# Model the seismic spectrum due to bedload transport in rivers The function calculates a seismic spectrum as predicted by the model of Tsai et al. (2012) for river bedload transport. The code was written to R by Sophie Lagarde and integrated to the R package 'eseis' by Michael Dietze. ```r model_bedload( gsd, d_s, s_s, r_s, q_s, h_w, w_w, a_w, f = c(1, 100), r_0, f_0, q_0, e_0, v_0, x_0, n_0, n_c, res = 100, adjust = TRUE, eseis = FALSE, ... ) ``` ## Arguments - `gsd`: `data frame` grain-size distribution function. Must be provided as data frame with two variables: grain-size class (in m, first column) and wgt/vol percentage per class (second column). See examples for details. - `d_s`: `Numeric` value, mean sediment grain diameter (m). Alternative to `gsd`. - `s_s`: `Numeric` value, standard deviation of sediment grain diameter (m). Alternative to `gsd`. - `r_s`: `Numeric` value, specific sediment density (kg / m^3) - `q_s`: `Numeric` value, unit sediment flux (m^2 / s) - `h_w`: `Numeric` value, fluid flow depth (m) - `w_w`: `Numeric` value, fluid flow width (m) - `a_w`: `Numeric` value, fluid flow inclination angle (radians) - `f`: `Numeric` vector, frequency range to be modelled. If of length two the argument is interpreted as representing the lower and upper limit and the final length of the frequency vector is set by the argument `res`. If `f` contains more than two values it is interpreted as the actual frequency vector and the value of `res` is ignored. - `r_0`: `Numeric` value, distance of seismic station to source - `f_0`: `Numeric` value, reference frequency (Hz) - `q_0`: `Numeric` value, ground quality factor at `f_0`. "Reasonable value may be `20`" (Tsai et al. 2012). - `e_0`: `Numeric` value, exponent characterizing quality factor increase with frequency (dimensionless). "Reasonable value may be `0`" (Tsai et al. 2012). - `v_0`: `Numeric` value, phase speed of the Rayleigh wave at `f_0` (m/s). Assuming a shear wave velocity of about 2200 m/s, Tsai et al. (2012) yield a value of 1295 m/s for this parameter. - `x_0`: `Numeric` value, exponent of the power law variation of Rayleigh wave velocities with frequency - `n_0`: `Numeric` vector of length two, Greens function displacement amplitude coefficients. Cf. N_ij in eq. 36 in Gimbert et al. (2014) - `n_c`: `Numeric` value, option to include single particle hops coherent in time, causing spectrum modulation due to secondary effects. Omitted is no value is specified, here. Usual values may be between 2 and 4. - `res`: `Numeric` value, output resolution, i.e. length of the spectrum vector. Default is 1000. - `adjust`: `Logical` value, option to adjust PSD for wide grain-size distributions, according to implementation by Tsai et al. (2012). - `eseis`: `Character` value, option to return an eseis object instead of a data frame. Default is `FALSE`. - ``...``: Further arguments passed to the function. ## Returns `eseis` object containing the modelled spectrum. ## Details The model uses a set of predefined constants. These can also be changed by the user, using the `...` argument: * `g = 9.81`, gravitational acceleration (m/s^2) * `r_w = 1000`, fluid specific density (kg/m^3) * `k_s = 3 * d_50`, roughness length (m) * `log_lim = c(0.0001, 100), limits of grain-size distribution functiontemplate (m)` * `log_length = 10000, length of grain-size distribution function template` * `nu = 10^(-6)`, specific density of the fluid (kg/m^3) * `power_d = 3`, grain-size power exponent * `gamma = 0.9`, gamma parameter, after Parker (1990) * `s_c = 0.8`, drag coefficient parameter * `s_p = 3.5`, drag coefficient parameter * `c_1 = 2 / 3`, inter-impact time scaling, after Sklar & Dietrich (2004) When no user defined grain-size distribution function is provided,the function calculates the raised cosine distribution function as defined in Tsai et al. (2012) using the default range and resolution as specified by `log_lim` and `log_length` (see additional arguments list above). These default values are appropriate for mean sediment sizes between 0.001 and 10 m and log standard deivations between 0.05 and 1. When more extreme distributions are to be used, it is necessary to either adjust the arguments `log_lim` and `log_length` or use a user defined distribution function. The adjustment option (implemented with package version 0.6.0) is only relevant for wide grain-size distributions, i.e., `s_s` > 0.2. In such cases, the unadjusted version tends to underestimate seismic power. ## Examples ```r ## calculate spectrum (i.e., fig. 1b in Tsai et al., 2012) p_bedload <- model_bedload(d_s = 0.7, s_s = 0.1, r_s = 2650, q_s = 0.001, h_w = 4, w_w = 50, a_w = 0.005, f = c(0.1, 20), r_0 = 600, f_0 = 1, q_0 = 20, e_0 = 0, v_0 = 1295, x_0 = 0.374, n_0 = 1, res = 100, eseis = TRUE) ## plot spectrum plot_spectrum(data = p_bedload, ylim = c(-170, -110)) ## define empiric grain-size distribution gsd_empiric <- data.frame(d = c(0.70, 0.82, 0.94, 1.06, 1.18, 1.30), p = c(0.02, 0.25, 0.45, 0.23, 0.04, 0.00)) ## calculate spectrum p_bedload <- model_bedload(gsd = gsd_empiric, r_s = 2650, q_s = 0.001, h_w = 4, w_w = 50, a_w = 0.005, f = c(0.1, 20), r_0 = 600, f_0 = 1, q_0 = 20, e_0 = 0, v_0 = 1295, x_0 = 0.374, n_0 = 1, res = 100, eseis = TRUE) ## plot spectrum plot_spectrum(data = p_bedload, ylim = c(-170, -110)) ## define mean and sigma for parametric distribution function d_50 <- 1 sigma <- 0.1 ## define raised cosine distribution function following Tsai et al. (2012) d_1 <- 10^seq(log10(d_50 - 5 * sigma), log10(d_50 + 5 * sigma), length.out = 20) sigma_star <- sigma / sqrt(1 / 3 - 2 / pi^2) p_1 <- (1 / (2 * sigma_star) * (1 + cos(pi * (log(d_1) - log(d_50)) / sigma_star))) / d_1 p_1[log(d_1) - log(d_50) > sigma_star] <- 0 p_1[log(d_1) - log(d_50) < -sigma_star] <- 0 p_1 <- p_1 / sum(p_1) gsd_raised_cos <- data.frame(d = d_1, p = p_1) ``` ## References Tsai, V. C., B. Minchew, M. P. Lamb, and J.-P. Ampuero (2012), A physical model for seismic noise generation from sediment transport in rivers, Geophys. Res. Lett., 39, L02404, doi:10.1029/2011GL050255. ## Author(s) Sophie Lagarde, Michael Dietze

model_bedload function