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 Measuring sediment flux in rivers remains a significant problem in studies of landscape evolution. Recent studies suggest that observations of seismic noise near rivers can help provide such measurements, but the lack of models linking observed seismic quantities to sediment flux has prevented the method from being used. Here, we develop a forward model to describe the seismic noise induced by the transport of sediment in rivers. The model provides an expression for the power spectral density (PSD) of the Rayleigh waves generated by impulsive impacts from saltating particles which scales linearly with the number of particles of a given size and the square of the linear momentum. After incorporating expressions for the impact velocity and rate of impacts for fluvially transported sediment, we observe that the seismic noise PSD is strongly dependent on the sediment size, such that good constraints on grain size distribution are needed for reliable estimates of sediment flux based on seismic noise observations. The model predictions for the PSD are consistent with recent measurements and, based on these data, a first attempt at inverting seismic noise for the sediment flux is provided.
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 The transport of coarse sediment by rivers sets the pace of landscape evolution by controlling channel morphology and the rates of bedrock incision [e.g., Whipple, 2004]. Moreover, accurate predictions of sediment flux are needed for diverse applications including sedimentation engineering, river restoration, and flood hazard mitigation. Most models for bed load sediment transport are empirical and typically rely on data from flume experiments where sediment flux is at the transport capacity [e.g., Fernandez Luque and van Beek, 1976]. In many mountain streams, sediment flux is under-capacity, however, and is governed by the sediment supply from upstream and neighboring hillslopes, for which no bed load-flux models exist [e.g.,Whipple, 2004]. Our inability to accurately model bed load transport stems from a lack of measurements during floods in steep rivers where traditional measurement techniques (e.g., sediment traps) are extremely difficult, if not impossible, to apply.
 A potential solution to this data gap is to use acoustic or seismic energy from bed load particle impacts as a proxy for sediment flux [Govi et al., 1993; Barton et al., 2006; Burtin et al., 2008; Hsu et al., 2011]. High-frequency (>1 Hz) seismic noise near rivers has been shown to correlate with river discharge [Burtin et al., 2008, 2010, 2011; Hsu et al., 2011], and the observed increase of seismic noise with increasing flow depth has, in all cases, been partly attributed to particles impacting bedrock. Burtin et al.  show that the majority of this seismic noise observed in Nepal is generated in reaches of the Trisuli River with high gradients. Similar to Burtin et al. , Hsu et al.  argue that observed hysteresis in high frequency seismic power relative to river discharge during major storms in Taiwan is related to sediment transport. However, none of the above mentioned studies were able to directly convert measurements of seismic noise into bed load flux because a theoretical underpinning that relates the two has yet to be developed.
 Our goal is to derive a mechanistic model for seismic noise generated by river sediment-bed impacts using as simple and generic of parameterizations as possible that still incorporate the relevant first-order physics. The primary idea of this model is that each sediment particle impacts the river bed and creates a force impulse that then excites seismic waves that travel to nearby seismic stations. With many particles, each impacting the bed at random times relative to the others, one can calculate the total noise power spectral density (PSD) observed at a given station. To attempt to make comparisons with the observations discussed in the Introduction, we make model predictions over the frequency range 1 Hz <f < 1000 Hz.
2.1. Seismic Impact Model
 In this forward model, it is assumed that individual grains of diameter D each impact the river bed with speed wi, assumed normal to the bed, where wi depends on D and other fluvial parameters, as will be discussed in Section 2.2. For a single particle, the impact force can be described by an elastic contact problem, of which the simplest case is perfectly elastic Hertzian contact [Johnson, 1987]. For this case, the maximum force amplitude F0 and time of contact Δt are given by well known expressions [Johnson, 1987]. For D < 2 m, wi > 0.1 m/s, and a typical elastic modulus (E = 5 ⋅ 1010 Pa), then Δt ≲ 10−3 s, and the impact can be assumed to be instantaneous relative to the frequency range of interest, and impart an impulse equal to
where m = ρsVp is the mass of the particle, Vp is the particle volume, and ρs is rock density. If the impact is not perfectly elastic, I could be as much as a factor of 2 smaller (when perfectly inelastic), in which case I ≈ mwi. A force history for a single particle's instantaneous impulse can then be expressed as F1(t) = Iδ(t) where δ(t) is the Dirac delta function.
 Given a force history F(t) at location x0, the ground velocity (t) at location x is given in the frequency domain by
where F(f) ≡ [F(t)] is the Fourier transform of F(t), and G(t) is the displacement Green's function. Since wi is vertically incident, Rayleigh waves are expected to be the dominant waves excited [e.g., Sanchez-Sesma et al., 2011]. For a horizontally homogeneous medium, and assuming an approximate Rayleigh-wave sensitivity that decays with depth proportional toe−kz, we can follow Aki and Richards and approximate the amplitude of the Rayleigh-wave Green's function as
where vcis the Rayleigh-wave phase velocity,vu is group velocity, k ≡ 2πf/vc is the angular wavenumber, r ≡ |x − x0| is the source-station distance, andQ is the (dimensionless) quality factor.
 To use equation (3), we must have estimates of the frequency-dependentvc, vu and Q. For vc and vu, we use values of average shear wave speed for a typical generic rock site (and in the frequency range of interest) given as
where v0 = 2206 m/s, z0 = 1000 m, and α = 0.272 [Boore and Joyner, 1997]. Using these values, and approximations as above, one can solve for vc as
where vc0 = [(2πz0f0)−αv0Γ(1 + α)]1/(1−α), ξ = α/(1 − α), and Γ(x) is the gamma function (see auxiliary material). With values as given above, vc0 = 1295 m/s, f0 = 1 Hz, ξ = 0.374. Furthermore, under the same approximations, it can be shown that vu = vc/(1 + ξ) = 0.73vc. On the other hand, Q is typically assumed to be of the form of Q = Q0(f/f0)η [e.g., Erickson et al., 2004]. Anderson and Hough  suggest that for the relatively high frequencies and shallow depths of interest, Q0 ≈ 20 and η = 0 are reasonable approximations, and we use these values throughout, despite known spatial variations in Q [e.g., Campbell, 2009].
 To describe the rate of impacts, we define n(D) to be the number of particles with grain size D, per unit length of river per unit D. If the average time between impacts of each particle is ti(D), then ∫Dn ⋅ ti−1dD is the total rate of impacts per unit length of river. In Section 2.2, we relate n/ti to sediment flux.
 In the simplest version of the model, we assume that all impacts occur randomly spaced in time, i.e. F(t) = I ∑jNδ(t − tj), with random tj, and N being the number of impacts. For random tj, one can show that F(f) = F1(f) ≡ I, so that the sum of impacts does not affect the shape of the force spectrum, and the force amplitude squared grows linearly with N. We can now express the power spectral density (PSD) of a station's velocity time series (per unit grain size D) to be
where the integral is along the full length of river R, and 1(t) is the ground velocity due to F1(t).
and β ≡ 2πr0(1 + ξ)f1+ξ−η/(vc0Q0f0ξ−η) are dimensionless. As shown in the auxiliary material, χ(β) can be approximated as
It may be noted that equation (7) scales with frequency roughly as f4.9exp[−2πr0f1.4/(Q0vc0f00.4)].
 Finally, the total PSD, PvT(f), is given as an integral over the grain size distribution
Using equation (9) in equation (7) and substituting into equation (10) then yields an algebraic expression for the total PSD, PvT(f), as a function of frequency f, grain size distribution and other model parameters.
 The random impact model discussed above can be made more realistic by including the correlated impacts of the same particle. If Nc hops of a single particle occur before the impact time becomes significantly different from an integer multiple of the timescale between impacts, ti, then the forcing
can be used instead of F1(t) to approximate an average over timescale Ncti. This then results in a frequency modulation of equation (7) equivalent to multiplying equation (7) by
Since there is expected to be a large variance in hop times [e.g., Lamb et al., 2008a], Nc is likely a relatively small number, i.e. Nc≲ 4, and therefore only has a second-order effect on the model (seeSection 3).
2.2. Fluvial Components
 To drive the seismic model, relationships are needed to predict the rate and velocity of streambed impacts by fluvially transported particles. Here we follow recent work that characterized these processes in the context of bedrock incision [e.g., Sklar and Dietrich, 2004; Turowski et al., 2007; Lamb et al., 2008a]; in particular, we use the model of Lamb et al. [2008a] because it explicitly solves for particle fall velocity. Owing to the strong dependency of seismic energy on particle size and impact velocity (i.e., equation (7) scales as m2wi2), we focus on seismic energy generated from saltating particles alone and neglect particles that are rolling or sliding along the bed, particles suspended in the flow, and viscous damping of particles impacts by the fluid [cf. Lamb et al., 2008a]. The rate of particle impacts per unit channel length (for a given grain size) can be calculated from
where Wis the average channel-bed width,qbD is the volumetric sediment flux per unit grain size D per unit channel width traveling as bed load, Ubis the vertically-averaged streamwise particle velocity andHb is the bed load layer height, sis the depth-averaged particle settling velocity, andVp is the particle volume. C1 ≈ 2/3 accounts for the fact that the total time between impacts should also include the particle ejection or rise time as well as the fall time [Sklar and Dietrich, 2004].
 The depth-averaged bed load velocity and layer height are given as empirical expressions bySklar and Dietrich  derived from several different bed load studies. The best fit relationships are
where R = (ρs − ρf)/ρf, ρs ≈ 2700 kg/m3 and ρf = 1000 kg/m3 are the sediment and fluid densities, respectively, g = 9.8 m/s2 is the acceleration due to gravity, τ* ≡ u*2/(RgD), u* is the bed shear velocity, and H is the total flow depth. Uis the depth-averaged flow velocity calculated asU = 8.1u*(H/ks)1/6 [Parker, 1991], where ks = 3D50 [e.g., Kamphius, 1974], and the critical value of the Shields stress (τ*c) is the value of τ* at the threshold of particle motion found from Lamb et al. [2008b] for the median grain size D50. For other grain sizes, we calculate τ*c = τ*c50(D/D50)−γ, where γ ≈ 0.9 [Parker, 1990]. The bed shear velocity is calculated assuming steady and uniform flow as u* = , where θis the channel-bed slope angle.
 The particle impact velocity normal to the bed can be calculated from a balance between the forces of gravity and drag for spherical particles [Lamb et al., 2008a] as
where wst = is the terminal settling velocity, and b ≡ 3CdρfHb/(2ρsDcosθ). The drag coefficient Cd depends on the particle Reynolds number and grain shape, and we calculate Cd from the empirical formula of Dietrich  for natural sediment (Corey Shape Factor = 0.8, Powers Roundness Scale = 3.5) (for 0.01 m < D < 0.6 m, Cd ranges from 1.4 to 0.5). The average settling velocity through the bed load layer can be calculated from the same force balance as above
 Finally, the average time between impacts for a given saltating particle can be calculated from Sklar and Dietrich  as
 In bedrock rivers, bed load flux qbD is determined by the supply of sediment from neighboring hillslopes and from upstream, and is the primary fluvial parameter that we attempt to constrain. However, with ample supply, the total flux qb ≡ ∫ qbDdD is limited by the river's transport capacity qbc, which can be calculated following Fernandez Luque and van Beek  as
3. Model Results
 Although our model is meant to be general, it is useful to explore the model results using parameters that scale roughly after a natural river. Herein we use the Trisuli River, which is one of the main trans-Himilayan rivers in central Nepal.Burtin et al.  attributed heightened seismic noise to sediment transport in a ≈25 km reach of the Trisuli River that is steep (θ ≈ 1.4°), relatively narrow (W ≲ 50 m), and rapidly incising (5 mm/yr) into the underlying bedrock [Lavé and Avouac, 2001]. Although water discharge was not measured locally, water depth was measured ≈50 km downstream [Burtin et al., 2008], from which we derive water depth at the location of interest using standard hydraulic geometry formulations (see auxiliary material). We assume sediment to be spherical (Vp = πD3/6) and estimate sediment size from grain size measurements of Attal and Lavé  for a reach of the nearby Marsyandi River with a similar drainage area and slope (see auxiliary material).
 The final model is produced when equation (13) for n/ti and equation (16) for wi are substituted into equation (7) along with the given expressions for Ub, Hb and s. equation (7) then predicts the observed seismic PSD (Pv) for a given D, H, θ, W, r0, and qbD. Before showing model results, we observe that once all expressions are substituted, equation (7) approximately scales as D3qbD for b ≲ 1 and constant τ*/τ*c. This implies that the seismic signal is strongly dependent on D, and that one must have good constraints on grain size distribution if qb is to be inferred from observations of Pv. Note that the fluvial parameterizations for n/ti and wi may be different than assumed here in rivers with large scale bedrock roughness where oblique bed impacts may cause impact velocities to scale with flow velocity U rather than settling velocity wst [e.g., Johnson and Whipple, 2010]. If true, this would cause equation (7) to scale approximately with D2 rather than D3, and non-vertical impact would also cause a higher fraction of Love waves to be generated than assumed.
 Using representative values from Lavé and Avouac , Attal and Lavé  and Burtin et al. , we obtain PSDs as function of frequency f as shown in Figure 1 for 2 choices of r0 and 3 choices of D. In these plots, it is assumed that all particles are of median size D = D50 and that qb = qbD describes the flux of these particles. We predict the general spectral features of such PSDs to be similar for any steady impact model where the frequencies of interest are less than one over the impact time (i.e., f < 1/Δt). As shown, the modification introduced by equation (12) creates a modulation in the PSDs with frequency spacing Δf = 1/ti, revealing a potential seismic signature of sediment size independent of sediment flux, but has no effect on a smoothed version of the PSDs. Compared with the PSDs in Figure 6 of Burtin et al. , the model is able to predict some aspects of the observations, including the general peak around ≈7 Hz, the sharper increase of the PSDs to this peak compared to the more gradual decrease at higher frequencies, and the higher PSD values at high frequencies (up to 15–20 Hz) for stations that are closer to the Trisuli River. The modulation introduced by equation (12) may also be observed by Burtin et al. , and could potentially be used as a constraint on the grain size distribution.
 To explore the dependence of equation (7) on fluvial parameters, in Figure 2 we plot Pv for fixed r0 = 600 m at f = 7 Hz (near the peak of the PSD) but with variable H, θ, D and qb. For fixed qb the PSDs have the somewhat unintuitive feature that they decrease both with increasing slope (θ) and with increasing flow depth (H) (solid lines of Figure 2a). This results from a larger hop height and velocity, which reduces the impact rate (equation (13)) in the fluvial framework used. However, qb may increase with both θ and H; for example, if we set qb = qbc then the PSDs increase with both θ and H (dashed lines of Figure 2a).
 So far, expressions have been evaluated and plotted for a single grain size, D. As noted above, the approximate D3 dependence of Pvimplies that larger grain sizes have a disproportionately larger effect on the seismic signal compared to smaller grain sizes. For comparisons with observations, then, it is important to use a grain size distribution that is realistic. Thus, instead of using a typical log-normal distribution of grain sizes, which has an unrealistically long tail at largeD, we introduce a new log-‘raised cosine’ distribution, which has almost the same shape as a log-normal distribution but has a cut-off at both large and smallD (see Figure 3a). The raised cosine distribution is defined by
and (x; μ, s) = 0 otherwise, and has equivalent mean, median, and variance as a normal distribution N(μ, σg2) if we choose s ≡ σg/ (where μ and σg are the mean and standard deviation of the normal distribution). To best fit the data of Attal and Lavé  for the region of interest, we choose D50 = 0.15 m and σg = 0.52 (dimensionless, see auxiliary material). The resulting log-‘raised cosine’ distributionp(D) ≡ (log[D]; log[D50], s)/D as well as the Pv resulting from this grain size distribution are plotted in Figure 3a. For this example, the peak in Pv occurs at D = 0.34 m and corresponds to D94, the 94th percentile grain size. Integrating Pv over all D results in the total PSD, which in this case is PvT = −150.3 dB. While we expect σg to be realistic, in Figure 3b, we show the sensitivity of the dominant grain size to variations in σg. As shown, the dominant grain size is typically far above the median grain size D50 except when σg is unrealistically small (<0.3).
 Finally, we make a preliminary attempt at inverting the observations of Burtin et al.  for the total bed load flux qb, without attempting to calibrate the model, and assuming the average estimated and measured fluvial and seismic parameters as before. As stated previously, this inversion relies heavily on adequate knowledge of grain size distribution, which is lacking, and despite other poorly constrained parameters, this is likely the cause of the largest uncertainty. Given the scaling of Pv with D3qbD ≡ D3pqb, and the dominant D being close to D94, if the grain size distribution changes with flow discharge through the monsoon season then the seismic PSD approximately constrains D943qb rather than qb alone. It is also unknown what fraction of the observed seismic noise may be attributable to water flow noise (or other environmental sources), but a large portion of the seismic signal is likely due to bed load [Burtin et al., 2008]. Assuming the grain size distribution is unchanging (as given above), and assuming the full seismic signal is due to bed load, qb as inverted from the seismic data (see Figure 3c) seems to scale approximately with qbc/5 but has the clear hysteresis inherent in the Burtin et al.  data. The predicted hysteresis in sediment flux as a function of discharge is supported by measurements of Gabet et al. in a neighboring river, and likely occurs as sediment supply from hillslopes is depleted near the end of the wet season. These observations are consistent with a supply-limited river, whereqb < qbc (Figure 3c), resulting in bedrock that is partially exposed [Sklar and Dietrich, 2004] and susceptible to rapid erosion [Lavé and Avouac, 2001]. The calculations presented show the feasibility of such an inversion and we expect the framework described here to be useful to constrain sediment flux from seismic observations at other rivers.
 The authors thank J.-P. Avouac, L. Bollinger, and J. Lavé for helpful comments, and thank J. Johnson and L. Sklar for thoughtful reviews. This research was partially supported by NSF grant EAR0922199 to MPL.
 The Editor thanks Joel Johnson and Leonard Sklar for their assistance in evaluating this paper.