Version: March 2. Revision Mark 4, Kate 5, Marc 0

For Journal of Geology: Geological Notes

A Critical Evaluation of Late Tertiary Accelerated Uplift Rates for the Eastern Cordillera, Central Andes of Bolivia

___________________________

Mark H. Anders, Kathryn M. Gregory-Wodzicki and Marc Spiegelman

Lamont-Doherty Earth Observatory of Columbia University, Palisades, NY 10964-8000 USA

(e-mail: manders@ldeo.columbia.edu)

ABSTRACT

Many papers have cited a recent fission-track study of the Eastern Cordillera of the Central Andes of Bolivia as evidence for a marked acceleration of uplift rates over the last 40 m.y. However, this interpretation of the data is not justified. The problem lies not with the actual apatite and zircon fission track data themselves, which we have no particular reason to question, but with the manipulation of these data. The use of the common variable "time" in a plot of uplift rate versus mean sample age results in a pronounced distortion, incorrectly interpreted as increasing uplift rates, which is solely a function of the variance in determining closure age in the two fission-track systems. In order to determine what can be inferred about exhumation history from the data, we constructed a simple quantitative model based on Bayesian inference. Unlike linear regression analysis of elevation-age plots, which treats apatite and zircon data as independent systems, this model defines the systems as paired controls, which provides additional constraints on exhumation history.

-MARK/C - PLEASE HELP WITH STATISTICAL TERMINOLOGY HERE

Results suggest that both an exponential or constant exhumation model fit the data. In the constant exhumation model, the first stage is a moderate unroofing rate of of 0.02 mm/yr from around 100 Ma to about 40 Ma. From about 24-40 Ma the unroofing rate is about 0.13 mm/yr, and then at 24 Ma increases to about 0.55 mm/yr. In the exponential model, exhumation rate increases from 0 mm/yr at 100 Ma to 0.3 mm/yr by 20 Ma, then reaches up to 0.7 mm/yr by the present. More zircon ages which include length data are needed to distinguish between these exhumation histories.

Introduction

The uplift history of the Central Andes is of interest to both climatic and tectonic studies. The Andes affect global climate because they form the only barrier to atmospheric circulation in the southern hemisphere and contain the second highest plateau on earth. Their uplift has probably changed patterns of precipitation, seasonal heating, and upper atmosphere flow, and increased rates of chemical weathering (Ruddiman and Kutzbach 1989; Hay 1996; Broccoli and Manabe 1997) . In fact, Raymo et al. (1988) cite the Andes as one of the three orogens that may have contributed to global cooling since the late Miocene.

The formation of the heart of the Andes, the ~ 4000 m high, 350 km wide Central Andean Plateau, is somewhat enigmatic, because it is associated with a non-collisional plate margin. Debate centers on the relative importance of crustal shortening versus magmatic addition in thickening the crust, and on the role of crustal processes versus mantle processes in producing uplift. Because elevation is a function of the thickness, density, and strength of the lithosphere, the uplift history of the Andes provides important constraints for these debates.

However, few quantitative studies of Andean uplift exist (for a review, see Gregory-Wodzicki, in press) . Of these studies, one of the most influential is that of Benjamin et al. (1987) . They measured apatite and zircon fission-track ages for the Huayna Potosi and Zongo plutons and associated metasediments from near La Paz in the Eastern Cordillera of Bolivia (figs. 1, 2).

A fission-track age records the time at which a mineral cooled through its closure temperature. The concept of a closure temperature is a bit of an oversimplification; instead of a single temperature threshold where fission tracks suddenly become stable, there exists instead a wide temperature zone with gradually decreasing annealing, called the partial annealing zone. Generally, this zone is between 60 - 110°C for apatite and 200-350°C for zircon (Gallagher et al., 1998).

By providing information on cooling, fission-track ages can be used to constrain the timing of exhumation and tectonic denudation in orogenic belts. For example, Benjamin et al. (1987) interpret the fission-track data from Bolivia to suggest that uplift rates of the Eastern Cordillera have increased exponentially since 40 Ma, beginning at rates of 0.1-0.2 mm/yr then increasing significantly around 10-15 Ma and reaching rates of 0.7 mm/yr by 3 Ma (Fig 3). Many subsequent studies have accepted this model of accelerating uplift (Table 1).

However, there are some serious problems with the interpretation of this data, besides those uncertainties typically associated with the analysis of fission-track ages (i.e. Hurford 1991; Gallagher et al. 1998) . First of all, as England and Molnar (1990) discuss, fission track data reflect the removal of overlying material, that is, exhumation, not surface uplift. Benjamin et al. (1987) make the assumption that an increase in exhumation rate reflects an increase in surface uplift, but this is not necessarily true. Though erosion rate does increase with elevation, it also varies with climate (Ritter 1988). For example, climate change from a non-glacial to glacial climate or a climate regime with frequent small storms to a climate regime with less frequent large storms could also increase erosion rates (Molnar and England 1990; Gregory and Chase 1994). Comparison with climate and uplift records need to be made before an increase in erosion is related to uplift.

Masek et al. (1994) further criticize the conclusions of Benjamin et al. (1987). They point out that the calculated uplift rates (more properly, exhumation rates) in figure 3 are correlated with the ages of the samples, resulting in a plot of 1/time versus time. Thus, they suggest that the plot does not provide evidence of steadily increasing erosion rates.

Here we expand on the brief discussion of Masek et al. (1994). We will show how figure 3 in Benjamin et al. (1987) is simply a representation of the variance in determining closure age rather than an indication of accelerating uplift rates.

Assessing the late Tertiary erosion rate of the Central Andes

Determining Erosion Rates

The techniques used by Benjamin et al. (1987) to assess erosion rates from fission-track ages, described in detail in Benjamin (1986), are complex enough to warrant some discussion here. A suite of samples was collected between 1520 m and 5100 m, and closure ages were determined for apatite and zircon separates. Wherever possible apatite and zircon closure ages were analyzed from the same sample, thus providing two control points. The data were then plotted as elevation versus age for each mineral and each rock type (figure 2 of Benjamin et al. (1987), our figure 2).

It is the manipulation of these to primary data to produce figure 3 in Benjamin et al. (1987) (our fig. 3) and the resulting interpretation of this plot that we will show leads to erroneous conclusions about exhumation rates. For the eleven apatite ages, Benjamin et al. (1986) calculate exhumation rate using a special case of the mineral pairs method. First, they subtract surface temperature, specified as 10°C, from the closure temperature of apatite, specified as 110 °C, to get the total temperature differential. They then divide the temperature differential by the geothermal gradient, which they assume to be a constant 30 °C through time, to get the depth to the closure threshold. Thus, for each apatite sample, the amount of exhumation is a constant, 3.33 km or 100°C/30°C/km.

The X-axis of Figure 3, the mean length of time for this amount of exhumation, is determined by dividing the closure age, tc, by 2. The Y-axis, exhumation, or "uplift" rate, is then determined by dividing the depth to the closure threshold, 3.33 km, by X, the mean age:

, (1)

For the four samples with both apatite and zircon ages, the total amount of exhumation is calculated by subtracting the apatite annealing temperature from the zircon annealing temperature (specified as 210 °C) to get the temperature difference, and then dividing by the geothermal gradient, also yielding 3.33 km. Uplift rate is determined by dividing 3.33 km by the difference in the sample ages:

, (2)

The relief method was used for five pairs of samples separated by at least 500 m. In this calculation, uplift rate is calculated by dividing the difference between the two sample elevations, and , by the difference in the sample ages:

, (3)

For further information about these caluclations, see Benjamin et al. (1986).

We do not discuss the the four zircon-apatite pair calculations or the five relief method calculations further, because they do not bear on the crux of our analysis.

-MARK - FIG 3 IS MISSING TWO OF THE RELIEF METHOD POINTS - I THINK THEY ARE THE TWO ON P.48 OF B'S THESIS

Artifact in Assessing Erosion Rates.

Before we discuss in detail how the calculation described above can lead to erroneous conclusions about erosion rates, we will present a generalized discussion of what we call the common variable problem.

Pearson (1897) was the first to point out the spurious effects of a common variable in assessing the correlation coefficient; since then numerous other authors have discussed the effect (see, Atchley et al. 1976; Atchley and Anderson 1978; Kenney; 1982; Jackson and Somers 1991; Schlager et al. 1998).

However, the problem associated with common variable plots is not restricted to just the calculation of a correlation coefficient as is commonly thought (Schlager et al., 1998), but also includes a distortion associated with calculating the regression slope. Anders et al. (1987) showed graphically and through computer simulation that the slope of a linear regression in log (Y/X) versus log (X) space is dependent on the variance in the Y variable. As the variance in Y increases, the slope of a linear regression drives toward -1. It follows from this that if X in such a log plot is fixed to a single value, then any variance in the value of X will be plotted on a line whose slope is -1. This is an intrinsic property of a plotting any data in the form of log (X/Y) versus log (X).

In the case of a non-log plot, the variance in X for a single value of Y will plot on a hyperbolic curve of 1/X. As with the case of the log plot, variance toward the low values of X will produce high values of Y/X which are a sole function of the variance in Y. For example, if X is the measurement of the age of something, any age determinations that underestimate the true age will result in higher values of X/Y.

- MARK - SO WHAT ABOUT THE DETERMINATIONS THAT OVERESTIMATE THE TRUE AGE?

Any regression or interpretation of data plotted on this kind of graph will lend itself to interpreting high variance in Y as an intrinsic property of an independent measurement of Y/X.

The plot of exhumation rate versus mean age as shown in figure 3 of Benjamin et al. (1987), our figure 3, encounters the same difficulties when the same value of X, or age, is used to determine both the independent and dependent variable. The only difference from the generalized case is that for some of the estimates the age or X term is multiplied by 1/2 which only effects the shape of the resulting plot, not the distortion discussed above.

Benjamin et al. (1987) concluded from the shape of the data in their figure 3 (our fig. 3) that the "curve indicates that uplift rates have been increasing exponentially for the past 40 m.y." This is an erroneous conclusion that, unfortunately, has been carried on in the literature concerning the uplift history of the central Andes (table 1). To illustrate our point, consider the transformation of the apatite points from a plot of amount eroded versus age to a plot of exhumation rate versus time (Figure 4). Transforming this horizontal line into Y/X versus X space (fig. 4) results in all values of Y falling on the hyperbolic curve labeled 3.33/x/2. As a result, all values of uplift rate must fall on this curve. Note that in figure 3 all the datum for sampling elevation except 2380 m fall on this curve. Recalculating this datum point from Benjamin (1986) adjusts it to the 3.33/x/2 curve shown in figure 4.

Another way to look at these transformed data is to suppose for a moment that the length of time for any sample to pass through 3.33 km to the surface is the same, i.e., there is a constant exhumation rate. However, a measurement of the closure age has, like all measurements, a variance associated with it. Any underestimation of the closure age will result in a skew toward high values of rate. The underestimated variance will appear as an "exponential" increase in rate. The greater the variance the greater the presumed rate of exhumation.

Another line of reasoning leads to the conclusion that "exponentially" increasing exhumation rates is an erroneous interpretation. Because the transect over which the samples were collected is assumed to be a cohesive block over the last 40 m.y., then an exponential increase in exhumation rates would result in the youngest, and thus topographically lowest points having progressively increasing exhumation rates. Note that in figure 3 the highest erosion rate is from the topographic mid-point (3300 m) of the transect. Moreover, there is no clear pattern of topographically lower sampling location having the highest exhumation or erosional unroofing rate.

- MARK -DOES THIS WORK? B ASSUMES THAT THE 110° ISOTHERM IS PARALLEL TO THE SURFACE. SO IF EROSION WAS UNIFORM OVER THE BLOCK, THEN ALL THE POINTS SHOULD HAVE REACHED THE SURFACE AT THE SAME TIME. THIS SCENARIO WOULD SUGGEST THAT THE SCATTER IN AGES IS DUE TO MEASUREMENT ERROR. IF WE BELIEVE THE AGES, THEN, IN B'S WORLD, THERE MUST HAVE BEEN DIFFERENTIAL EROSION THAT MOVED THE OLDER SAMPLES UP TO THE SURFACE MORE QUICKLY THAN THE YOUNGER SAMPLES, WHERE THEY DID NOT UNDERGO ANY PARTIAL ANNEALING

What the Data do Reveal about Uplift History

Linear Regression Models

One way to avoid the common variable problem is to plot sample elevation versus age, as in Figure 2, rather than elevation divided by age versus age. The typical approach to interpreting these plots is to fit regression lines to the data using age as the dependent variable. However, it is perhaps more useful to plot the amount of material eroded as a function of time, rather than the elevation, as we are interested in determining the exhumation rate; as discussed above, fission-track data do not record surface uplift.

The inverse of the slope in the amount eroded-age plots can be used to estimate mean exhumation rate if several assumptions are made, including: 1) the depth of the isotherms remained constant 2) the samples moved as a coherent block, and 3) the geothermal gradient remained constant.

The first assumption is problematic in areas with high relief and denudation rates over 1000mm/yr, such as the Southern Alps, because isotherms are affected by the shape of surface topography and denudation rate (Stüwe et al.,1994). This is less of a problem for the Central Andes, where erosion rates are lower (see below). The second assumption is probably reasonable, but it is rather unlikely that the geothermal gradient remained constant during active orogeny and denudation.

Two end-member models have been used to relate elevation data to amount of material eroded. In one, isotherms are assumed to be parallel to the earth's surface, and in another, they are assumed to be parallel to both the geoid and the mean surface elevation. In the first case, each point on the surface, regardless of its elevation, represents the same depth of erosion. This is the case modeled by Benjamin in his modified mineral pair method. The observed spread in ages would suggest either 1) age measurement error, 2) spatial variation in the geotherm after the samples passed the closure age, 3) variable erosion, or 4) variable movement within the block. However, this model of parallel isotherms is rather unlikely, as surface topography typically has much more roughness than isotherms. Thus, the problem with the apatite mineral pair data lie not only with the common variable plot, but also in the underlying assumptions.

In the second scenario, in which isotherms are parallel to the geoid and the mean surface elevation, the amount of material eroded, , is determined by the following equation from Brandon et al. (1998):

(4)

where is the depth from the mean surface elevation to the closure isotherm, is sample elevation, and is mean surface elevation. This transformation derives slopes which are equivalent to the commonly used age-elevation plots.

We applied this model to the Benjamin et al. (1987) data, and the results are shown in Figure X. The older zircons from between 43 - 117 Ma in age define a regression line with a slope of 0.02 mm/yr (r2 = 0.05, s = 41 Ma, F = 1.3), but as Benjamin et al (1987) point out, the interpretation of these datapoints is somewhat ambiguous. The ages could represent a time of slow exhumation and cooling, or they could represent a time of partial annealing in a paleo partial annealing zone, in which case the curve would contain no information about exhumation rates. Benjamin et al. (1987) suggest that the low slope of the curve argues for the paleo partial annealing zone hypothesis, but fission-track length data are needed to confirm this interpretation. Length data did not begin to be routinely collected until the mid-80's, and the Benjamin et al. (1987) study lacks this information.

The younger zircon samples define a line with a slope of 0.15 mm/yr (r2 = 0.65, s = 4.2 Ma, F = 12.3), and the apatite samples define a line with a slope of 0.54 mm/yr (r2 = 0.32, s = 2.6 Ma, F = 7.2). Workers have suggested that this increase in slope suggests an increase in erosion rates at around 10-15 Ma (Masek et al., 1994).

However, if the apatite closure isotherm was not parallel to the geoid, then the actual mean exhumation rate would be lower than that calculated from these slopes (Brown and Summerfield). For example, Kennan (2000) suggests that in the Zongo-Huayna Potosi study area, the apatite closure isotherm probably parallels the mean surface elevation, which drops to the northeast. Based on his calculations, the exhumation rate for the apatites is closer to 0.3 mm/yr. This suggests a much smaller increase in exhumation rate in the Miocene, if any, given the small datasets and large errors.

-MARC - SO WHAT DOES YOUR MODEL ASSUME ABOUT THE GEOTHERM?

Bayesian Models

Linear regression analysis is certainly a valid approach to interpreting age-exhumation plots, but treating the apatite and zircon data as independent fails to exploit potentially useful information: the fact that the zircons must record the same events as the apatites. In response to this problem, we developed a simple quantitative model using Bayesian inverse modeling, which considers the apatite and zircon data in conjunction. In Baysian modeling, fits are evaluated

-MARC - PLZ INSERT SOMETHING INTELLIGENT SOUNDING HERE.

In our model, a block of crust with thickness h = 3580 km, equal to the thickness of the sampled fission-track transect, is uplifted at a rate n through a distance b + g + h, where b is the distance between the apatite and zircon closure threshold, and g is the distance between the apatite closure threshold and the base of the crustal block (Fig 5).

We specified two end member models, one with a constant exhumation rate, and the other with an exponentially increasing exhumation rate. In the constant exhumation model, exhumation is negligible until time t*, when it undergoes a step increase to a higher exhumation rate, n:

(2)

where n = uplift velocity, n0 = velocity at time zero, and t = time.

-MARC - WE SHOULD BE CAREFUL ABOUT USING THE TERM UPLIFT, BECAUSE PEOPLE ARE WONT TO MISINTERPRET FISSION-TRACK DATA TO REFLECT SURFACE UPLIFT RATES RATHER THAN UNROOFING RATES.

In the exponentially increasing model:

(3)

where a = a constant.

Values for t*, n, a, b, and g were generated using a random walk, and those that fit the data of Benjamin et al. (1987) and Crough (1983) were kept as part of a family of possible solutions (Fig 6). We specified some constraints on the values of these parameters, but we preferred to keep these large to show the robustness of the solutions. A more detailed description of the model can be found in Spiegelman (in preparation).

Results and Discussion of Familty of Solutions

Both end-member models produced reasonable fits of the data (Fig 6). In the constant uplift model, erosion rate is minimal until around 40 Ma, then increases to between 0.3 to 0.6 mm/yr; note that we did not specify the age of the increase in exhumation, t*, 40 Ma was the best fit to the data as identified by the model. Because uplift is constant after the rate increase, the curve defined by the younger zircon data must be parallel to that through the apatite data. Another interesting observation is that a fairly large range of exhumation rates, 0.3 - 0.6 mm/yr, translates to a fairly narrow range in best fit elevation age-curves.

In order to see if the data was consistent with the increase in uplift around 10-15 Ma as suggested by Masek et al. (1994), we ran another version of the model which excluded the older zircon ages. The results suggest an increase in exhumation rate at around 24 Ma from 0.13 mm/yr to 0.55 mm/yr (Fig 7). Note that these values are the same as derived from the linear regression model.

The exponential model produced a curve similar to that of Figure 3, with uplift rates of about 0.1 mm/yr around 40 Ma, which increase to 0.2 by 20 Ma, and reach up to 0.6 mm/yr by the present. It is somewhat surprising that such a marked increase in exhumation would be consistent with such a modest curvature through the apatite data. However, the apatite data represent only a small span of time, 10.6 Ma, which makes the curvature difficult to discern.

The exponential model fits the older zircon data quite well. However, it is not this data that drive the exponential fit. If we exclude the oldest five zircon samples, which possibly contain no information about exhumation rate, an exponential increase is still consistant with the data (Fig 7).

This model is simplified in that it assumes that closure temperature is a discrete threshold. With track-length data, the model could be expanded to consider more complex thermal structures. It could also be expanded to two dimensions in order to model variation in the geotherm with exhumation.

Conclusions

In conclusion, a reinterpretation of the data of Benjamin et al. (1987) suggests either of two exhumation histories: 1) one with two step-increases in exhumation rate, from 0.02 mm/yr to 0.13 mm/yr at around 40 Ma, then from 0.13 mm/yr to 0.55 mm/yr at around 24 Ma, or 2) exponentially increasing uplift rates from 0.3 mm/yr by 20 Ma to up to 0.7 mm/yr by the present. A more thorough study with additional data collection is needed to distinguish between these two exhumation rate regimes.

ACKNOWLEDGMENTS

This work was supported by PRF 32194-AC2 and NSF EAR 99-02782 to MHA and NSF EAR 97-09114 to KMG. This is Lamont Doherty Earth Observatory contribution XXXX.

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TABLE 1. Citations of Benjamin et al. (1987).

As evidence of Accelerated Uplift/ High Uplift Rates As evidence of late Cenozoic Uplift

Raymo et al. (1988) Isacks et al. (1988)

Kono et al. (1989) Miller and Harris (1989)

Ruddiman et al. (1989) Wörner et al. (1992)

Seltzer (1990) Andriessen, 1995

Hurford (1991) Vandervoort et al. (1995)

Dewey and Lamb (1992) Allmendinger and Gubbels (1996)

McMillan et al. (1993) Chamberlin et al. (1999)

Hoke et al. (1994) Graeber and Asch (1999)

Laubacher and Naeser (1994) Schmitz et al. (1999)

MacFadden et al. (1994) Roperch et al. (2000)

Walsh (1994) As problematic

Zhou and Stüwe (1994) England and Molnar (1990)

Curry et al. (1995) Masek et al. (1994)

Muñoz and Charrier (1996) Kennan (2000)

Kley et al. (1997)

Lamb and Hoke (1997)

Okaya et al. (1997)

Somoza (1998)

Agemar et al. (1999)

Somoza et al. (1999)

Figure Captions

Figure 1. Location of Benjamin et al. (1987) fission-track study, on base of USGS 30 arc-second DEM as processed by the Cornell Andes Project.

Figure 2. Apatite and Zircon fission-track closure ages plotted against elevation of sampling site. Data from Benjamin et al. (1987) and Benjamin (1986). Solid lines are linear regression on the Y axes. This was done as opposed to the normal regression on X because the fit to the data was very close to that shown on figure 3 of Benjamin et al.(1987). Dashed line is a line corresponds to the thickness of uplifted rock as calculated in Benjamin (1986) and does not represent a sampling elevation. See text for further discussion.

Figure 3. Apatite and Zircon fission-track data from Benjamin (1986) and Benjamin et al. (1987). See text or Benjamin (1986) for technique used to convert fission-track data shown in figure 2 to data in this plot. Open circles are apatite fission track data. Solid circles are data from paired apatite and zircon samples. Open triangles are calculated using the relief method. Numbers correspond to topographic elevation of the sampling site as listed in Benjamin (1987).

Figure 4. Plot of transformations of three lines shown in fig. 2. The transformation were done in accordance with the techniques use to transform data in fig.2 to those data in fig. 3. See text for further discussion.

Figure 5. Diagram of model.

Figure 6. Results of Bayesian modeling.