La fuente piramidal

In the Quasi-M-closed setting, the focus shifts stronger to predictive Bayesian multi-model frameworks. This requires to account for limited observations with the Bayesian Bootstrap. For the two predictive methods, Pseudo-BMS/BMA and Bayesian Stacking, the bootstrapped results are decisive for the comparison bet- ween the three methods. The frameworks yield vastly different results, as it can clearly be seen in Figure 19.

BMS/BMA assumes to be employed in an M-closed setting and (incorrectly) identifies the zonated model (2a) as DGP. Knowing, that this model is not the exact representation of the actual DGP, we can still take it as the most plausible quasi-true model. This preference for the zonated model is similar to the result in Section 4.3.1, yet the model alternatives are rated differently. In the Quasi- M-closed setting, the simple homogeneous model is essentially discarded despite the conservative tendency of BMS/BMA in model selection. Instead, the more complex pilot-point model (3) remains as plausible candidate and even the most

complex geostatistical model (4) receives higher probability to be the (quasi-)true model than the simplest one. This shows that, despite the zonated model ap- pearing as presumably close resemblance of the true physical system, BMS/BMA struggles to identify it as DGP as clear as in an actual M-closed setting.

Pseudo-BMS/BMA 0 0.5 1 weights BMS/BMA 0 0.5 1 weights BS 0 1 2 3 4 5 6 # PTs 0 0.5 1 weights Homogeneous Zonated (inf.) Pilot-Point Geostatistical Pseudo-BMS/BMA + BB BS + BB 0 1 2 3 4 5 6 # PTs

Figure 19: Expected model weights over growing data size (number of included pumping tests, # PTs) for the Quasi-M-closed setting, i.e., with data observed from the sandbox aquifer. Top row: BMS/BMA; center row: Pseudo-BMS/BMA without (left) and with (right) Bayesian Bootstrapping (BB); bottom row: Bayesian Stacking (BS) without (left) and with (right) BB.

Pseudo-BMS/BMA yields a very different result in Quasi-M-closed when di- rectly compared to BMS/BMA and also in comparison to the result of Pseudo- BMS/BMA in M-closed (cf. Section 4.3.2). The tendency toward models of hig- her complexity is obvious and the simplest model (1) never stands a chance to be preferred. In non-consistent selection, the model that promises largest predictive capability while having a complexity that is still supported by the current amount data is preferred. Hence, even though a human expert might think that the zona- ted model is the closest resemblance of the physical sandbox, Pseudo-BMS/BMA finds this trade-off to be fulfilled best by the pilot-point model (3) over growing data. Over increasing amount of data, the tendency toward more complex models is clearly visible. While for two pumping tests, the zonated model is rated best, yet without strong advance, it loses ground in favor of the more complex alternatives - especially the pilot-point model. The most complex geostatistical model does

not gain weight but (contrarily to the BMS/BMA weights and to the results from M-closed) it also does not loose any. The Bayesian Bootstrap confirms the model weights as being robust over various instances of D from the true data distribution q(y|Mtrue). Hence, the most likely bootstrapped weights are almost equal to the

original Pseudo-BMS/BMA weights.

Bayesian Stacking for predictive model combination behaves completely differently from either selection framework. The conjoint inference of model weights via rating the combined ensemble K is more challenging than rating each model individually and afterwards evaluating the weights as in Pseudo-BMS/BMA: The maximization problem in Equation 20 has several local maxima as solutions for different instan- ces of D and Bayesian Stacking without bootstrapping yields only one. Hence, the BB shows a much larger effect, yielding the global maximum over the true data distribution q(y|Mtrue). The converged wBB are, again, more robust and allow for

a more reliable interpretation. The stabilized model weights distribution over four to six pumping tests supports the assumption of convergence. At each data stage, Bayesian Stacking seeks to find the combined average of models that covers the data best. While for a single pumping test this seems to be accomplished by a combination of the homogeneous and the zonated model, more complex models receive higher weight in the combination with growing data. Over two and three pumping tests, Bayesian Stacking moves toward a stable combination of essentially the pilot-point and the zonated model (with minor contribution of the homoge- neous one). It appears to combine the preferred models of both BMS/BMA and Pseudo-BMS/BMA. While the BB adjustment is not as strong for small data si- zes it has more impact for growing data when it is more difficult to find a global maximum. For four and more pumping tests, the stable model weights allow for averaging in the sense of real model combination. In an illustration like Figure 2, the true DGP can conceptually be located roughly between the pilot-point and zonated model. Physically, this can be interpreted as accounting for the fringes of the sand layers: The zonated model shows too stark contrasts between neig- hbouring zones and the PP shows a too smooth transition. The reality is probably somewhere in between, and the Bayesian Stacking weights reflect this.