# Evaporative Characteristic Length The functions `lc` and `lt` compute the **characteristic length** $L_c$ of **stage-I** evaporation from a soil and its ``target'' (expected) value $L_t$, used to constrain estimates of nonlinear parameters of the Van Genuchten-Mualem (VGM) model for water retention curves and hydraulic conductivity functions. utf8 ```r lc(alpha, n, tau, k0, e0, c0 = NULL, c3 = NULL, c4 = NULL) lt(n, tau, e0, c0, c1, c2, c3, c4) ``` ## Arguments - `alpha`: parameter $\alpha$ (inverse air entry pressure) of the VGM model, see `wc_model` and `hc_model`. For consistency with other quantities, the unit of $\alpha$ should be 1/meter [$m^-1$]. - `n`: parameter $n$ (shape parameter) of the VGM model, see `wc_model` and `hc_model`. - `tau`: parameter $\tau$ (tortuosity parameter) of the VGM model, see `hc_model`. - `k0`: saturated hydraulic conductivity $K_0$, see `hc_model`. If `k0` is missing or equal to `NA` in calls of `lc` then `k0` is approximated by the same relation as used for `lt`, see **Details**. For consistency with other quantities, the unit of $K_0$ should be meter/day [$m d^-1$]. - `e0`: a numeric scalar with the **stage-I** rate of evaporation $E_0$ from a soil, see **Details** and `soilhypfitIntro`. For consistency with other quantities, the unit of $E_0$ should be meter/day [$m d^-1$]. - `c0, c1, c2, c3, c4`: numeric constants to approximate the parameter $\alpha$ and the saturated hydraulic conductivity $K_0$ when computing $L_t$, see **Details** and `control_fit_wrc_hcc`. For consistency with other quantities, the following units should be used for the constants: * `c1`: $m^-1$, * `c3`: $m d^-1$. The remaining constants are dimensionless. ## Details The **characteristic length** of **stage-I evaporation** $L_c$ (Lehmann et al., 2008, 2020 ) is defined by $$ L_\mathrm{c}(\boldsymbol{\nu}, K_0, E_0) = \frac{\frac{1}{\alpha \,§ n} \left(\frac{2n-1}{n-1}\right)^\frac{2n-1}{n}}{1 + \frac{E_0}{K_\mathrm{eff}}}L_c(\nu, K_0, E_0) =1 / (\alpha * n) * ((2n-1)/(n-1))^((2n-1)/n) / (1 / ((1 + E_0/K_eff))) $$ where $\boldsymbol{\nu}^\mathrm{T} = (\alpha, n, \tau)$ are the nonlinear parameters of the VGM model, $K_0$ is the saturated hydraulic conductivity, $E_0$ the stage-I evaporation rate and $K_eff = 4 * K_VGM(h_crit; \nu)$ is the effective hydraulic conductivity at the critical pressure $$ h_\mathrm{crit} = \frac{1}{\alpha} \,\left(\frac{n-1}{n}\right)^\frac{1-2n}{n},h_crit = 1/\alpha * ((n-1)/n)^((1-2n)/n), $$ see `hc_model` for the definition of $K_VGM(h; K_0, \nu)$. The quantity $L_t$ is the expected value (``target'') of $L_c$ for given shape ($n$) and tortuosity ($\tau$) parameters. To evaluate $L_t$, the parameters $\alpha$ and $K_0$ are approximated by the following relations $$ \widehat{\alpha} = g_\alpha(n; c_0, c_1, c_2) =c_1 \, \frac{n - c_0}{1 + c_2 \, (n - c_0)},hat\alpha = g_\alpha(n; c_0, c_1, c_2) = c_1 * (n - c_0) / (1 + c_2 * (n - c_0)), $$ $$ \widehat{K}_0 = g_{K_0}(n; c_0, c_3, c_4) =c_3 \, (n - c_0)^{c_4}.hat K_0 = g_{K_0}(n; c_0, c_3, c_4) = c_3 * (n - c_0)^c_4. $$ The default values for $c_0$ to $c_4$ (see argument `approximation_alpha_k0` of `control_fit_wrc_hcc`) were estimated with data on African desert regions of the database **ROSETTA3** (Zhang and Schaap, 2017 ), see Lehmann et al. (2020) for details. ## Returns A numeric scalar with the characteristic evaporative length (`lc`) or its expected value (`lt`). ## References Lehmann, P., Assouline, S., Or, D. (2008) Characteristic lengths affecting evaporative drying of porous media. **Physical Review E**, 77 , 056309, tools:::Rd_expr_doi("10.1103/PhysRevE.77.056309") . Lehmann, P., Bickel, S., Wei, Z., Or, D. (2020) Physical Constraints for Improved Soil Hydraulic Parameter Estimation by Pedotransfer Functions. **Water Resources Research** 56 , e2019WR025963, tools:::Rd_expr_doi("10.1029/2019WR025963") . Zhang, Y. , Schaap, M. G. 2017. Weighted recalibration of the Rosetta pedotransfer model with improved estimates of hydraulic parameter distributions and summary statistics (Rosetta3). **Journal of Hydrology**, 547 , 39-53, tools:::Rd_expr_doi("10.1016/j.jhydrol.2017.01.004") . ## Author(s) Andreas Papritz papritz@retired.ethz.ch . ## See Also `soilhypfitIntro` for a description of the models and a brief summary of the parameter estimation approach; `fit_wrc_hcc` for (constrained) estimation of parameters of models for soil water retention and hydraulic conductivity data; `control_fit_wrc_hcc` for options to control `fit_wrc_hcc`; `soilhypfitmethods` for common S3 methods for class `fit_wrc_hcc`; `vcov` for computing (co-)variances of the estimated nonlinear parameters; `prfloglik_sample` for profile loglikelihood computations; `wc_model` and `hc_model` for currently implemented models for soil water retention curves and hydraulic conductivity functions; ## Examples ```r # use of \donttest{} because execution time exceeds 5 seconds # estimate parameters of 4 samples of the Swiss forest soil dataset # that have water retention (theta, all samples), saturated hydraulic conductivity # (ksat) and optionally unsaturated hydraulic conductivity data # (ku, samples "CH2_4" and "CH3_1") data(swissforestsoils) # select subset of data sfs_subset <- droplevels( subset( swissforestsoils, layer_id %in% c("CH2_3", "CH2_4", "CH2_6", "CH3_1") )) # extract ksat measurements ksat <- sfs_subset[!duplicated(sfs_subset$layer_id), "ksat", drop = FALSE] rownames(ksat) <- levels(sfs_subset$layer_id) colnames(ksat) <- "k0" # define number of cores for parallel computations if(interactive()) ncpu <- parallel::detectCores() - 1L else ncpu <- 1L # unconstrained estimation (global optimisation algorithm NLOPT_GN_MLSL) # k0 fixed at measured ksat values rsfs_uglob <- fit_wrc_hcc( wrc_formula = theta ~ head | layer_id, hcc_formula = ku ~ head | layer_id, data = sfs_subset, param = ksat, fit_param = default_fit_param(k0 = FALSE), control = control_fit_wrc_hcc( settings = "uglobal", pcmp = control_pcmp(ncores = ncpu))) summary(rsfs_uglob) coef(rsfs_uglob, lc = TRUE, gof = TRUE) # constrained estimation by restricting ratio Lc/Lt to [0.5, 2] # (global constrained optimisation algorithm NLOPT_GN_MLSL) # k0 fixed at measured ksat values rsfs_cglob <- update( rsfs_uglob, control = control_fit_wrc_hcc( settings = "cglobal", nloptr = control_nloptr(ranseed = 1), pcmp = control_pcmp(ncores = ncpu))) summary(rsfs_cglob) coef(rsfs_cglob, lc = TRUE, gof = TRUE) # get initial parameter values from rsfs_cglob ini_param <- cbind( coef(rsfs_cglob)[, c("alpha", "n")], ksat ) # constrained estimation by restricting ratio Lc/Lt to [0.5, 2] # (local constrained optimisation algorithm NLOPT_LD_CCSAQ) # k0 fixed at measured ksat values rsfs_cloc <- update( rsfs_uglob, param = ini_param, control = control_fit_wrc_hcc( settings = "clocal", nloptr = control_nloptr(ranseed = 1), pcmp = control_pcmp(ncores = ncpu))) summary(rsfs_cloc) coef(rsfs_cloc, lc = TRUE, gof = TRUE) op <- par(mfrow = c(4, 2)) plot(rsfs_uglob, y = rsfs_cglob) on.exit(par(op)) op <- par(mfrow = c(4, 2)) plot(rsfs_uglob, y = rsfs_cloc) on.exit(par(op)) ```

soilhypfit_evaporative_length function