#### 3.1 Viscoelastic Deformation From Baseline Changes

[12] The measurements of eight baselines from the 1973–2000 trilateration and GPS surveys (Figure 3) were used to model the lithospheric rheology of the Hebgen Lake fault zone (data along with their 1-*σ* errors are summarized in the Supporting Information). Geodetic sites and baselines were chosen because: (1) They were occupied at least four times in the 1973–1987 trilateration or 1987–2000 GPS survey periods; (2) They are nearly orthogonal to the trace of the 1959 fault rupture so that relative postseismic motion between the hanging-wall and footwall of the fault can be best sampled; (3) The ground motion observations have relatively long time spans from 13 years up to 27 years. Note that three baselines, AIRP-HOLM, AIRP-BIGN, and BIGN-LION, have combined trilateration and GPS measurements overlapping in 1987 (Figure 3). *Savage et al*. [1996] determined that EDM-measured baselines were systematically longer, by a factor of 0.283 ± 0.100 parts per million, than the same baselines measured by GPS. We thus applied this correction to combine the two different geodetic observations into a consistent dataset (Table S1).

[13] Effects other than viscoelastic relaxation, such as afterslip and poroelastic rebound, may also contribute to the observed postseismic deformation. Afterslip can be the result of relaxation of a stress perturbation within the velocity-strengthening region when an earthquake propagates into that region from below [*Marone et al*., 1991], while poroelastic rebound can be produced by the postseismic relaxation of pore-fluid pressure gradients induced by the coseismic volume change of the country rock around the fault [e.g., *Peltzer et al*., 1998]. *Pollitz et al*. [2000], however, concluded that both afterslip and poroelastic rebound were minor contributions when considering the postseismic deformation pattern 3 years after the 1992 Lander earthquake. Because our geodetic observations were collected beginning 14 years after the 1959 Hebgen Lake main shock, these effects were not considered significant for our modeling analysis.

[14] Because geodetic measurements of surface deformation capture effects related to both time-invariant (tectonic) and transient (here the postseismic) processes, the former needs to be evaluated and removed when the latter is used to model the lithospheric rheology [e.g., *Hammond et al*., 2009]. We also assumed that the steady state tectonic deformation across the Hebgen Lake fault zone has been mainly the uniaxial southwest extension of the eastern Snake River Plain [*Puskas et al*., 2007], where an average extensional strain rate of ~0.027 *μ*strain/yr was estimated (see the Supporting Information).

#### 3.2 Rheologic Modeling

[15] This study first implemented a two-layer model for the lithospheric rheology beneath the Hebgen Lake fault zone (Figure 4), with an elastic layer overlying a viscoelastic layer and a viscoelastic half-space. A linear Maxwell rheology was assumed for both the viscoelastic layer and half-space (see the Supporting Information). This working model of lithospheric rheology is consistent with a seismically accepted crustal model that the Conrad and Moho discontinuities are the rheologic and chemical boundaries separating the upper crust (elastic layer), lower crust (viscoelastic layer), and upper mantle (viscoelastic half-space), respectively.

[16] Figure 4 shows a working dislocation model for the *M*_{w} 7.3 Hebgen Lake rupture based on the seismic moment tensor analysis of *Doser* [1985] and the geodetic models of *Barrientos et al*. [1987]. The elastic moduli were derived from the average crust and upper mantle *P*-wave velocities of the Hebgen Lake-Yellowstone region [*Smith et al*., 1989]. Two generally east to east-southeast striking fault segments ruptured during the main shock: (1) the southern Hebgen Lake segment is 18 km long, 15 km wide, with a dip of 70° SW and an average slip of 5.5 m, and (2) the northern Red Canyon segment is 18 km long, 12 km wide, with a dip of 50° SW and a slip of 4.6 m.

[17] The depths and viscosities of the two layers in Figure 4 were evaluated to best fit the temporal changes of baseline lengths across the Hebgen Lake fault (Figure 3). To do this we applied a Monte Carlo approach [e.g., *Spada*, 2001] in which a large set of a priori possible rheologic models were randomly selected for producing predictions of postseismic deformation field. Misfits between these forward results and the observations were then calculated and used to identify best fit models whose misfits are smaller than a given bound.

[18] A set of 20,000 rheologic models was generated spanning the range of plausible lithospheric conditions. For each model, four parameters were randomly selected in the ranges of 15–20 km and ≤40 km for the bottom depths of the two layers, *d*_{1} and *d*_{2}, respectively, and 10^{18}–10^{22} Pa-s for both viscosities, *η*_{1} and *η*_{2} (Figure 4). The algorithm VISCO1D [*Pollitz*, 1997] was run for each model to estimate predicted rates of baseline change (see further discussions and examples of VISCO1D in the Supporting Information). For each baseline, a chi-squared, *χ*^{2}, function was employed to evaluate the misfit

- (1)

where and are the observed and predicted rates of baseline changes, is the one standard deviation uncertainty of the observation, and *N* is the number of surveys of the baseline. For each baseline in Figure 3, rheologic models with calculated *χ*^{2} misfits within the lowest 5% interval were accepted. The models with acceptable fits to all eight baselines were considered to be the best fit models.

[19] Two best fit rheologic models, shown in Figure 5, were derived from the deformation measurements. For the baseline group Yellowstone (YS) that includes four baselines adjacent to the Yellowstone calderas (Figure 5a), the best fit models are characterized by a 9 km thick viscoelastic layer at 17 km depth with mean viscosities of 2 × 10^{19} Pa-s and 5 × 10^{18} Pa-s for the viscoelastic layer and half-space, respectively. The distributions of best fit models and the fitted curves of this group are shown in Figure 6. The mean viscosity of the lower crust is about four times higher than that of the upper mantle, and this difference is statistically significant based on a Student's *t*-test: the two-tailed probability *p* is 0.0012, indicating that there is a 98.8% chance of significantly different means. Here we define, by convention, the statistically significant difference of two means as being *p* < 0.05.

[20] Models for the baseline group Hebgen Lake fault (HLF) with baselines straddling the central and northwest Hebgen Lake fault zone (Figure 5b) reveal a thicker viscoelastic layer, ~15 km, and higher mean viscosities, 3 × 10^{21} and 2 × 10^{19} Pa-s for the lower crust layer and the upper mantle half-space, respectively, than those from the YS baseline group (Figure 7). These differences indicate a lateral variation of lithospheric rheology from northwest to the southeast in the Hebgen Lake-Yellowstone area, generally consistent with the large heat flow change from the Yellowstone caldera (>200 mW/m^{2}) to the Hebgen Lake fault zone (~130 mW/m^{2}, see later discussions).

[21] Note that the viscoelastic modeling algorithm used in this study [*Pollitz*, 1997] does not allow for fault dislocations to extend into the viscoelastic layer. Thus, the bottom depth of the elastic layer *d*_{1}, assumed as the upper bound of the brittle-ductile transition zone, has to be deeper than 15 km as constrained by the Hebgen Lake dislocation model (Figure 4). This constraint is consistent with the observed focal depth maxima distribution of the local seismicity (B-B′ in Figure 2). Possible spatial variations in the thickness of the brittle upper crust, which would be thinner than 15 km for areas with shallow seismicity and high surface heat flow, cannot be resolved. Nonetheless, we suggest that the lateral variation of rheologic properties beneath the Hebgen Lake fault zone implied by the GPS and trilateration observations are consistent with the local tectonic characteristics, in which shallow earthquake focal depths in addition to high temperatures and magmatic deformation sources of the Yellowstone calderas correspond to a much hotter and thus weaker lower crust and upper mantle [*Smith and Braile*, 1994].