Theory and equations

Parameter estimation

The inverse problem estimates model parameters by minimizing the differences between observed and simulated values Kool et al. (1987); Hopmans et al. (2002); Šimůnek and Hopmans (2002). The algorithm of HYDRUS 4.17 Šimůnek et al. (2013) was reconstructed from its manual and output; the objective function, the weighting, the statistics and the first iteration reproduce HYDRUS 4.17.

Objective function

Φ(b) = Σ_i=1ⁿ ( W_i [ F_i − F_i(b) ] )², W_i = w_i v_j

with the vector b of the estimated parameters, the observations F_i (types 0–15, see observations), the model values F_i(b), the weight w_i of the observation and the weight v_j of its measurement set j. The measurement sets are the observation types (retention points belong to the water contents). The set weights are:

no internal weighting: v_j = 1;
weighting by mean ratio: v_j = the smallest mean of the absolute values of all sets divided by the mean of set j;
weighting by standard deviation: v_j = 1/(√n_j σ_j) with the standard deviation σ_j of set j.

Prior information on parameters (types 7–11) enters the same sum. The model values are taken at the observation times, which are fixed time levels of the simulation.

Marquardt–Levenberg iteration

Φ is minimized with the Marquardt–Levenberg method Marquardt (1963): from the sensitivity (Jacobian) matrix J of the model values with respect to the parameters, computed with forward differences (1 % of each parameter), each iteration solves

(J^T W J + λ D) Δb = J^T W r

for the change Δb of the parameters, with the residuals r, the diagonal D of J^TWJ and the Marquardt parameter λ (0.02 at the start, divided by 10 at the start of every iteration and multiplied by 10 after a failed step). Parameters are kept within their constraints. The iteration ends when no parameter changes by more than 1 %, after MIT iterations, or when Φ cannot be reduced further.

Rusteau iteration

With adaptive time steps in every run, Φ is not a smooth function of the parameters: a small change of a parameter changes the time steps, and Φ jumps by up to a few per cent. The derivatives then carry large errors, and the HYDRUS iteration can stop far from the minimum at a point that depends on the last bits of the arithmetic. Rusteau therefore (by default)

iterates in stages on frozen time levels: within a stage all runs use the time levels of the run at the start of the stage, so Φ is smooth and the derivatives use increments of 0.1 %; a new stage starts from the end point of the previous one while that lowers Φ (computed with adaptive time steps) by more than 0.1 %;
starts from further starting points (8 by default, in parallel threads), placed deterministically with a Halton sequence Halton (1960) within the constraints (log-uniformly over more than a decade) or within a factor of four of the initial estimates, and keeps the lowest Φ.

The HYDRUS 4.17 iteration (adaptive time steps in every run, one starting point) is available for comparisons. In all twelve inverse examples of HYDRUS-1D the Rusteau iteration ends at the same or a lower Φ than HYDRUS 4.17.

Statistics

From the last iteration Šimůnek and Hopmans (2002):

the covariance matrix C = s² (J^TWJ)⁻¹ with the mean square error s² = Φ/(n − p) (n observations, p parameters);
the standard errors SE_k = √C_kk and the 95 % confidence intervals b_k ± t_{0.975, n−p} SE_k (Student's t);
the correlation matrix a_kl = C_kl/√(C_kkC_ll); |a| near 1 means that the two parameters cannot be estimated independently from the data;
R² of the regression of fitted on observed values;
ME = Σ W r / n, MAE = Σ |W r| / n, RMSE = √(Σ (W r)² / (n − 1));
the information criteria AIC = n ln(Φ₀/n) + 2p Akaike (1974) and BIC = n ln(Φ₀/n) + p ln n Schwarz (1978) with the objective function Φ₀ at the start of the last iteration (as HYDRUS 4.17 prints them).

Good practice

Estimate as few parameters as possible; fix those known from measurements.
Combine different kinds of data (e.g. outflow and retention points, or pressure heads and water contents).
Give physically sensible constraints, especially with several starting points.
Check the correlation matrix and the confidence intervals (results).

Literature cited in this topic

Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control 19(6): 716–723. doi:10.1109/TAC.1974.1100705
Halton, J. H. (1960). On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numerische Mathematik 2: 84–90. doi:10.1007/BF01386213
Hopmans, J. W., J. Šimůnek, N. Romano, and W. Durner (2002). Inverse methods. In: J. H. Dane and G. C. Topp (eds.), Methods of Soil Analysis, Part 4, Physical Methods, SSSA Book Series No. 5, Soil Science Society of America, Madison, WI, pp. 963–1008.
Kool, J. B., J. C. Parker, and M. Th. van Genuchten (1987). Parameter estimation for unsaturated flow and transport models – A review. Journal of Hydrology 91(3–4): 255–293. doi:10.1016/0022-1694(87)90207-1
Marquardt, D. W. (1963). An algorithm for least-squares estimation of nonlinear parameters. Journal of the Society for Industrial and Applied Mathematics 11(2): 431–441. doi:10.1137/0111030
Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics 6(2): 461–464. doi:10.1214/aos/1176344136
Šimůnek, J., and J. W. Hopmans (2002). Parameter optimization and nonlinear fitting. In: J. H. Dane and G. C. Topp (eds.), Methods of Soil Analysis, Part 4, Physical Methods, SSSA Book Series No. 5, Soil Science Society of America, Madison, WI, pp. 139–157.
Šimůnek, J., M. Šejna, H. Saito, M. Sakai, and M. Th. van Genuchten (2013). The HYDRUS-1D Software Package for Simulating the One-Dimensional Movement of Water, Heat, and Multiple Solutes in Variably-Saturated Media, Version 4.17. HYDRUS Software Series 3, Department of Environmental Sciences, University of California Riverside, Riverside, CA, USA, 307 pp. https://www.pc-progress.com/en/Default.aspx?hydrus-1d

All sources: References.