Deriving rock uplift histories from data-driven inversion of river profiles

Transcription

1 GSA DATA REPOSITORY Deriving rock uplift histories from data-driven inversion of river profiles Christoph Glotzbach 1 1 Institute of Geology, Leibniz University Hannover, Callinstr. 30, Hannover, Germany, glotzbach@geowi.uni-hannover.de DATA REPOSITORY DR1: Methodological details The rate of change in elevation of the river profile depends on the rock uplift U and erosion rate E e.g., Whipple and Tucker, 1999), and thus the evolution equation can be written as: z 1) t = U + E x,t x,t The lower cases in equation 1) indicate that the uplift and erosion rate can vary with time t) and location x). Erosion counteracts uplift and degrades uplifted river profiles. Based on geomorphological investigations, fluvial erosion laws have been implemented, such as the transport-limited or detachment-limited model e.g., Whipple and Tucker, 1999). Here the so-called stream power model is used, which assumes that the rate of incision is primarily controlled by discharge and channel slope, which is used as a proxy for stream power e.g. Whipple and Tucker 1999). The erosional coefficient K) is laterally and temporally constant in the presented applications, but the program also allows to vary K laterally. Temporal variations of discharge and K induced by orbital cycles have little effect on the presented river profile modelling, because the length and timescales of the model is much larger than the climate variations e.g. Paul et al. 2014). The concavity m r /n r ) and exponent of the river gradient n r ) are free parameters in all models and allowed to vary from and from 2/3 to 5/3, respectively e.g., Whipple, 2001). All models start with flat river profiles, which are set to the present-day outlet elevation of individual trunk streams. The latter is invariant during modelling and used as model boundary condition. The presented model enables to take into account the flexural isostatic uplift as a result of changes in topography. The isostatic uplift is modelled by solving the biharmonic equation similar to the approach in PECUBE Braun et al., 2012): D 4 Δu x 4 + D 4 Δu y 4 + 2D 4 Δu x 2 y 2 = ΔρgΔu + ρ 0gΔe 2) where D is the lateral constant flexural rigidity given by: ET e 3 D = 121 v 2 ) T e is the equivalent elastic thickness of the plate, E is the Young s modulus, v is the Poisson s ratio, g is the acceleration due to gravity and Δe and Δu are the increment in

2 surface topography and isostatic uplift over a time step Δt of computation. An efficient Fourier-transform approach was used to solve the so-called thin elastic plate equation Nunn and Aires, 1988). The isostatic uplift is added to the input rock uplift and thus either increase or decrease the overall rock uplift. The complex interaction between the transient topographic evolution and the input uplift history result in complex uplift paths that show distinct differences compared to the step-like uplift paths modelled without isostasy cf. Fig. DR3D and Fig. DR4D). The misfit of modelled and observed data is regulating the inversions, and is calculated with the log-likelihood lnl) function for Gaussian-distributed errors: n 2 ln2π) o lnl = + lnσ 2 i i p i 4) i=1 σ i where p i is the predicted value elevation, thermochronological age, nuclide concentration), and o i and σ i is the observed value and 1σ error of observation i and n is the total number of observations. The lnl of each observation class elevation of river nodes, cosmogenic nuclide concentrations, thermochronological ages and fission track length distributions) is normalised divided) by the number of observations and the sum of normalised lnls regulates the inversion, avoiding preferential fitting of the largest dataset. Analytical uncertainties of thermochronological and cosmogenic nuclide data are well known, ranging from approximately 5 to 20%. As an example the mean errors of the apatite fission track ages and 10 Be concentrations from the Sila massif are 10 and 6%, respectively. The vertical absolute height error is less than 16 m for 90% of the global shuttle radar topography mission SRTM) data e.g. Rabus et al. 2003), but river profiles extracted from steeply incised streams will be associated with much larger errors. I used an arbitrary height error of 75 m, which corresponds to a mean relative error of 10%. The neighbourhood algorithm inversion was used to efficiently explore the mulidimensional parameter space and extract statistical parameters Sambridge 1999a,b). This study used the parameterisation suggested by Glotzbach et al. 2011). Three main parameters control the convergence behaviour of the inversion: i) the number of iterations N i ), ii) the number of models generated at each iteration N m ) and iii) the number of Voronoi cells re-sampled at each iteration N r ). In this study N i is 200 inversion test) and 400 Sila massif), N m is 100 and N r is 90. The rather high resampling ratio 0.9) led to a rather slow convergence and ensures that the full parameter space is explored. Bayesian estimates such as 1D marginal PDFs are first derived separately for each dataset and finally they are combined with: PC D 1,D 2,D 3 ) =a n =3 n =1 PC D n ) PC) 2 5) where PC) is the prior probability of parameter C and PC D n ) are the posterior probabilities of independent datasets, here the river profile, the cosmogenic and thermochronological data. River long profiles were extracted from shuttle radar topography mission SRTM) digital elevation models DEM) with a vertical accuracy of ±16 m. Only those stream

3 segments with an accumulation area >1 km 2 were extracted from the DEM, and smoothed with a moving average window of 1 km. The resulting river profile is resampled to 1 km distance between successive river nodes. The Sila massif model consists of 129 river nodes, and each river node is connected to a 500 m long hillslope discretised into 10 nodes 50 m spacing). The computation time of a single forward model of the Sila massif is 16.4 CPU seconds on a 2.66 GHz Quad-Core Intel Xeon), whereas the solution of the riverhillslope model takes 7 seconds, the calculation of 10 Be concentration at all river and hillslope nodes takes 8 seconds and the 1D thermal modelling takes 1.4 seconds. Increasing the number of river nodes increases only sligthly the computational time of the river-hillslope model Fig. DR1); the calculation of 261 river nodes takes 10 seconds compared to 7 seconds for 129 river nodes CPU seconds y = 0.025x Number of river nodes Fig. DR1: Solution time of a the river-hillslope model for a single forward model as a function of the number of river nodes.

4 DR2: Testing the performance of the model Two synthetic datasets river profile, cosmogenic and thermochronological data) were generated with 1.) increasing uplift rates and 2.) decreasing uplift rates. In both examples, river nodes were extracted from the Acher catchment in the northern Black Forest in southwestern Germany. 1.) The synthetic data is modelled with an uplift rate of 0.1 km/ma from 50 to 1 Ma, followed by 0.5 km/ma uplift until present. The river profile was modeled assuming that river incision is proportional to stream power and the erosional coefficiency K) and exponents of discharge m) and slope n) are 1*10-6 yr -1, 0.5 and 1, respectively. The resulting river profile is characterised by a downstream increase in steepness with a distinct knickzone at 6-10 km upstream of the bounding node Fig. DR2A). Recorded erosion rates and cooling paths were used to calculate at all model nodes 63 river nodes and 630 hillslope nodes) the concentration of cosmogenic 10 Be and at three locations apatite U-Th)/He ages and apatite fission track ages with corresponding length distributions Fig. DR2A-C and DR3A-C). The synthetically generated analytical data was used for inverse modeling of seven parameters: P1) the time of change in uplift rate, P2-3) the uplift rate before and after P1, P4) the erosional coefficient K), P5) the exponent of slope n), P6) the concavity index m/n) and P7) the geothermal gradient. Convergence of the likelihood function is achieved after running a few thousands models in a few hours. Inversion results obtained by inverting only the river profile without analytical data reveal that single parameters, such as the exponent n and the concavity index m/n) can be successfully retrieved Fig. DR2D). The complete uplift history, especially the timing of the increase in uplift rate, however, can only be reconstructed by the integrated inversion of all datasets including the thermochronological and cosmogenic nuclide data Fig. DR2D).

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ig. DR2: Results of inverse modeled synthetic river profile, thermochronological and cosmogenic 10 Be data. Synthetic data was modeled with an erosional coefficiency K) and exponents of discharge m) and slope n) of 1*10-6 yr -1, 0.5 and 1, and 0.1 km/ma uplift from 50 to 1 Ma, followed by 0.5 km/ma until present. A) Synthetic river profile, apatite fission track age and length distribution, apatite U-Th)/He age and cosmogenic 10 Be nuclide concentrations of sample localities. The best-fit modeled data is shown in comparison to the synthetic data. B) Erosion rate evolution of modeled river nodes of the trunk stream for the last 1.2 Ma. C) Modeled cooling curves of three sample localities along the trunk stream used to calculate thermochronological data. D) 1D marginal distributions of forward models with their mean value and standard deviations. The black distributions are derived from an inversion based on all data river profile plus analytical data) and the red distributions are from an inversion based solely on the river profile. 2.) The synthetic data is modelled with an uplift rate of 1.0 km/ma from 20 to 1 Ma, followed by 0.1 km/ma uplift until present. The river profile was modeled assuming an erosional coefficiency K) of 1*10-7 yr -1 and exponents of discharge m) slope n) of 0.5 and 1.25, respectively. The resulting river profile is characterised by a upstream increase in steepness Fig. DR3A). Erosion rates and cooling paths were extracted from similar sample locations as in first example and used to calculate the concentration of cosmogenic 10 Be and apatite U-Th)/He ages and apatite fission track ages with corresponding length distributions Fig. DR3A-C). The synthetically generated analytical data was used for inverse modeling of seven parameters: P1) the time of change in uplift rate, P2-3) the uplift rate before and after P1, P4) the erosional coefficient K), P5) the exponent of slope n), P6) the concavity index m/n) and P7) the geothermal gradient. Convergence of the likelihood function is achieved after running a few thousands models. Extracted 1D marginal distributions of the

6 posterior probability distribution predict most of the free parameter correctly, except for the erosional coefficient k) and the exponent of slope n r ). The uplift history, however, is correctly retrieved by the inverse modelling Fig *7*89 K : : : : : : NQ,-./01 S4,TO,EB9 F : : : : )P,4+5- A0+<7,-E0+, Ø=1.13 SD=0.37 )GR-. -08?Q1 : F % L G5A+5-A0+<7,-E0+, : : : : : : F : %? Ø=1.01 SD=0.14 4*M,4-<45=*7,-08--/0 % K*A80EB,-=45+-65OEH L *E*8*07-<45=*7,-08--/0 : N Ø=0.15 ),+<,408O4,-.JG1 L % F : : : : U : : : G % )*+,-.>01 % )*+,-./01 Ø=2.8*10-6 SD=1.8* *7*89 % )*+,-./012-3 Ø=0.80 SD=0.07 : : ;<7*=8-.>+?/012-3 Ø=0.45 SD=0.03 : ;<7*=8-.>+?/012-3 % Ø=28.3 SD=4.4 ;<7*=8-.>+?/01 D 12-3 : : : *7*89-.C D : : F G5EB-*EH,C-+ 4?E : : % II-.JG?>+12-3F )*+,-./01 Fig. DR3: Results of inverse modeled synthetic river profile, thermochronological and cosmogenic 10 Be data. Synthetic data was modeled with an erosional coefficiency K) and exponents of discharge m) and slope n) of 1*10-7 yr -1, 0.5 and 1.25, and 1.0 km/ma uplift from 20 to 1 Ma, followed by 0.1 km/ma until present. A) Synthetic river profile, apatite fission track age and length distribution, apatite U-Th)He age and cosmogenic 10 Be nuclide concentrations of sample localities. The best-fit modeled data is shown in comparison to the synthetic data. B) Erosion rate evolution of modeled river nodes of the trunk stream for the last 1.2 Ma. C) Modeled cooling curves of three sample localities along the trunk stream used to calculate thermochronological data. D) 1D marginal distributions of forward models with their mean value and standard deviations.

7 DR3: Inversion results of the Sila massif taking into account isostasy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ig. DR4: Results of inverse modelled Trionto river profile, thermochronological and cosmogenic 10 Be data, taking into account uplift caused by flexural isostasy with the following parameters: elastic thickness of 5 km, density of the crust and mantle 2700 and 3200 kg/m 3, Young s modulus 1*10 11 Pa, Poisson ratio of A) Observed and best-fit river profile, apatite fission track age, mean apatite fission track length MTL) and cosmogenic 10 Be nuclide concentrations of sample localities. The observed river profile is color coded according to lithology green: metamorphic rocks, blue: Mesozoic-Eocene sediments, red: granite). The best-fit model yield a normalized misfit of 19 for the following parameters P1-P9): 16.3 Ma, 0.86 Ma, 1.90 mm/a, 0.07 mm/a, 1.45 mm/a, 4.12*10-6, 0.69, 0.63, 21.3 C/km. B) 1D marginal distributions of the posterior probability distribution with their mean values and standard deviations. The lower right diagram shows the 2D marginal distribution of the uplift history derived by combining parameters P1-P5. The thick black and red line are the total rock uplift input uplift plus uplift caused by changes in topography) for the best-fit model for the lowest red) and higher black) river node. C) 2D marginal distributions of the posterior probability distribution of parameter pairs that show obvious tradeoffs. A% AG A A E P P/07A1.0Q*N13 9 M0 9 K1D L>) M,4K1DG L>) M,4K1D A A A% A A A % I.3*12J,4K1D A A A% A A D9/E,E.).-FD9/E,E.).-F A A A A A A A% L>).8-12;3MJ,4 A A A I.3*12J,4K1D A A N>/0* K1DC A A A A A A% A D9/E,E.).-F Probability *10-3 )

8 DR4: River profile evolution of the Trionto river, Sila massif, Italy % +,-./ ) * ) % 607/28-9:14;*<081,9=,/3>45 Fig. DR5: Modeled river profile at 20, 15, 10, 5, 1, 0.5, 0 and in 2 Ma of the Trionto river for the best-fit model with following parameters: i) an erosional coefficient k) of 4.36*10-6, ii) an exponent n of 1.25, iii) a concavity index m/n) of 0.40, iv) a geothermal gradient of 25.5 C/km and v) an initial uplift of 1.5 km/ma until 17.5 Ma, followed by 0.08 km/ma uplift until 0.9 Ma and finally 0.78 km/ma uplift. References cited in the Data Repository Braun, J., van der Beek, P.A., Valla, P., Robert, X., Herman, F., Glotzbach, C., Pedersen, V., Perry, C., Simon-Labric, T., and Prigent, C., 2012, Quantifying rates of landscape evolution and tectonic processes by thermochronology and numerical modeling of crustal heat transport using PECUBE. Tectonophysics, v , Glotzbach, C., van der Beek, P.A., and Spiegel, C., 2011, Episodic exhumation and relief growth in the Mont Blanc massif, Western Alps from numerical modelling of thermochrology data. Earth Planet. Sci. Lett., v. 304, Nunn, J., and Aires, J., 1988, Gravity anomalies and flexure of the lithosphere at the Middle Amazon Basin, Brazil. J. Geophys. Res., v. 93, Paul, J.D., Roberts, G.G., and White, N., 2014, The African landscape through space and time. Tectonics, v. 33, doi: /2013tc Rabus, B., Eineder, M., Roth, A., and Bamler, R., 2003, The shuttle radar topography mission a new class of digital elevation models acquired by spaceborne radar. ISPRS J. Photogramm. Remote Sens., v. 57, Sambridge, M., 1999a, Geophysical inversion with a neighbourhood algorithm I. Searching a parameter space. Geophys. J. Int., v. 138, Sambridge, M., 1999b, Geophysical inversion with a neighbourhood algorithm II.

9 Appraising the ensemble. Geophys. J. Int., v. 138, Whipple, K.X., and Tucker, G.E., 1999, Dynamics of the stream-power river incision model: Implications for height limits of mountain ranges, landscape response timescales, and research needs. J. Geophys. Res., v. 140, p Whipple, K.X., 2001, Fluvial landscape response time: how plausible is steady-state denudation? Am. J. Sci., v. 301, p