Forward or inverse use.

Generally, a model runs in the forward mode. That is, we have input data (initial conditions, parameter values, etc.) and the model executes to predict a result, such as an organismal or population growth rate. In other cases, we have a result and we want to determine the causative factors. In this case, the model is used in the inverse mode.

One example has been alluded to another page, the canopy analyzer and my almost concurrent attempt to do the same. Here, we can measure the penetration of light beneath a plant canopy in various directions, and we can infer the leaf area index of the canopy above, a structural parameter.

A more complex example is afforded by models that attempt to estimate the transpiration rate of vegetated land surfaces from satellite measurements of reflected radiation and surface temperature. The reflected radiation, primarily, is used to estimate both the total radiation absorbed by the surface and the amount of vegetation. The surface temperature is used to estimate how the energy was partitioned between generating sensible heat and latent heat (evapotranspiration, ET). Cooler surfaces are partitioning more of the energy to ET. The physics and biology of the situation is complex. One major complication in a simple method, called SEBAL (Surface Energy BAlance Land; Bastiaanssen et al., 1998), is that the satellite measures one average temperature, but there are two major (average or class) temperatures, that of the exposed soil and that of the vegetation. Their variations impy very different things for ET. Nonetheless, the model has some success, particularly if some rather clever corrections are applied; my graduate student, Isabella Mariotto, is finishing this work.

A rather different inverse model is one I developed for estimating the total leaf area on individual plants, from digital images that show the leaves and the gaps between them. (The gap distribution is the result we want to invert to leaf area estimates.) The very simplest formulation is that the gap fraction is a simple exponential function of leaf area index, LAI, in various sections of the image. That, plus the physical area of these sections, will let us estimate the total area. The reality is hugely complex. Identifying what is leaf and what is not leaf by color can be extremely hard, given that leaves vary in color, some pixels are mixed leaf and non-leaf, and the soil below can be receiving green-tinged light scattered off leaves. Also, the color balance of the incident sunlight varies with time of day and season, or the camera may have biases in recording colors. Plant images may overlap from poorly chosen view angles. There are solutions, which required much work. Another problem is that the simple gap models assume that leaves are very randomly distributed in space, whereas real leaves are clumped, and on several spatial scales. Again, there are several routes to infer this clumping. Various other problems are addressed in the final model, which is rather accurate and much less work than destructive harvesting. It also allows repeated measures on any plant.

References

W. G. Bastiaanssen,. M., M. Menenti, R. A. Feddes, and A. A. M. Holtslag. 1998. A remote sensing surface energy balance algorithm for land (SEBAL). 1. Formulation. J. Hydrol. 212/213: 198-212.

J. M. Welles and J. M. Norman. 1991. Instrument for indirect measurement of canopy architecture. Agron. J. 83: 818-825.

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