The supplementary materials contain 7 figures and a discussion on the relationship between residual cross-correlations and those of the innovations.

Conditions under which the cross-correlations between the innovations are asymptotically preserved by the residuals.

Let be the downstream measurement of the th variable at time the -dimensional vector covariates that include the corresponding upstream measurement and its lags, i.e., and that the following stochastic regression models hold (with the intercept and outliers omitted for simplicity):

,(A1)

where is a -dimensional coefficient vector and the regression errors follow an order autoregressive process, i.e.

(A2)

with the ’s being independent and identically distributed random variables of zero mean and positive variance , and they are independent of past ’s and current and past ’s. The ’s are known as the innovations, and may be used as the proxy for the measurement of the th variable due to the processes occurring within the reservoir in the th period. The main object of study consists of, say, regressing on Since the ’s are unobservable, they will be replaced by the residuals, denoted by , obtained from fitting the model defined by (A1) and (A2), separately for each the variables. We shall assume that the unknown parameters can be consistently estimated with the estimation error being of order ) , where is the sample size and the coefficient estimates are denoted by and . For instance, this holds if the model defined by (A1) and (A2) is estimated by the method of conditional least squares, under some mild regularity conditions (Klimko and Nelson, 1978). We show below that the large-sample joint distribution of the regression coefficient estimates using the residuals is identical to that using the latent innovations under the following additional conditions:

(C1) The vector process is a stationary, ergodic process with finite second moments.

We show the validity of the claim by proving below that

.(A3)

The residuals are computed by the following two equations where

,(A4)

(A5)

After some algebra, it can be shown that

Hence, , because, for instance, with

,

as is asymptotically normally distributed by the martingale central limit theorem (Billingsley 2013, Theorem 18.1), and approaches , with increasing sample size , in view of (C1).

Supplementary Figure captions:

Figure S1. Adjusted measurements as residuals of fitted transfer function models.

Figure S1. Model diagnostics of the transfer function model (6) fitted to the square root of N. The top sub-figure is the residual time plot, the middle sub-figure the residual autocorrelation function (ACF) and the bottom sub-figure shows the p-value of Ljung-Box test assessing the whiteness of the residuals based on the first lags of residual ACF, for ranging from 13 to 24. All p-values are greater than 5%, suggesting that the fitted transfer function model provides a good fit to the N data.

Figure S3. Model diagnostics of the transfer function model (6) fitted to pH (P). Same convention as in Supplementary Figure 1. The p-values are slightly higher than 5%, indicating an adequate fit to the data.

Figure S4. Model diagnostics of the transfer function model (6) fitted to total alkalinity (A). Same convention as in Supplementary Figure 1. The Ljung-Box tests suggest the model provides a good fit to the data.

Figure S5. Model diagnostics of the transfer function model (6) fitted to total hardness (H). Same convention as in Supplementary Figure 1. The Ljung-Box tests suggest the model provides a good fit to the data.

Figure S6. Model diagnostics of the transfer function model (6) fitted to the natural log of TSS. Same convention as in Supplementary Figure 1. The Ljung-Box tests suggest the model provides a good fit to the data.

Figure S7. Model diagnostics for the final model (8) fitted with all data.

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