### Modeling

We use numerical models both to strengthen and complement our data analysis,and to perform (numerical) experiments to test ideas and understanding. In a deeper sense, all our equations and laboratory equipment reflect our conceptual models of how systems work. The once-sharp line between experiment and modeling has blurred, and the two are now recognized as complementary tools in our investigation of the natural world.

Equations, spreadsheets, and programs like CXTfit are all models in some sense, but the simulation models we develop and use are models in a more obvious sense. The main environments we use for numerical modeling are spreadsheets, MATLAB, FreeFEM++, and COMSOL. We also use some ・canned" programs such as Hydrus (both 1D and 2D), and use code developed by colleagues (e.g., Flerchinger's SHAW model). Finally, we write some models from scratch, in Pascal, C++, FORTRAN, and other languages.

We use models for both forward problems and inverse problems. In a forward problem, we know the various parameters (say, the relevant soil properties) and we want to predict some outcome (say, how long before a wetting front reaches 30 cm). In an inverse problem, we have some data (say, soil temperatures with depth and time) and we want to estimate the values of the relevant parameters (say, the thermal diffusivity).

**Some modeling that we have performed over the last decade:**

1.Coupled heat and water movement in soil cores with varying boundary conditions. This was largely based on the Nassar and Horton formulation for coupled heat and water movement.

2.Coupled heat and water movement with freezing, largely based on the SHAW model.

3.Coupled heat, water, and chemical movement in both wettable and non-wettable soils.

4.Change in thermal properties as capillary bridges form between osculating spheres.

5.Imbibition into rock cores, using the formal analogy between diffusion and imbibition. This work used random walk methods and showed that the slope change observed in the imbibition curves of some rocks was due to low pore connectivity.

6.Diffusive movement of nonreactive solutes in low-connectivity rock. This is approached both using random walk methods in pore-scale network models, and using finite difference methods informed by percolation theory.

7.Determination of reflection positions in TDR signals with second order BMO

### Second Order BMO

**Determination of Reflection Positions in TDR Signals**

Second order BMO (bounded mean oscillation) analysis is a numerical analysis method for determining reflection positions in TDR signals. Second order BMO is based on the physical assumption that the electromagnetic wave passing through a boundary of materials with different dielectric constants leads to a non-smooth change in the TDR signal. The non-smooth changes, where the second order BMO reaches the local maxima, are used to determine the reflection positions. A stable numerical scheme developed by Wang et al. (2014) is presented. MATLAB (MathWorks Inc.) is required to operate the scheme.

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