## 3 Methodology
### Prediction Models  
We calibrate and assess five multivariate prediction models, each of which try to accurately predict multibeam WD using SDRD data and seabed-type, as two predictor variables. Preliminary analyses indicated that spatial autocorrelation effects are present and that the relationship between WD and SDRD changes across the study waters. With these initial findings in mind, we choose the following predictors for evaluation: (1) MLR; (2) GWR; (3) KED-GN; (4) KED-LN; and (5) GWRK (_i.e._, kriging with GWR as its trend component). Essentially, MLR assumes stationary (constant) relationships between the response and predictor data, as does KED-GN. On the other hand, GWR, KED-LN and GWRK each model the same relationships, as non-stationary. In addition, KED-GN, KED-LN and GWRK each account for spatial autocorrelation effects, whilst MLR and GWR do not. Thus, MLR is the only non-spatial model of the five; and only KED-LN and GWRK model both non-stationary relationshipsand autocorrelation effects. A summary of these properties is given in Table 2. The five study predictors are now formally described. For a response variable z and predictor variables \(y_{1},y_{2},\cdots,y_{k}\), the MLR model has this form: \(z_{\alpha}=\chi_{0}+\sum_{k=1}^{K}\chi_{k}\,y_{\alpha k}+\tau_{\alpha}\) with sample data denoted by \(\alpha=1,\cdots,n\); and where \(\tau_{\alpha}\) are the residual data. Using ordinary least squares to find the estimator \(\hat{\chi}\), a prediction from MLR at a target location \(\mathbf{x}\) is found from:  
\[\hat{z}_{MLR}\left(\mathbf{x}\right)=\chi_{0}+\sum_{k=1}^{K}\chi_{k}\,y_{k} \left(\mathbf{x}\right) \tag{2}\]  
The GWR model can be defined as \(z_{\alpha}=\chi_{0}(\mathbf{x}_{\alpha})+\sum_{k=1}^{K}\chi_{k}\left(\mathbf{ x}_{\alpha}\right)y_{\alpha k}+\tau_{\alpha}\), where \(\chi_{k}(\mathbf{x}_{\alpha})\) is a realisation of the continuous function \(\chi_{k}(\mathbf{x})\) at sample location \(\alpha\). In particular, a localised MLR is calibrated at any location \(\mathbf{x}\), where observations close to \(\mathbf{x}\) are geographically weighted (GW) according to some kernel function. Thus, GWR parameters are estimated using a weighted least squares (WLS) approach with weights changing according to location. A prediction from GWR at a target location \(\mathbf{x}\) is found from:  
\[\hat{z}_{GWR}\left(\mathbf{x}\right)=\chi_{0}\left(\mathbf{x}\right)+\sum_{k =1}^{K}\chi_{k}\left(\mathbf{x}\right)y_{k}\left(\mathbf{x}\right) \tag{3}\]  
For this study, the weighting matrix in GWR is specified using a bi-square kernel whose optimal bandwidth is found in an adaptive form using leave-one-out cross-validation. Here, the squared prediction error (or cross-validation (CV) score) is calculated for a range of bandwidths and the bandwidth that gives the minimum CV score is considered optimal. All forms of kriging stem from the linear predictor \(\hat{z}(\mathbf{x})-m(\mathbf{x})=\sum_{\alpha=1}^{n}\lambda_{\alpha}\left( \mathbf{x}\right)[z(\mathbf{x}_{\alpha})-m(\mathbf{x}_{\alpha})]\), where \(\lambda_{\alpha}(\mathbf{x})\) is the kriging weight assigned to \(z(\mathbf{x}_{\alpha})\); and where \(m(\mathbf{x})\) and \(m(\mathbf{x}_{\alpha})\) are the means of the random variables \(Z(\mathbf{x})\) and \(Z(\mathbf{x}_{\alpha})\), respectively. The prediction error is also defined as a random variable \(\hat{Z}(\mathbf{x})-Z(\mathbf{x})\), where the variance of this prediction error \(\sigma_{E}^{2}(\mathbf{x})=VAR\{\hat{Z}(\mathbf{x})-Z(\mathbf{x})\}\) is minimized under the constraint that \(\mathrm{E}\{\hat{Z}(\mathbf{x})-Z(\mathbf{x})\}=0\). The weights \(\lambda_{\alpha}(\mathbf{x})\) are found by solving a system of linear equations calibrated with parameters of the covariogram, which is a model of the data's spatial autocorrelation.