Method 1: Direct Sampling 87 

The Direct Sampling (DSm) method investigates the ability of the M1D final parameter distributions to predict the observed 2D data and involves direct use of

random selection of 2000 non-repeating sets was chosen from the M1D distributions and each set was run in the M2D model.

There was, however, one caveat with selection of the 2000 sample sets, which introduced bias into the second half of the selected sets (sets 1001 – 2000). The MCMC algorithm searches for the minimum objective function value given the parameters and, once a minimum is found, tends to stay within that parameter space. Many of the final sets from the M1D model found optimal values of n in M2 (n2) between the lower limit (n2 = 1) and n2 = 1.1 such that >60% of the first 1000 DSm sample sets had n2 values between 1.0 and 1.1 (Figure 4-6). Sample sets were not nearly so constrained with respect to other parameters as they were with n2 and most other parameters ranged through much of the a priori distributions. However, I wanted to avoid undersampling the full range of

n2 and so, for the second 1000 sample sets. I constrained sampling to a priori sets where

n2 > 1.1. Although this limited some other parameters, the second 1000 sets covered nearly the full range of the a priori distributions for all but four paramters: θS2, KS2, α3, and KS3. The most extreme case was KS3 where σfor the n2-constrained set is two orders of magnitude lower than when n2 is unconstrained (Figure 4-6). It is important to note that the distribution of the M1D objective function values (i.e., how well the parameter sets fit the data in the 1D model) was not significantly different between the first and second 1000 parameter sets.

Figure 4-6: M1D parameter distributions of full sets (solid grey) and n2- constrained sets (red outline).

Each of the 2000 DSm parameter sets were run in the M2D model and final distributions of NLLψ, NLLθ, and NLLΔt are shown in Figure 4-7, separated into the first 1000 samples and second 1000 samples. When n2 > 1.1 (n2-constrained), the distributions of NLLψ, NLLθ, and NLLΔt are more narrow and are shifted to lower values (i.e., better fits to the observed data), but cover the same range of NLL when n2 is unconstrained. This implies that constraining n2 is not significantly limiting the model’s ability to fit the observed data while avoiding oversampling parameter space with high NLL values (i.e., poor-fitting parameters). Minimum values and NLLψ, NLLθ, and NLLΔt were 159, 67, and 15.8, respectively and were found in the first, second, and first 1000 sets (minimum value of NLLθ from the second 1000 sets was 68.7). The M2D predicted ψ(t), θ(t), and Δtbndry

from each best-fit data set (i.e., minimum NLLψ, NLLθ, and NLLΔt ) are shown in Figures 4-8 and 4-9. The success of the M2D model at predicting the observed ψ(t) data provides validation of model consistency between the M1D and M2D models.

Figure 4-7: Distributions of NLLψ, NLLθ, and NLLΔt from the direct sampling

Figure 4-8: Optimal fits to ψ(t) and θ(t) from minimum NLLψ, NLLθ, and NLLΔt

parameter sets of the DSm method (note different y-axis for N3 and N4); shaded regions show ±1σ data error.

From the ψ(t) and θ(t) data fits (Figure 4-8), we see that minimum sets from NLLψ and NLLΔt fit the observed ψ(t) and θ(t) data equally well, although neither set accurately predicts twf of θ(t) data below 60 cm and both predict earlier twf than what is observed. The minumum NLLθset accurately predicts θ(t) data and closely predicts ψ(t) data for all sensors except AT4, where it is predicting much later twf than what is observed. The implications of improper fitting of AT4 will be discussed below.

In Figure 4-9, I show the observed and calculated Δtbndry data for all three

the observed Δtbndry data except near the beginning of the GPR line (left side of model). Near the end of the model time (t = 30 hr), after steady-state had been reached,

differences between observed and calculated Δtbndry are 2.2 – 3.3 ns, which is similar to error in manually-calculated Δtbndry above. This is somewhat surprising given that ψ(t) and θ(t) data are closely matched with the same parameter sets, specifically the initial and steady-state θ(t), which are responsible for Δθ and will have the greatest influence over

Δtbndry. I withhold further investigation of Δtbndry fits until the next section where

optimization is focused on, among other things, directly minimizing NLLΔt.

Figure 4-9: Optimal fits to Δtbndry from minimum NLLψ, NLLθ, and NLLΔt

parameter sets of the DSm method; shaded regions show ±1σdata error. From the results of the DSm investigation, I make three conclusions about the ability of the M1D parameter distributions to predict M2D data: 1) constraining n2 > 1.1 does not inhibit optimization of NLLψ or NLLθ (i.e., the model’s ability to predict ψ(t) or

space, where more of the poor-fitting parameter sets are found. 2) Lack of abundant correlation between NLLψ and NLLθ (Figure 4-7) suggests that there may not be an optimal data set that fits all ψ(t) and θ(t) data equally well (especially when considering ψ(t) in AT4). 3) The model-predicted ψ(t) and θ(t) data show that the DSm sets more accurately predict ψ(t) data than θ(t) or Δtbndry data, which is expected given that the DSm sets were sampled from distributions that fit ψ(t) data in the M1D model. In the next section, I attempt to directly minimize NLLθ, NLLΔt, ΣNLL using a multi-start direct search approach.