Subcarpeta C Compuesta

Figure 4.12 shows the mean and standard deviation of elevations for the base case and all factorial point cases.

a. Mean elevations for each case and time slice; the bold black line marks the base case data.

b. Standard deviation of elevations for each case and time slice, the base case data in black bold. M e an e le vation, m 117.0 117.5 118.0 118.5 119.0 119.5 120.0 120.5 121.0 0 10 20 30 40 50 60 70 80 90 100 Tim e , 000s yrs

Standard de viaton of e le vations , m

44.8 45.0 45.2 45.4 45.6 45.8 46.0 46.2 0 10 20 30 40 50 60 70 80 90 100 Tim e , 000s yrs

Figure 4.12, a and b: Mean and standard deviation of elevations for the base case and all factorial point cases in the central composite design sample.

The plots show that the base case response always lies broadly in the middle of each range of the factorial point results throughout the simulations, the range in the latter increasing over time. It is helpful to recall here that the factorial part of the central composite design comprises the 128 simulations using values corresponding to the +1 and -1 levels for each parameter, listed in Table 3.5, and used in the design level combinations listed in Appendix D, and actual value combinations listed in Appendix E. For both metrics plotted here, the range of results from these combinations at 100,000 years is still quite small, being only about 1.2 m for each. Expressed as percentages of the base case result, the ranges are c.

1.0% for the mean elevation data, and c. 2.6% for the standard deviation data.

Although there is some variation within the results for each of these metrics, they say little about any differences between the modelled landscapes generated by the factorial point and base case samples. In particular, in a geomorphological investigation, it can be easily envisaged that a LEM would be used to simulate the evolution of a landscape from a past time to the present, in the manner presented in Figures 1.2 or 1.3 (but not presuming the equifinality shown in those examples!). In such a situation, the present landscape forms the reference with which the simulations are to be compared. It is appropriate, therefore, to consider how the factorial point simulations would compare with a reference landscape. In

this instance, however, we do not have a real landscape to use as a ‘target’ for the simulations, and in the absence of such, the base case is used as the reference instead. It will be appreciated that the use of a the base case as a common reference provides a constant condition for comparative purposes, as would indeed be true if a real landscape were being considered. Other metrics may therefore be devised which calculate directly a measure of the differences between each factorial point sample result and the base case. In this instance, the first step is to calculate the difference data, by subtracting the elevations of the base case, cell for cell, from those of each example parameter case. The difference data can then be analysed and manipulated to show the differences in various ways. As

examples, Figure 4.13a shows the standard deviations of these differences, and Figure 4.13b the mean of the absolute values of the differences.

a. Standard deviations data, for each time slice. b. Means of absolute differences data, for each time slice.

Std de viation of diffe re nce s in e le vation be tw e e n bas e cas e and all othe rs , m

0.0 0.5 1.0 1.5 2.0 2.5 0 10 20 30 40 50 60 70 80 90 100 Tim e , 000s yrs

M e an abs olute diffe re nce in e le vation be tw e e n bas e cas e and all othe rs , m

0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 10 20 30 40 50 60 70 80 90 100 Tim e , 000s yrs

Figure 4.13, a and b: Standard deviation of differences and mean of absolute differences in elevation between the base case and each factorial point case in the central composite

design sample.

In both plots, the range in the results clearly increases over time. However, it is difficult to assess what the spreads in the data at 100,000 years actually indicate, or how the sensitivity or otherwise of the metrics should be assessed. For example, if expressed as percentages of the mean elevation from the base case, the range in the data is c. 1.5% for the standard deviation of elevation differences at 100,000 years, and c. 0.8% for the mean of absolute differences at the same time, which suggests low sensitivity. These metrics also give no indication of how well the base case and other landscapes differ from each other over space. For example, a big standard deviation of the differences could mean that most of the

contrast, that the differences are large, but confined to a only small areas of the landscape. The mean absolute difference data could be interpreted similarly.

To try to incorporate an indication of differences over space, a different metric is devised, called the percentage topographic difference metric, or more simply the ‘topographic metric’. This is calculated from the elevation difference data of the two landscapes, and requires a simple routine for counting the number of cells where the absolute difference is less than one metre. One metre is selected here as this is comparable with the error in the initialising DEM of the Smith River catchment (subsection 3.5.1 q.v.). Moreover, it is also considered quite demanding, in that a researcher would probably be very satisfied to have been able to generate a modelled landscape after 100,000 years which agrees with a real one to within ±1 m at every point.

Rather than express the total differences as a number of cells, the metric is calculated here as a proportion of the total catchment area, and converted to a percentage. The metric can thus be understood as stating that “x % of the landscape generated by parameter case i lies within ±1 m of the base case …”. This greatly assists interpretation of the differences between the landscapes, as compared with the other elevation-based metrics considered here. The topographic metric results calculated in this way are plotted in Figure 4.14.

Topographic m e tr ic, % (s e e te xt) 70 75 80 85 90 95 100 0 10 20 30 40 50 60 70 80 90 100 Tim e , 000s yrs

Figure 4.14: Results for the topographic difference metric for each factorial case in the central composite design sample.

The figure shows that the spread in the results for the topographic metric increases during the simulations. At 100,000 years, the topographic metric ranges between about 73% and 94%, so some of the landscapes appear to be very similar to the base case, whilst others

differ markedly from it. The higher sensitivity of the topographic metric, compared with the mean elevation and standard deviation of elevations (Figure 4.12), is clearly evident.

In summary, of the elevation metrics considered here, the author decided that the

topographic metric was the probably the most useful, and chose this as a suitable subject for metamodelling. Discussion now turns to drainage density and related metrics.