figures 1b and 1d clearly illustrates the advantage of knowing seafloor age when confirming thermal isostasy as a significant geodynamic process. However, in a detailed analysis, Nagihara et al. [1996] use the heat flow – bathymetry analysis to assess the nature of cooling and extent of reheating of old ocean basins in the Atlantic Ocean. By examining bathymetry as a function of heat flow, they were able to discriminate between ocean cooling models, which they argue is not possible by examining the heat flow and bathymetry data in the traditional sense as a function of age. Therefore one might explore using heat flow and elevation to study thermal isostasy on the continents where there are difficulties in using age [Rao et al., 1982]. 2.2. Continental Thermal Isostasy [11] The thermal state of continental lithosphere is also described by a set of geotherms. Whereas oceanic geotherms may be identified in terms of age, continental geotherms do not correlate well with rock age or thermotectonic age. Instead, the most important controlling parameter for continental geotherms is surface heat flow [Pollack and Chapman, 1977b; Chapman and Pollack, 1977; Blackwell, 1977; Lachenbruch and Sass, 1977; Rao et al., 1982]. [12] Steady state, 1-D, conductive continental geotherms computed for surface heat flow values between 40 and 100 mW m2 are shown in Figure 2a (geotherm construction is discussed in section 3.3). A thermal isostatic curve for the continents can be derived from this geotherm family. We use a reference geotherm corresponding to a surface heat flow of 40 mW m2 and assign a lithosphere having this thermal state an elevation of 0 km. Although the zero-
elevation reference heat flow at 40 mW m2 is assumed here, the actual zero elevation could easily be set at another value. This initial assumption for the zero-elevation reference heat flow is refined further by Hasterok and Chapman [2007]. The solid curve in Figure 2b shows a theoretical thermal isostatic relation for the continents using equation (1) and a coefficient of thermal expansion of 3.0 105 K1 within the crust, consistent with the values for major crustal forming rocks including granite and gabbro [Roy et al., 1981]. Within the mantle, the value of thermal expansion used is 3.2 105 K1, which falls between the range of predicted values (3.04 – 3.47 105 K1 [Afonso et al., 2005]). Within the range of surface heat flow for continental tectonic provinces (35 –100 mW m2), the predicted elevation range resulting from thermal isostasy is approximately 3 km. This prediction is similar to the oceanic bathymetry difference between hot ridges and cold abyssal plains. [13] Continental thermal isostasy, however, is frequently obscured by compositional effects (Figure 2b). There is little correlation between observed elevation of 36 North American tectonic provinces and the continental thermal isostatic curve. Much of this scatter is the result of variations in crustal density and thickness between the different tectonic provinces. [14] It is well known that lateral variations in crustal thickness and density cause variations in elevation [Pratt, 1855; Airy, 1855]. The elevation effect due to compositional variation can be estimated by a simple isostatic calculation using continental extremes. A mountainous region with a crustal thickness of 50 km and a density of 2800 kg m3 (granodiorite) would have an elevation of 5 km higher than a rift province with a crustal thickness of 25 km and a density of 2900 kg m3 (gabbro). Both columns assume a
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Adding the elevation adjustment to the observed elevation, we arrive at the final adjusted elevation, eadj, given by eadj ¼ eobs þ DeC :
Figure 3. Cartoon illustrating parameters used in compositional correction for elevation. MSL is the mean sea level, and LOC is the level or depth of compensation. The observed crustal columns (left and right) are adjusted to an arbitrary standard (crustal thickness of 39 km and density 2850 kg m3). similar mantle density of 3340 kg m3. The potentially large compositional effect can easily mask the effect of thermal isostasy. Therefore, in order to isolate the effect of thermal isostasy in the continents, it is necessary first to remove compositional isostatic effects.
3. Methods 3.1. Normalizing Compositional Elevation [15] Compositional variations involving both crustal thickness and density are removed with a simple isostatic correction to the observed elevation. The adjustment normalizes any crustal column to an arbitrary crustal standard. For this study, the standard crustal thickness, h0c, is 39 km, and the standard crustal density, r0c, is 2850 kg m3. These values represent the average crustal thickness and density of the North American provinces as determined by Hasterok and Chapman [2007]. The lithospheric mantle density chosen for this study is 3340 kg m3 based on the xenolith derived estimate by Griffin et al. [1999] for Proterozoic lithosphere. [16] Consider a crustal column with a crustal thickness, hc, and crustal density, rc, at an elevation of eobs (Figure 3). If the density and thickness of the observed crustal column are adjusted to match the standard density and thickness, the elevation would change by an amount DeC [Han and Chapman, 1995] given by rc0 rc hc 1 ; DeC ¼ hc0 1 rm rm
ð2Þ
where rm is the density of the mantle. If the province sits below sea level (e.g., continental shelf), an additional term involving the water depth (where eobs is the bathymetry) and seawater density, rw, is required and the elevation adjustment becomes rc0 rc eobs rw hc 1 DeC ¼ hc0 1 : rm rm rm
ð3Þ
ð4Þ
[17] Three important physical parameters must be estimated in order to make the compositional adjustment: eobs, h c and rc . The observed elevation is obtained from GTOPO30 for elevations above sea level [U.S. Geological Survey (USGS), 1997], and for elevations below sea level the bathymetry data set of Smith and Sandwell [1994, 1997]. Crustal thickness is obtained from 1-D whole crustal VP models [Chulick and Mooney, 2002]. Crustal density is estimated by using empirical velocity-density (VP-r) relationships. 3.2. VP -r Relationships [18] We use two VP-r relationships to estimate density within the crust. Crustal densities are first computed using the linear VP-r relationship derived by Christensen and Mooney [1995] given by r ¼ a þ bVP ;
ð5Þ
where a and b are empirically derived regression coefficients based on a linear least squares inversion of laboratory VP-P-T-r measurements for 19 rock types. When computed crustal densities fall below 2500 kg m3, densities are estimated using the Nafe-Drake curve [Barton, 1986] as suggested by numerous studies [e.g., Ravat et al., 1999; Christensen and Mooney, 1995; Zoback and Mooney, 2003]. [19] Several VP-r relationships were tested to determine the most reasonable conversions from velocity to density including the Nafe-Drake relationship presented by Barton [1986], a pressure-dependent model by Ravat et al. [1999] and a pressure-/temperature-dependent model by Christensen and Mooney [1995]. The linear relationship by Christensen and Mooney [1995] (equation (5)) produced the least scatter in the adjusted elevation when compared to the thermal isostatic model shown in Figure 2b. The linear relationship, when compared with the nonlinear formulation from Christensen and Mooney, also yields the best fit to crustal forming rocks as shown in their paper. Therefore the linear relationship of Christensen and Mooney [1995] is used in this study. [20] Christensen and Mooney [1995] give the regression coefficients to their linear VP-r relationship, a and b in equation (5), at five discrete P-T conditions. Their laboratory VP-r data are determined at pressures extending from 143 MPa to 1.43 GPa (depth equivalent of 5 to 50 km). The VP data are extrapolated to temperatures computed along thermal gradients of 10, 20 and 30 K km1 at 5 km depth intervals. Because crustal temperatures and thicknesses may fall outside the modeled P-T conditions, the VP-r relationships are extended to include a larger range of likely temperature-depth space for the continental crust. Seismic velocity data are extrapolated using the simple formulation described in Appendix A. Once the laboratory data set is extended, regression coefficients for the linear relationship are computed using a linear least squares inversion of the VP and density data excluding monomineralic and
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Figure 4. A nomogram contouring the compositional elevation adjustment for crustal thickness and crustal density. The dashed lines which intersect at hc =39 km and rc = 2850 kg m3 represent the chosen standard crustal column. Hypothetical provinces labeled Rift, Thrust, and LIP (large igneous province) are discussed in the text. WYC is the Wyoming Craton. Contour interval is 0.5 km. extrusive igneous rocks as suggested by Christensen and Mooney [1995]. [21] One fear with using a temperature-dependent VP-r relationship is the removal of the thermal effect on density and therefore the thermal isostatic response of the crust. Although the VP-r relationships determined by Christensen and Mooney [1995] are pressure and temperature dependent, the effects of temperature on density are not incorporated. 3.3. Geotherm Models [22] Geotherm computations are used in the velocitydensity conversion to develop the thermal isostatic model. There exist a number of geotherm models of the continental crust [Blackwell, 1977; Lachenbruch and Sass, 1977; Chapman, 1986; Chapman and Furlong, 1992]. Many other models are based on locally derived estimates of thermal conductivity and/or heat production. Although locally and xenolith derived models may be best in the region for which they are developed, they may be poor estimates when extended to other regions. Therefore we feel it is best to use a consistent geothermal model based on the fewest parameters necessary to demonstrate the usefulness of this method. We chose the method of Chapman [1986] and Chapman and Furlong [1992] to compute our geotherms because they yield a set of geotherms that are well documented, commonly used, and yield temperatures which fall between other warm and cool models. [23] The properties of the geotherm model include a lithospheric structure divided into an upper and lower crust and mantle lithosphere. Thermal conductivity within the crust is determined using a pressure-/temperature-dependent relationship of Chapman [1986]. Thermal conductivities
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within the upper and lower crust are initially set as 3.0 and 2.65 W m1 K1, respectively, corresponding generally to felsic and mafic crystalline rocks at STP conditions. Mantle lithospheric thermal conductivities are computed using the method of Schatz and Simmons [1972], which includes a radiation term at temperatures above 500 K. Surface heat production is determined assuming a 40% radiogenic contribution to surface heat flow with a characteristic depth of 8 km [Pollack and Chapman, 1977a]. This characteristic depth is also used as the decay length of an exponentially decreasing function used to compute heat generation within the upper crust [Lachenbruch, 1970]. Heat production of the lower crust is set as 0.4 mW m3 consistent with many studies of exposed granulite terranes [Pinet and Jaupart, 1987]. Mantle heat production is assumed to be 0.02 mW m3, a lower value justified by Chapman [1986] and Chapman and Furlong [1992]. Temperatures within the asthenosphere are computed using an adiabat with a potential temperature of 1300°C and increase with depth of 0.3°C km1. This adiabat is similar to the average current mantle adiabat described by Thompson [1992]. Details of the geotherm calculation and boundary conditions used are given by Chapman [1986]. [24] The thermal structure in some regions differs from the geotherm models used in this study as a result of nonsteady state processes and differences in thermophysical properties. However, we have chosen to keep our thermal models as simple as possible, interpreting the more difficult and uncertain parameters, i.e., heat production and thermal conductivity, as residuals from a reference thermal isostatic model. [25] Precedents for choosing simple Earth models and interpreting anomalies with respect to the simple models exist in other branches of geophysics. For example, seismologists interpret mantle tomograms, in terms of velocity deviations from a reference model (e.g., PREM). Independent information about lithospheric properties (rifts, cratons, etc.) is not incorporated into the standard model. Instead, anomalous regions emerge in the analysis and the anomalies are interpreted in terms of properties and processes that differ from the standard model. Likewise, gravity fields are interpreted in terms of simple models based on free-air, Bouguer, and isostatic assumptions. Again, local knowledge is incorporated in the anomaly interpretation stage.
4. Sensitivity Tests and Uncertainty [26] To quantify the functional relationship between heat flow and elevation, we must determine the uncertainty in adjusted elevation. Geodynamic processes commonly perturb elevation by 10 s to a few 100 m. Therefore errors introduced in computing adjusted elevation must be as small as possible. This study aims to reduce processing uncertainties, resulting from the conversion of a layered VP model to an average density, to
