# Characterization of Export Regimes in Concentration–Discharge Plots via an Advanced Time-Series Model and Event-Based Sampling Strategies

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Regime and Memory Model (RMM)

_{k}and variance ${v}_{k}$. Furthermore, we assume that k

_{t}follows (over time) an auto-regressive model of order one (AR(1), [46]) in order to account for the dominant characteristic time scale of release and chemical turnover in the catchment. Thus, the AR(1) model is commensurate with the source strength and includes effects such as hysteresis or event-to-event variations of source availability.

_{ch}, and the mean and variance of k, respectively, m

_{k}and ${v}_{k}$. The characteristic time is defined as

_{k}, ${v}_{k}$ are the well-known moment relations for the lognormal distribution [48]:

^{3}] or [M/T] rather than for $Y=\mathrm{log}\left(k\right)$. Therefore, with only four parameters (α, m

_{k}, v

_{k}, and t

_{ch}), we can represent concentrations versus time.

#### 2.2. Bayesian Parameter Inference of Full Time-Series Data

_{ch}, m

_{k}, v

_{k}, and by using a given time series for Q

_{t}. Thus, the next step is to fit the model to the observed data of concentration C

_{t}. Due to the stochastic randomness implied for C

_{t}by k

_{t}(explicit by the white noise ε

_{t}in Equation (2)), this is achieved by choosing parameter values such that the resulting model is most likely to generate good-fitting random C

_{t}time series. This parameter choice is subject to a list of plausibility arguments for admissible parameter values.

_{ch}, m

_{k}, v

_{k}) and

**y**denotes the measurement data (

**Q**,

**C**). The prior $p\left(\mathit{\theta}\right)$ represents our belief (or knowledge) about the parameters before seeing any data and has to be chosen by us (Table 1). The first term on the right-hand side, $p\left(\mathit{y}|\mathit{\theta}\right)$, is the likelihood of observing the data for the given parameters. The term in the denominator, p(

**y**), is independent of the parameters $\mathit{\theta}$, and therefore does not need to be computed. The result is the posterior distribution $p\left(\mathit{\theta}|\mathit{y}\right)$, which expresses our updated (calibrated) belief about the parameters after seeing the data

**y**.

#### 2.2.1. Prior Distributions

_{ch}, we assumed that concentrations in the river present roughly weekly cycles (e.g., due to weekly cycles in the release of wastewater treatment plants, which are in turn triggered by cycles in known activities) with a mean equal to four days (m

_{tch}= 95.7 h) and a standard deviation of three days (s

_{tch}= 71.7 h). Thus, the 95% confidence interval was about [1 d, 14 d]. Because t

_{ch}must be positive, we chose once again a lognormal distribution. The corresponding parameters of the lognormal distribution LN(μ, σ) followed from respective copies of Equations (6) and (7), resulting in μ

_{tch}= 4.35 and s

_{tch}= 0.65.

_{k}, and variance v

_{k}of the apparent release rate k, so we chose uniform distributions. We will discuss the mean m

_{k}first. For α = 0, it corresponds to the long-term mean of nitrate concentrations. For an average discharge rate of 0.87 m

^{3}/s (Section 3), if we rounded this to ≈1 m

^{3}/s, then concentration C = k (Equation (1)) for both extreme values of α. Therefore, we can interchange C and k in the following discussion.

_{k}) within the river were between pristine water (≈1 mg/L) and the effective average of the undiluted, agriculturally shaped groundwater in the Ammer catchment (≈40 mg/L, as observed as integral catchment output during baseflow by Schwientek et al. [20]). Finally, we will discuss the standard deviation $\sqrt{{v}_{k}}$ of k. For α = 0, it controls the amplitude of fluctuations in nitrate concentrations. We assumed a uniform distribution between 1 and 6.32, because we could expect both an almost constant k and a dynamic k.

#### 2.2.2. Likelihood

**Y**

_{t}and

**Y**

_{t}

_{−l}, where l is the lag distance along time between two observations. For example, the data we used comprehended 6604 hourly samples. Between the third and the first samples, the lag time was 2. Therefore, if the whole time series is taken into account, then the lag time between two observations is l = 1. Thinning out to half the sampling rate corresponds to l = 2. Irregular sampling triggers individual lag values between consecutive data values.

_{ch}, m

_{k}, and v

_{k}) were currently proposed trial values. We then wanted to compute the likelihood for data from the time series

**y**= (

**Q**,

**C**) for the trial values $\mathit{\theta}$. Note that

**y**can include the whole high-frequency series of data or only a short list of observations at various lag times.

**Q**and

**C**into a time series of k

_{t}. Then, by applying the log, we obtained a time series of Y

_{t}. Next, we calculated $\lambda =1/{t}_{ch}$ (Equation (5)), and then c and ${\sigma}_{\epsilon}$ from m

_{Y}and v

_{Y}. We did so by first solving Equations (6) and (7), and then Equations (3) and (4). Before computing the depreciated increment between two observations, we wanted to get a

**Y**

_{0}zero-mean AR(1); therefore, we removed the mean m

_{Y}(Equation (3)) from

**Y**:

**X**

_{l}can be computed:

**X**

_{l}are mutually independent, as they are sums of non-overlapping segments from the white-noise series ε

_{t}. The variance and corresponding standard deviation of the depreciated increment are, respectively:

**X**

_{l}, which in turn is given by the multivariate normal distribution:

_{t}is the total number of samples of the considered series (here n

_{t}= 6604 when using l = 1). In Equation (13), we can use the simple product rule due to independence within the time series of

**X**. Finally, the likelihood term $\mathrm{p}\left({\mathit{X}}_{\mathit{l}}|\mathit{\theta}\right)$ in Equation (13) is given by

_{l}**X**implied by the chosen priors and the above equations.

_{l}#### 2.2.3. Sampling from the Posterior

_{l}, and $p\left(\alpha ,{\mathit{\theta}}_{-\alpha},{X}_{l}\right)$ is the prior of the parameters and X

_{l}. For simpler discussion, we reduced the sample to a point-estimate by computing the posterior mean from the posterior samples:

_{full}from the full data set, and values ${\alpha}_{post,d}$ for various sampling designs d.

#### 2.3. Retrospective Optimal Design of Experiments (ODE)

**D**is a set of considered design strategies, d is any considered design, d

_{opt}is the best design among

**D**, and ${\varnothing}_{\alpha}\left(d\right)$ is the squared error in estimating α for design d (as ${\alpha}_{post,d}$) when compared to α

_{full}:

_{full}(α inferred with full time-series data). We also assessed whether ${\alpha}_{post,d}$ had little remaining uncertainty as expressed by intervals in its distribution $\mathrm{p}\left(\alpha |{\mathit{y}}_{d}\right)$ according to Equation (15).

^{3}/s (which corresponds to a value 25% below the mean discharge of the whole time series) and a high discharge rate to be Q > 2.5 m

^{3}/s.

## 3. Catchment and Data

^{2}. The geology is dominated by limestone of the Middle Triassic (“Oberer Muschelkalk”) and gypsum-bearing mudstone of the Upper Triassic (“Gipskeuper”). Both formations, particularly the limestone, are strongly karstified, and the Ammer River is primarily fed by karst springs, most of them being situated close to the main stem river [52]. In line with the large storage capacity of the karst system, the permeable rocks of the catchment lead to a dampened hydrologic variability with a very strong and steady baseflow. Mean annual low flow (0.44 m

^{3}/s) is as high as 50% of the long-term average flow (0.87 m

^{3}/s) (retrievable from https://www.hvz.baden-wuerttemberg.de/, accessed on 14 May 2020). Nevertheless, pronounced and sharp flood peaks typically occur during the summer season. They are caused by surface runoff generation on impervious urban areas in the upper catchment [20,53]. Usually, due to steep recessions, baseflow is attained within a few hours after the flood peaks. In total, 17% of the catchment area is urbanized and 71% is occupied by agriculture, 66% of which is used for arable land [20].

^{3}/s) was similar to the long-term average (0.87 m

^{3}/s). The maximum Q (27.5 m

^{3}/s) represents an exceptional flood with a return period of about 20 years. The average C was 31.5 mg/L, but a range between 9.5 and 44.7 mg/L was captured.

^{3}/s ($lo{g}_{10}Q$ = −0.30–0.00) during the recorded time series. During these baseflow periods, flow is predominantly fed from the large karst storage and, at rather constant rates, from wastewater treatment plant effluents, and thus exhibits stable nitrate concentrations around an average of 31.5 mg/L ($lo{g}_{10}C$ = 1.50). Most floods occur in the summer period and are the result of convective precipitation events. They typically have very short durations with steep recessions and cause a dilution of the nitrate concentrations. This may be explained by a fast runoff component dominated by a heavy precipitation event on sealed urban surfaces that carry low nitrate concentrations.

## 4. Results

#### 4.1. RMM Simulations

_{ch}), while the other three parameters are inferred by Bayesian inference using the full-time series data. Table 3 includes the inferred hyper-parameters used to generate the concentration time series shown in Figure 2.

^{®}Core™ i7-7700 CP @3.60 GHz, 32 GB of RAM), which makes it a very affordable and efficient model.

#### 4.2. Characterization with Extensive Time-Series Data

#### 4.3. Characterization with Sparse Data (Sampling Strategies)

## 5. Discussion

#### 5.1. Comparison of the RMM to the Linear Regression Approach

#### 5.2. Performance of Sampling Strategies

#### 5.3. Comparison to Other Studies

_{3}, K, Cl) depending on when samples were observed. The C–Q relationship of these solutes was differently impacted when discharge changes were caused by individual peak events or by longer time scales. These studies observed opposite export regimes based on when measurements were observed. Contrarily, the RMM shows robustness (Table 5) since it does not result in “opposite” export regimes depending on when samples are taken.

## 6. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Residuals in the C–Q Regression of the Ammer Catchment

**Figure A1.**Residuals along time (top), moving average of residuals (middle), moving variance of residuals (bottom). The window size is 24 samples (equivalent to one day). Note that the moving average is not always equal to zero, and the moving variance is not constant.

## Appendix B. Sampling Strategies

#### Appendix B.1. Time Frequency Sampling Strategies

d | Q (m^{3}/s) | C (mg/L) | t (d) | l |

1 | 1.29 | 30.44 | 55.00 | 1321 |

0.69 | 34.09 | 110.00 | 2641 | |

0.62 | 26.78 | 165.00 | 3961 | |

1.23 | 34.09 | 220.00 | 5281 | |

0.81 | 35.20 | 275.00 | 6601 | |

2 | 1.50 | 24.97 | 27.50 | 661 |

1.29 | 30.44 | 55.00 | 1321 | |

0.62 | 35.93 | 82.50 | 1981 | |

0.69 | 34.09 | 110.00 | 2641 | |

0.42 | 33.30 | 137.50 | 3301 | |

0.62 | 26.78 | 165.00 | 3961 | |

1.41 | 29.97 | 192.50 | 4621 | |

1.23 | 34.09 | 220.00 | 5281 | |

0.89 | 34.00 | 247.50 | 5941 | |

0.81 | 35.20 | 275.00 | 6601 | |

3 | 1.00 | 20.91 | 13.75 | 331 |

1.50 | 24.97 | 27.50 | 661 | |

0.76 | 24.84 | 41.25 | 991 | |

1.29 | 30.44 | 55.00 | 1321 | |

0.92 | 31.03 | 68.75 | 1651 | |

0.62 | 35.93 | 82.50 | 1981 | |

0.62 | 35.74 | 96.25 | 2311 | |

0.69 | 34.09 | 110.00 | 2641 | |

0.69 | 33.18 | 123.75 | 2971 | |

0.42 | 33.30 | 137.50 | 3301 | |

0.36 | 32.66 | 151.25 | 3631 | |

0.62 | 26.78 | 165.00 | 3961 | |

0.85 | 23.03 | 178.75 | 4291 | |

1.41 | 29.97 | 192.50 | 4621 | |

1.70 | 30.78 | 206.25 | 4951 | |

1.23 | 34.09 | 220.00 | 5281 | |

1.14 | 29.66 | 233.75 | 5611 | |

0.89 | 34.00 | 247.50 | 5941 | |

0.89 | 33.50 | 261.25 | 6271 | |

0.81 | 35.20 | 275.00 | 6601 | |

4 | 6.68 | 24.59 | 6.87 | 166 |

1.00 | 20.91 | 13.75 | 331 | |

0.70 | 24.74 | 20.62 | 496 | |

1.50 | 24.97 | 27.50 | 661 | |

1.51 | 30.79 | 34.37 | 826 | |

0.76 | 24.84 | 41.25 | 991 | |

0.67 | 27.09 | 48.12 | 1156 | |

1.29 | 30.44 | 55.00 | 1321 | |

0.57 | 31.94 | 61.87 | 1486 | |

0.92 | 31.03 | 68.75 | 1651 | |

0.68 | 35.42 | 75.62 | 1816 | |

0.62 | 35.93 | 82.50 | 1981 | |

0.55 | 35.77 | 89.37 | 2146 | |

0.62 | 35.74 | 96.25 | 2311 | |

0.62 | 30.36 | 103.12 | 2476 | |

0.69 | 34.09 | 110.00 | 2641 | |

0.62 | 31.25 | 116.87 | 2806 | |

0.69 | 33.18 | 123.75 | 2971 | |

0.69 | 36.21 | 130.62 | 3136 | |

0.42 | 33.30 | 137.50 | 3301 | |

0.42 | 33.14 | 144.37 | 3466 | |

0.36 | 32.66 | 151.25 | 3631 | |

0.62 | 22.75 | 158.12 | 3796 | |

0.62 | 26.78 | 165.00 | 3961 | |

0.77 | 5.84 | 171.87 | 4126 | |

0.85 | 23.03 | 178.75 | 4291 | |

1.14 | 27.59 | 185.62 | 4456 | |

1.41 | 29.97 | 192.50 | 4621 | |

1.14 | 33.77 | 199.37 | 4786 | |

1.70 | 30.78 | 206.25 | 4951 | |

1.41 | 35.94 | 213.12 | 5116 | |

1.23 | 34.09 | 220.00 | 5281 | |

1.05 | 34.45 | 226.87 | 5446 | |

1.14 | 29.66 | 233.75 | 5611 | |

0.97 | 32.40 | 240.63 | 5776 | |

0.89 | 34.00 | 247.50 | 5941 | |

0.97 | 34.10 | 254.38 | 6106 | |

0.89 | 33.50 | 261.25 | 6271 | |

0.81 | 34.10 | 268.13 | 6436 | |

0.81 | 35.20 | 275.00 | 6601 |

#### Appendix B.2. River Discharge Frequency Sampling Strategies:

^{3}) for this period. Then, we evenly distributed 5, 10, 20, and 40 samples based on the river discharge.

d | Q (m^{3}/s) | C (mg/L) | t (d) | l |

5 | 4.08 | 28.55 | 44.92 | 1079 |

0.55 | 33.66 | 115.46 | 2772 | |

1.21 | 28.43 | 183.83 | 4413 | |

1.14 | 34.87 | 221.50 | 5317 | |

0.81 | 34.40 | 275.13 | 6604 | |

6 | 1.36 | 24.16 | 27.54 | 662 |

4.08 | 28.55 | 44.92 | 1079 | |

0.62 | 36.60 | 78.71 | 1890 | |

0.55 | 33.66 | 115.46 | 2772 | |

0.48 | 29.58 | 161.50 | 3877 | |

1.21 | 28.43 | 183.83 | 4413 | |

1.90 | 23.95 | 203.50 | 4885 | |

1.14 | 34.87 | 221.50 | 5317 | |

0.89 | 34.90 | 246.54 | 5918 | |

0.81 | 34.40 | 275.13 | 6604 | |

7 | 1.58 | 20.00 | 13.54 | 326 |

1.36 | 24.16 | 27.54 | 662 | |

10.82 | 12.53 | 36.08 | 867 | |

4.08 | 28.55 | 44.92 | 1079 | |

0.54 | 27.05 | 58.42 | 1403 | |

0.62 | 36.60 | 78.71 | 1890 | |

1.21 | 24.39 | 98.67 | 2369 | |

0.55 | 33.66 | 115.46 | 2772 | |

0.55 | 31.46 | 138.17 | 3317 | |

0.48 | 29.58 | 161.50 | 3877 | |

1.30 | 8.38 | 170.37 | 4090 | |

1.21 | 28.43 | 183.83 | 4413 | |

1.41 | 32.88 | 193.12 | 4636 | |

1.90 | 23.95 | 203.50 | 4885 | |

1.41 | 36.06 | 211.54 | 5078 | |

1.14 | 34.87 | 221.50 | 5317 | |

1.14 | 30.98 | 233.62 | 5608 | |

0.89 | 34.90 | 246.54 | 5918 | |

0.89 | 34.00 | 260.13 | 6244 | |

0.81 | 34.40 | 275.13 | 6604 | |

8 | 1.91 | 23.66 | 7.37 | 178 |

1.58 | 20.00 | 13.54 | 326 | |

0.71 | 24.35 | 21.12 | 508 | |

1.36 | 24.16 | 27.54 | 662 | |

0.95 | 31.89 | 34.67 | 833 | |

10.82 | 12.53 | 36.08 | 867 | |

1.98 | 19.82 | 38.04 | 914 | |

4.08 | 28.55 | 44.92 | 1079 | |

0.96 | 30.94 | 49.58 | 1191 | |

0.54 | 27.05 | 58.42 | 1403 | |

0.55 | 29.70 | 68.25 | 1639 | |

0.62 | 36.60 | 78.71 | 1890 | |

0.55 | 35.13 | 88.92 | 2135 | |

1.21 | 24.39 | 98.67 | 2369 | |

0.69 | 30.93 | 107.00 | 2569 | |

0.55 | 33.66 | 115.46 | 2772 | |

0.62 | 33.32 | 126.62 | 3040 | |

0.55 | 31.46 | 138.17 | 3317 | |

0.42 | 32.32 | 153.33 | 3681 | |

0.48 | 29.58 | 161.50 | 3877 | |

7.73 | 14.64 | 168.58 | 4047 | |

1.30 | 8.38 | 170.37 | 4090 | |

1.21 | 17.76 | 176.62 | 4240 | |

1.21 | 28.43 | 183.83 | 4413 | |

3.06 | 24.54 | 188.83 | 4533 | |

1.41 | 32.88 | 193.12 | 4636 | |

1.14 | 33.03 | 198.08 | 4755 | |

1.90 | 23.95 | 203.50 | 4885 | |

2.01 | 28.50 | 207.33 | 4977 | |

1.41 | 36.06 | 211.54 | 5078 | |

1.23 | 35.48 | 216.17 | 5189 | |

1.14 | 34.87 | 221.50 | 5317 | |

0.97 | 35.97 | 227.79 | 5468 | |

1.14 | 30.98 | 233.62 | 5608 | |

1.05 | 32.20 | 240.08 | 5763 | |

0.89 | 34.90 | 246.54 | 5918 | |

0.97 | 32.70 | 253.13 | 6076 | |

0.89 | 34.00 | 260.13 | 6244 | |

0.81 | 34.90 | 267.50 | 6421 | |

0.81 | 34.40 | 275.13 | 6604 |

#### Appendix B.3. Low Q Sampling Strategies

^{3}/s (which corresponds to a value 25% below the mean discharge of the whole time series). Then, we chose different low events (uncorrelated samples) equidistantly.

d | Q (m^{3}/s) | C (mg/L) | t (d) | l |

9 | 0.47 | 29.29 | 44.25 | 1063 |

0.69 | 32.35 | 79.46 | 1908 | |

0.62 | 34.46 | 109.46 | 2628 | |

0.55 | 34.09 | 136.67 | 3281 | |

0.66 | 32.90 | 261.38 | 6274 | |

10 | 0.51 | 25.48 | 17.96 | 432 |

0.52 | 29.60 | 44.21 | 1062 | |

0.65 | 30.33 | 64.37 | 1546 | |

0.62 | 34.91 | 79.12 | 1900 | |

0.62 | 34.26 | 92.79 | 2228 | |

0.55 | 33.75 | 109.33 | 2625 | |

0.69 | 33.18 | 123.75 | 2971 | |

0.48 | 33.10 | 136.50 | 3277 | |

0.36 | 34.43 | 149.17 | 3581 | |

0.69 | 5.78 | 171.92 | 4127 | |

11 | 0.46 | 26.78 | 8.29 | 200 |

0.51 | 25.48 | 17.96 | 432 | |

0.47 | 24.97 | 31.33 | 753 | |

0.52 | 29.60 | 44.21 | 1062 | |

0.56 | 34.22 | 54.08 | 1299 | |

0.65 | 30.33 | 64.37 | 1546 | |

0.52 | 36.25 | 72.50 | 1741 | |

0.62 | 34.91 | 79.12 | 1900 | |

0.55 | 36.45 | 86.46 | 2076 | |

0.62 | 34.26 | 92.79 | 2228 | |

0.62 | 27.33 | 102.37 | 2458 | |

0.55 | 33.75 | 109.33 | 2625 | |

0.55 | 33.19 | 117.42 | 2819 | |

0.69 | 33.18 | 123.75 | 2971 | |

0.55 | 34.16 | 130.17 | 3125 | |

0.48 | 33.10 | 136.50 | 3277 | |

0.55 | 32.95 | 142.83 | 3429 | |

0.36 | 34.43 | 149.17 | 3581 | |

0.30 | 33.35 | 155.58 | 3735 | |

0.69 | 5.78 | 171.92 | 4127 | |

12 | 0.52 | 32.26 | 3.25 | 79 |

0.46 | 26.78 | 8.29 | 200 | |

0.54 | 33.24 | 11.83 | 285 | |

0.51 | 25.48 | 17.96 | 432 | |

0.60 | 27.06 | 24.92 | 599 | |

0.47 | 24.97 | 31.33 | 753 | |

0.67 | 28.10 | 35.50 | 853 | |

0.52 | 29.60 | 44.21 | 1062 | |

0.64 | 28.27 | 50.42 | 1211 | |

0.56 | 34.22 | 54.08 | 1299 | |

0.56 | 35.21 | 60.62 | 1456 | |

0.65 | 30.33 | 64.37 | 1546 | |

0.56 | 39.31 | 69.17 | 1661 | |

0.52 | 36.25 | 72.50 | 1741 | |

0.68 | 34.43 | 75.75 | 1819 | |

0.62 | 34.91 | 79.12 | 1900 | |

0.55 | 33.32 | 83.29 | 2000 | |

0.55 | 36.45 | 86.46 | 2076 | |

0.55 | 37.39 | 89.62 | 2152 | |

0.62 | 34.26 | 92.79 | 2228 | |

0.62 | 34.89 | 95.96 | 2304 | |

0.62 | 27.33 | 102.37 | 2458 | |

0.62 | 30.44 | 106.08 | 2547 | |

0.55 | 33.75 | 109.33 | 2625 | |

0.62 | 29.91 | 114.17 | 2741 | |

0.55 | 33.19 | 117.42 | 2819 | |

0.62 | 34.02 | 120.58 | 2895 | |

0.69 | 33.18 | 123.75 | 2971 | |

0.62 | 30.54 | 126.92 | 3047 | |

0.55 | 34.16 | 130.17 | 3125 | |

0.48 | 34.31 | 133.33 | 3201 | |

0.48 | 33.10 | 136.50 | 3277 | |

0.55 | 34.84 | 139.67 | 3353 | |

0.55 | 32.95 | 142.83 | 3429 | |

0.42 | 31.52 | 146.00 | 3505 | |

0.36 | 34.43 | 149.17 | 3581 | |

0.30 | 31.59 | 152.33 | 3657 | |

0.30 | 33.35 | 155.58 | 3735 | |

0.62 | 25.06 | 163.83 | 3933 | |

0.69 | 5.78 | 171.92 | 4127 |

#### Appendix B.4. High Q Sampling Strategies

^{3}/s. Then, for one peak event, we chose equidistant samples (correlated samples). When no more observations were available of one peak, we proceeded to obtain samples of a second peak event, and so on.

d | Q (m^{3}/s) | C (mg/L) | t (d) | l |

13 | 3.12 | 27.72 | 35.79 | 860 |

17.23 | 27.55 | 35.83 | 861 | |

23.31 | 23.36 | 35.87 | 862 | |

27.54 | 17.22 | 35.92 | 863 | |

25.90 | 14.02 | 35.96 | 864 | |

14 | 3.12 | 27.72 | 35.79 | 860 |

17.23 | 27.55 | 35.83 | 861 | |

23.31 | 23.36 | 35.87 | 862 | |

27.54 | 17.22 | 35.92 | 863 | |

25.90 | 14.02 | 35.96 | 864 | |

21.23 | 12.01 | 36.00 | 865 | |

15.22 | 12.48 | 36.04 | 866 | |

10.82 | 12.53 | 36.08 | 867 | |

8.43 | 14.31 | 36.12 | 868 | |

6.66 | 15.32 | 36.17 | 869 | |

15 | 3.12 | 27.72 | 35.79 | 860 |

17.23 | 27.55 | 35.83 | 861 | |

23.31 | 23.36 | 35.87 | 862 | |

27.54 | 17.22 | 35.92 | 863 | |

25.90 | 14.02 | 35.96 | 864 | |

21.23 | 12.01 | 36.00 | 865 | |

15.22 | 12.48 | 36.04 | 866 | |

10.82 | 12.53 | 36.08 | 867 | |

8.43 | 14.31 | 36.12 | 868 | |

6.66 | 15.32 | 36.17 | 869 | |

5.40 | 16.12 | 36.21 | 870 | |

4.27 | 16.98 | 36.25 | 871 | |

3.50 | 16.80 | 36.29 | 872 | |

2.73 | 22.40 | 168.42 | 4043 | |

5.21 | 20.37 | 168.46 | 4044 | |

7.50 | 18.87 | 168.50 | 4045 | |

7.73 | 16.18 | 168.54 | 4046 | |

7.73 | 14.64 | 168.58 | 4047 | |

8.93 | 11.02 | 168.62 | 4048 | |

9.18 | 9.49 | 168.67 | 4049 | |

16 | 3.12 | 27.72 | 35.79 | 860 |

17.23 | 27.55 | 35.83 | 861 | |

23.31 | 23.36 | 35.87 | 862 | |

27.54 | 17.22 | 35.92 | 863 | |

25.90 | 14.02 | 35.96 | 864 | |

21.23 | 12.01 | 36.00 | 865 | |

15.22 | 12.48 | 36.04 | 866 | |

10.82 | 12.53 | 36.08 | 867 | |

8.43 | 14.31 | 36.12 | 868 | |

6.66 | 15.32 | 36.17 | 869 | |

5.40 | 16.12 | 36.21 | 870 | |

4.27 | 16.98 | 36.25 | 871 | |

3.50 | 16.80 | 36.29 | 872 | |

2.73 | 22.40 | 168.42 | 4043 | |

5.21 | 20.37 | 168.46 | 4044 | |

7.50 | 18.87 | 168.50 | 4045 | |

7.73 | 16.18 | 168.54 | 4046 | |

7.73 | 14.64 | 168.58 | 4047 | |

8.93 | 11.02 | 168.62 | 4048 | |

9.18 | 9.49 | 168.67 | 4049 | |

10.74 | 7.84 | 168.71 | 4050 | |

12.14 | 5.65 | 168.75 | 4051 | |

10.74 | 7.26 | 168.79 | 4052 | |

11.57 | 4.24 | 168.83 | 4053 | |

11.57 | 3.30 | 168.87 | 4054 | |

8.20 | 2.12 | 168.92 | 4055 | |

6.41 | 1.74 | 168.96 | 4056 | |

4.83 | 1.62 | 169.00 | 4057 | |

3.17 | 1.51 | 169.04 | 4058 | |

2.60 | 1.47 | 169.08 | 4059 | |

3.57 | 38.64 | 6.71 | 162 | |

6.29 | 42.54 | 6.75 | 163 | |

7.03 | 31.31 | 6.79 | 164 | |

7.82 | 22.64 | 6.83 | 165 | |

6.68 | 24.59 | 6.87 | 166 | |

5.29 | 21.10 | 6.92 | 167 | |

4.15 | 21.33 | 6.96 | 168 | |

3.07 | 22.49 | 7.00 | 169 | |

2.99 | 23.01 | 7.04 | 170 | |

2.75 | 25.78 | 7.08 | 171 |

#### Appendix B.5. Low and High Q Sampling Strategies

d | Q (m^{3}/s) | C (mg/L) | t (d) | l |

17 | 0.52 | 32.26 | 3.25 | 79 |

0.46 | 26.78 | 8.29 | 200 | |

3.12 | 27.72 | 35.79 | 860 | |

17.23 | 27.55 | 35.83 | 861 | |

23.31 | 23.36 | 35.87 | 862 | |

18 | 0.52 | 32.26 | 3.25 | 79 |

0.46 | 26.78 | 8.29 | 200 | |

0.54 | 33.24 | 11.83 | 285 | |

0.51 | 25.48 | 17.96 | 432 | |

0.60 | 27.06 | 24.92 | 599 | |

3.12 | 27.72 | 35.79 | 860 | |

17.23 | 27.55 | 35.83 | 861 | |

23.31 | 23.36 | 35.87 | 862 | |

27.54 | 17.22 | 35.92 | 863 | |

25.90 | 14.02 | 35.96 | 864 | |

19 | 0.52 | 32.26 | 3.25 | 79 |

0.46 | 26.78 | 8.29 | 200 | |

0.54 | 33.24 | 11.83 | 285 | |

0.51 | 25.48 | 17.96 | 432 | |

0.60 | 27.06 | 24.92 | 599 | |

0.47 | 24.97 | 31.33 | 753 | |

0.67 | 28.10 | 35.50 | 853 | |

0.52 | 29.60 | 44.21 | 1062 | |

0.64 | 28.27 | 50.42 | 1211 | |

0.56 | 34.22 | 54.08 | 1299 | |

3.12 | 27.72 | 35.79 | 860 | |

17.23 | 27.55 | 35.83 | 861 | |

23.31 | 23.36 | 35.87 | 862 | |

27.54 | 17.22 | 35.92 | 863 | |

25.90 | 14.02 | 35.96 | 864 | |

21.23 | 12.01 | 36.00 | 865 | |

15.22 | 12.48 | 36.04 | 866 | |

10.82 | 12.53 | 36.08 | 867 | |

8.43 | 14.31 | 36.12 | 868 | |

6.66 | 15.32 | 36.17 | 869 | |

20 | 0.52 | 32.26 | 3.25 | 79 |

0.46 | 26.78 | 8.29 | 200 | |

0.54 | 33.24 | 11.83 | 285 | |

0.51 | 25.48 | 17.96 | 432 | |

0.60 | 27.06 | 24.92 | 599 | |

0.47 | 24.97 | 31.33 | 753 | |

0.67 | 28.10 | 35.50 | 853 | |

0.52 | 29.60 | 44.21 | 1062 | |

0.64 | 28.27 | 50.42 | 1211 | |

0.56 | 34.22 | 54.08 | 1299 | |

0.56 | 35.21 | 60.62 | 1456 | |

0.65 | 30.33 | 64.37 | 1546 | |

0.56 | 39.31 | 69.17 | 1661 | |

0.52 | 36.25 | 72.50 | 1741 | |

0.68 | 34.43 | 75.75 | 1819 | |

0.62 | 34.91 | 79.12 | 1900 | |

0.55 | 33.32 | 83.29 | 2000 | |

0.55 | 36.45 | 86.46 | 2076 | |

0.55 | 37.39 | 89.62 | 2152 | |

0.62 | 34.26 | 92.79 | 2228 | |

3.12 | 27.72 | 35.79 | 860 | |

17.23 | 27.55 | 35.83 | 861 | |

23.31 | 23.36 | 35.87 | 862 | |

27.54 | 17.22 | 35.92 | 863 | |

25.90 | 14.02 | 35.96 | 864 | |

21.23 | 12.01 | 36.00 | 865 | |

15.22 | 12.48 | 36.04 | 866 | |

10.82 | 12.53 | 36.08 | 867 | |

8.43 | 14.31 | 36.12 | 868 | |

6.66 | 15.32 | 36.17 | 869 | |

5.40 | 16.12 | 36.21 | 870 | |

4.27 | 16.98 | 36.25 | 871 | |

3.50 | 16.80 | 36.29 | 872 | |

2.73 | 22.40 | 168.42 | 4043 | |

5.21 | 20.37 | 168.46 | 4044 | |

7.50 | 18.87 | 168.50 | 4045 | |

7.73 | 16.18 | 168.54 | 4046 | |

7.73 | 14.64 | 168.58 | 4047 | |

8.93 | 11.02 | 168.62 | 4048 | |

9.18 | 9.49 | 168.67 | 4049 |

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**Figure 1.**(

**a**) Hourly observed log data (nitrate concentrations and discharge rates) at the Ammer catchment between July 2011 and April 2012 [20]; (

**b**) regression slope of $lo{g}_{10}C$ versus $lo{g}_{10}Q$. Points within the red circle in (

**b**) illustrate an event of the data that presents memory. Concentration, C, units are in mg/L, and discharge rates, Q, in m

^{3}/s.

**Figure 2.**Concentration time-series data generated with our simple stochastic model for α = 0, α = 1, t

_{ch}= 20, and t

_{ch}= 500. Q values are based on Q of Figure 1a. Concentration, C, units are in mg/L and discharge rates, Q, in m

^{3}/s.

**Figure 3.**Concentrations interpolated by the RMM (black line) using the inferred values in Table 4 and assimilated data (red points). Daily observed data are shown with white points. Data is assimilated daily for the first 90 days, every 4 days from 90 days to 180 days, and every 8 days from 180 days to 240 days. Predictions (colored lines) of the RMM are shown for a time larger than 240 days. The upper and lower CI are also shown in panel (

**a**) (grey lines).

**Figure 4.**Prior and posterior distributions of $\alpha $ for designs based on time and river discharge frequency and event-based strategies. The red star shows the posterior value ${\alpha}_{full}$ and black stars show the posterior mean values ${\alpha}_{post,d}$ inferred by the designs.

**Figure 5.**Comparison of observed samples to linear regression approach for sampling strategies considering event-based designs with 40 observations.

**Figure 6.**Comparison of observed samples to linear regression approach for d = 5 and d = 9 with only five samples.

Parameter (Unit) | Distribution | Mean | Standard Deviation | Min. Value | Max. Value |
---|---|---|---|---|---|

α (/) | uniform | - | - | 0.0 | 1.0 |

t_{ch} (h) | lognormal LN(μ, σ) = LN (4.35, 0.65) | 95.7 | 71.7 | - | - |

${m}_{k}$$\left(\frac{mg}{l}{\left(\frac{{m}^{3}}{h}\right)}^{\alpha}\right)$ | uniform | - | - | 1 | 40 |

$\sqrt{{v}_{k}}$$\left(\frac{mg}{l}{\left(\frac{{m}^{3}}{h}\right)}^{\alpha}\right)$ | uniform | - | - | 1 | 6.32 |

d | Strategy | Description | # Samples |
---|---|---|---|

1 | Time frequency | Samples are taken uniform in time: every 55 d | 5 |

2 | every 27.5 d | 10 | |

3 | every 13.75 d | 20 | |

4 | every 6.875 d | 40 | |

5 | River discharge frequency | Samples are taken uniform in flow discharge: every 4,434,944 m^{3} | 5 |

6 | every 2,217,472 m^{3} | 10 | |

7 | every 1,108,736m^{3} | 20 | |

8 | every 554,368 m^{3} | 40 | |

9 | Low Q | Samples are taken at low discharge rates (Q < 0.67 m^{3}/s) that belong to different peak events (uncorrelated samples). We choose them equidistantly. | 5 |

10 | 10 | ||

11 | 20 | ||

12 | 40 | ||

13 | High Q | Samples are taken at high discharge rates (Q > 2.5 m^{3}/s) for one peak event (correlated samples). When no more observations are available of one peak, we proceed to grab samples of a second peak event and so on. | 5 |

14 | 10 | ||

15 | 20 | ||

16 | 40 | ||

17 | Low and High Q | Samples are taken at low and high discharge rates following previous Low Q and High Q strategies. Half of the samples are at low Q except for d = 17 in which only two samples are at low Q. | 5 |

18 | 10 | ||

19 | 20 | ||

20 | 40 |

**Table 3.**Inferred hyper-parameters values when α or t

_{ch}are fixed using the full time-series data.

Fixed Hyper-Parameter | ||||
---|---|---|---|---|

α = 0 | α = 1 | t_{ch} = 20 | t_{ch} = 500 | |

Inferred Hyper-Parameter | ||||

α (/) | - | - | 0.218 | 0.016 |

t_{ch} (h) | 105.9 | 42.1 | - | - |

${m}_{k}$$\left(\frac{mg}{l}{\left(\frac{{m}^{3}}{h}\right)}^{\alpha}\right)$ | 31.0 | 23.6 | 27.1 | 17.5 |

$\sqrt{{v}_{k}}$$\left(\frac{mg}{l}{\left(\frac{{m}^{3}}{h}\right)}^{\alpha}\right)$ | 7.0 | 28.3 | 1.9 | 39.8 |

**Table 4.**Inferred values of the Ammer catchment using the whole time-series data (a total of 6604 measurements).

Parameter (Unit) | Inferred Value |
---|---|

${\alpha}_{full}$ (/) | 0.011 |

${t}_{ch,full}$ (h) | 106.4 |

${m}_{k,full}$$\left(\frac{mg}{l}{\left(\frac{{m}^{3}}{h}\right)}^{\alpha}\right)$ | 30.8 |

$\sqrt{{v}_{k,full}}$$\left(\frac{mg}{l}{\left(\frac{{m}^{3}}{h}\right)}^{\alpha}\right)$ | 2.6 |

**Table 5.**Summary of α and range of CI 95% for each design for the RMM and linear regression model after applying ODE. Highlighted in green and grey are the best and worst sampling strategies of the RMM, respectively.

RMM | Linear Regression | |||||||||

d | # Samples | ${\alpha}_{full}$ | CI 2.5% | CI 97.5% | Range 95% | ${\alpha}_{reg}$ | CI 2.5% | CI 97.5% | Range 95% | |

Full | 6604 | 0.011 | 0.002 | 0.02 | 0.018 | 0.180 | 0.167 | 0.194 | 0.027 | |

Prior | 0 | 0.5 | 0.025 | 0.950 | 0.950 | |||||

d | # samples | ${\alpha}_{post,d}$ | CI 2.5% | CI 97.5% | Range 95% | ${\alpha}_{reg,d}$ | CI 2.5% | CI 97.5% | Range 95% | |

Time frequency | 1 | 5 | 0.351 | 0.013 | 0.911 | 0.898 | −0.093 | −0.696 | 0.509 | 1.206 |

2 | 10 | 0.256 | 0.012 | 0.673 | 0.661 | 0.119 | −0.095 | 0.333 | 0.429 | |

3 | 20 | 0.153 | 0.008 | 0.401 | 0.393 | 0.109 | −0.073 | 0.291 | 0.364 | |

4 | 40 | 0.113 | 0.006 | 0.287 | 0.281 | 0.055 | −0.144 | 0.253 | 0.397 | |

River discharge frequency | 5 | 5 | 0.202 | 0.01 | 0.537 | 0.527 | 0.094 | −0.087 | 0.275 | 0.363 |

6 | 10 | 0.206 | 0.011 | 0.513 | 0.502 | 0.131 | −0.035 | 0.298 | 0.333 | |

7 | 20 | 0.278 | 0.052 | 0.526 | 0.474 | 0.259 | 0.041 | 0.477 | 0.436 | |

8 | 40 | 0.269 | 0.139 | 0.397 | 0.258 | 0.262 | 0.141 | 0.383 | 0.243 | |

Low Q | 9 | 5 | 0.344 | 0.012 | 0.934 | 0.922 | −0.242 | −0.832 | 0.348 | 1.180 |

10 | 10 | 0.479 | 0.028 | 0.969 | 0.941 | 0.981 | −1.080 | 3.041 | 4.121 | |

11 | 20 | 0.564 | 0.072 | 0.967 | 0.896 | 0.550 | −0.380 | 1.480 | 1.860 | |

12 | 40 | 0.265 | 0.017 | 0.656 | 0.64 | 0.311 | −0.175 | 0.796 | 0.971 | |

High Q | 13 | 5 | 0.083 | 0.005 | 0.200 | 0.195 | 0.208 | −0.264 | 0.679 | 0.943 |

14 | 10 | 0.078 | 0.003 | 0.182 | 0.178 | 0.117 | −0.251 | 0.484 | 0.735 | |

15 | 20 | 0.067 | 0.002 | 0.200 | 0.198 | 0.103 | −0.094 | 0.300 | 0.394 | |

16 | 40 | 0.025 | 0.001 | 0.087 | 0.086 | 0.020 | −0.447 | 0.488 | 0.935 | |

Low & High Q | 17 | 5 | 0.076 | 0.005 | 0.155 | 0.15 | 0.040 | −0.045 | 0.126 | 0.171 |

18 | 10 | 0.074 | 0.004 | 0.171 | 0.167 | 0.102 | 0.020 | 0.184 | 0.163 | |

19 | 20 | 0.072 | 0.006 | 0.163 | 0.156 | 0.164 | 0.097 | 0.232 | 0.135 | |

20 | 40 | 0.134 | 0.049 | 0.224 | 0.175 | 0.227 | 0.176 | 0.278 | 0.102 |

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## Share and Cite

**MDPI and ACS Style**

Gonzalez-Nicolas, A.; Schwientek, M.; Sinsbeck, M.; Nowak, W. Characterization of Export Regimes in Concentration–Discharge Plots via an Advanced Time-Series Model and Event-Based Sampling Strategies. *Water* **2021**, *13*, 1723.
https://doi.org/10.3390/w13131723

**AMA Style**

Gonzalez-Nicolas A, Schwientek M, Sinsbeck M, Nowak W. Characterization of Export Regimes in Concentration–Discharge Plots via an Advanced Time-Series Model and Event-Based Sampling Strategies. *Water*. 2021; 13(13):1723.
https://doi.org/10.3390/w13131723

**Chicago/Turabian Style**

Gonzalez-Nicolas, Ana, Marc Schwientek, Michael Sinsbeck, and Wolfgang Nowak. 2021. "Characterization of Export Regimes in Concentration–Discharge Plots via an Advanced Time-Series Model and Event-Based Sampling Strategies" *Water* 13, no. 13: 1723.
https://doi.org/10.3390/w13131723