Los principios del Sistema de Financiación de las Comunidades Autónomas de

Sample preparation and statistical steps followed the same approach as Vale et al. (2016b). Source samples were dried at 40°C followed by light disaggregation and sieving to retain the < 63 μm fraction. Suspended sediment samples were wet sieved through to 63 μm, and then dried at 40°C. All samples were then weighed into crucibles for XRF analysis, and combusted at 850°C overnight to oxidize all elements and combust any organics. 2 g of the sample was mixed with 6 g of lithium tetraborate and fused into glass discs. Recovery of 2 g of sample was not possible for some of the suspended sediment samples, so in order to make up the necessary quantity, a measured quantity of purified SiO2 was added as required. This allowed the

element sample concentrations to be derived after accounting for dilution. A straight SiO2 disc was also made for analysis to quantify the introduction of any possible contaminations. The glass discs were analysed using a Panalytical Axios 1kW X-ray Fluorescence Spectrometer (XRF) for SiO2, TiO2, Al2O3, Fe2O3, MnO, MgO, CaO, Na2O, K2O, P2O5, Ba, Rb, Sr, Zr, Nb, Y,V, Cr, and Ni concentration. The glass discs were retained and a follow up analysis was conducted using a Agilent 7700 Series Inductively Coupled Plasma-Mass Spectra with an attached Laser Ablation unit (LA-ICP-MS) for Sc, V, Cr, Co, Ni, Cu, Zn, Ga, Rb, Sr,Y, Zr, Nb, Cs, Ba, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu, Hf, Ta, Pb, Th, and U. Where there are both XRF and LA-ICP-MS element concentrations the XRF data was given preference over LA-ICP-MS values because the XRF values provided more consistent results.

11 9 Fig. 36: Sediment s o ur ce Char acter is tics ; (A) Muds tone; ( B ) Hill subs urf a ce; (C ) Hill Surf ac e; (D)

Channel Bank; (E) Mountain Range;

(F ) G ra ve l Te rr ac e; (G ) Lo ess ; (H) Limest one

5.5.3 Statistical Discrimination

In order to select appropriate tracers the geochemical concentrations must show conservative behaviour from source to the sampling point. The bracket test provides a basic test that each tracer falls within the geochemical range of the potential sources. If they do not pass this test (within a 15 % measurement error) they were removed. The measurement error was larger for the flood samples than the samples from Section 4 since the lower sample yield resulted in greater uncertainty. Next, the two-step method after Collins et al. (1998) was employed. This first step relied on the non-parametric Kruskal-Wallis Η test to evaluate statistically significant differences of individual tracers between the selected source groups. The test was carried out on the concentrations of each analysed element to identify the statistically significant discriminants for subsequent analysis. A 95.0% confidence interval (α level of 0.05) was used for the critical p-value.

The second step used Discriminant Function Analysis (DFA) which allows for prediction of group membership based on linear combinations of predictor variables (Eq. 13)

ܦ ൌ ݒ_ଵܺ_ଵ൅ ݒ_ଶܺ_ଶ൅ ݒ_ଷܺ_ଷ൅Ǥ Ǥ Ǥ ݒ_௜ܺ_௜൅ ܽ (Eq. 13) Where D = discriminant function

v = the discriminant coefficient a = a constant

i = the number of predictor variables.

A stepwise minimization of Wilk’s Lambda multivariate DFA was conducted using SPSS (IBM Corp., 2012). Wilk’s Lambda is a measure of the between-group variability to within-group variability whereby minimizing the value reduces the sample grouping. It is guided by ‘F’ values which determine the entry and removal of variables as a measure of the extent to which an individual variable contributes to group prediction. Default values of 3.84 (probability = 0.5) and 2.71 (probability = 0.1) are used for F to enter and F to remove respectively.

5.5.4 Multivariate Mixing Model

The successful elements from the discriminant function analysis were incorporated into a multivariate mixing model to estimate the relative proportions of sediment source contributions from the time-sequence of suspended sediment samples. This was done

following the approach outlined by Collins et al. (1997), whereby equations are created for each element which relate the source proportions to the sediment sink element concentration. The relative proportions of each source group are estimated through minimizing the sum of the residual (objective function) for the element concentrations through least squares. Previously, Vale et al. (2016b) employed several mixing model variations in the Manawatu Catchment, and found that mean estimates of sediment proportions were comparable between the models. However, given that the model was required to estimate for each hourly sample, a less time-dependent model was preferred and the Collins et al. model provided shorter run-times in comparison. The following mixing model after Walling et al.

(1999), Collins et al. (1997), Owens et al. (1999) was selected:

ܴ_௘௦ ൌ σ ൬஼೔ିσ೘ೕసభ௑ೕௌ೔ೕ ஼೔ ൰ ௡ ௜ୀଵ ଶ (Eq. 14)

ܴ௘௦ = the sum of squares of the residuals

݊ = the number elements in the composite fingerprint (e.g. P2O5)

ܥ_௜=the concentration of element (i) in the sediment sink sample

݉= the number of source groups (e.g., mudstone, hill surface etc.)

ܺ_௜ = the relative proportion from source group (j) to the sediment sink sample

ܵҧ_௜௝ = the mean concentration of element (i) from the sample in source group (j). The mean was used in the basic model as there was not a significant difference observed between medians and mean data and thus suggesting minimal affect from outliers.

The model adheres to two constraints that must be satisfied to produce realistic values. The first constrains each source group proportion to being a positive value between 0 and 1, i.e.

Ͳ ൑ ܲ௜ ൑ ͳ (Eq. 15)

The second constrains the sum of all source group contributions to be equal to 1, i.e.

σ௡ ܲ_௜ൌ ͳ

௜ୀଵ (Eq. 16)

The mixing model source estimates are calculated based on the mean and standard deviation of each source group employing a Monte Carlo approach over 5000 replications.

The optimization of the model solution was conducted using the solver extension in Microsoft Excel. The optimization method employed was the ‘Generalized Reduced Gradient (GRG)

Nonlinear’ using the 200 population size multi-start parameter. The multi-start method automatically runs repeated iterations using different random starting values for the decision variables thereby providing a selection of locally optimal solutions of which the best can be selected as a likely globally optimal solution. Auto-scaling was used to negate any significant differences in scales between variables. The Constraint Precision denotes value that cannot be exceeded between the constraint value and reference cell, which was set to 0.0001. The convergence value was set to 0.001 and outlines the limit of relative change which can occur during the last 5 iterations before the model is regarded as converging on the optimal solution.