Subcarpeta C Individual

As previously discussed, the parameter space defined in section 3.3.3 was sampled using a central composite design. Specifically, the design incorporated a 210V 3

−

fractional factorial, which allows strongly clear estimation of all main effects and clear estimation of all two- factor interactions (subsection 3.3.3). To determine the design levels of the parameters to be used in each simulation, and in particular in the fractional factorial part of the design, each parameter was assigned a design number, and these are shown in Table 3.6.

Table 3.6: Design numbers assigned to each parameter, used in forming the design matrix.

Parameter Design number Parameter Design number kw 1 nci 6 mw 2 tci 7 kd 3 scr 8 kf 4 kb 9 τc 5 τb 10

It should be noted that the allocation of design numbers shown above is quite arbitrary, as it is immaterial to layout of the design which parameter is allocated to which number; any of the parameters could therefore have been allocated to any of the design numbers. All that is required is that the design generators are applied correctly and consistently. Thus, if kd is

denoted by design number 3, and scr by design number 8, then these numbers must be

applied, consistently in each row, throughout the preparation of both the design matrix and the planning matrix. The same point holds for the other parameters.

The design generators for fixing the design matrix levels are explained in Wu and Hamada (2000, p199). The generators were applied first to determine the design levels for scr, using:

8 = 3456 3.28 To determine the design levels for kb, the following was used:

9 = 13457 3.29 Finally, the design levels for τb were determined using:

(10) = 12467 3.30 By convention, the design level of factors numbered 10 or higher is placed in brackets.

The full design matrix comprises 1 simulation with all parameters at their central values, 20 simulations using the star points, and the 128 simulations which make up the fractional factorial, giving a total of 149 simulations to undertake the initial sampling of the parameter space. The full design matrix is too long to produce here, and appears in Appendix D, and the planning matrix, which shows the actual values used in the simulations, appears in Appendix E, together with an explanation of how the design levels are translated into actual parameter values. The parameter cases used in the simulations are also numbered, for ease of reference.

For all simulations, the simulation period was chosen arbitrarily as 100,000 years. Although this period is too short to enable the landscape to evolve to an equilibrium state (such

evolution would take perhaps two weeks of computer time for each simulation), it was thought to be sufficiently long to reveal the effects of both the faster processes, such as landsliding and fluvial transport, and the slower processes, such as weathering and slow mass movement.

These points conclude Chapter 3, and step 1 of the methodology as summarised in Figure 3.1. The simulations with GOLEM (step 2) can now be conducted, and this prepares the way for analysis of the results, selection of the metrics, and derivation of the metamodels (steps 2 to 4). These matters are presented in Chapter 4.

CHAPTER 4: MODEL OUTPUT, CHOICE OF METRICS

AND DERIVATION OF THE METAMODELS

4.1 INTRODUCTION

The author’s aim in this chapter is to present the next steps of the methodology, namely the output from the simulations with GOLEM using the central composite design, the choice of metrics, and the derivation of the metamodels (steps 2, 3 and 4 in Figure 3.1, q.v.). The chapter therefore includes detailed presentation of the output from GOLEM obtained from the main experiment, focusing particularly on results from the base case simulation i.e. the parameter case where all of the parameters are at their base case values. Different output metrics are discussed and contrasted, from which four are selected as subjects for the metamodels. An approach to deriving initial metamodels for these metrics is also outlined, covering the type of preliminary model to be fitted, and the requirements to ensure both a satisfactory fit to the data and model parsimony.

Besides the choice of metrics and metamodel derivation, the work needed during this phase of the research also required additional simulations, used in testing the preliminary

metamodels for three of the metrics, and subsequently in deriving the final form of the metamodel for each metric. In this respect, problems encountered in trying to derive a preliminary metamodel for the fourth metric, using the same approach applied in deriving the preliminary metamodels for the other three, led to a revision of the general approach to the regression analysis and the type of model to be fitted. This revision in turn allowed final metamodels for all four metrics to be developed satisfactorily, so the revision of the method is also explained. A particular feature of the revised regression approach was the proposed application of a bootstrap to the metamodel predictions. The principles of the bootstrap and the manner of its inclusion in the overall methodology are therefore also provided in this chapter.

Throughout the chapter, where possible, metrics and regression output data are summarised in plots and tables, supported by appendices with additional information where appropriate. Since output from the warm up and base case simulations is of key importance, the output from these simulations is discussed first and in some detail, beginning with evolution of topography.