# Intercomparing LSTM and RNN to a Conceptual Hydrological Model for a Low-Land River with a Focus on the Flow Duration Curve

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}within northwestern Germany and the Netherlands. The river flows 371 km from its spring to the outlet in the Northern Sea with a relatively low elevation difference of only 135 m and a general flow direction from southeast to northwest. The catchment’s climate is defined by the humid–temperate western wind zone of Central Europe with pronounced, but not very long, cold seasons. Mild winters, cool summers and abundant precipitation characterize this Atlantic-influenced region, whereas easterly wind conditions result in drier conditions, warmer summers as well as colder winters. Mean long-term annual precipitation is about 800 mm; mean annual temperature is between 8.5, and 9 °C and mean annual potential evapotranspiration (pET) is around 490 mm. The discharge of the Ems in most years is characterized by flood events in winter and a low-flow period from June to October. The high-flow phase usually lasts from December to March [26].

#### 2.2. Data

_{c}-value for the HBV model before calibration.

#### 2.3. Models

#### 2.3.1. HBV

#### 2.3.2. RNN

#### 2.3.3. LSTM

#### 2.4. Model Set-Up

#### 2.4.1. RNN and LSTM Forcing Data

^{3}/s) into the sub-catchment. This forcing data are comparable to the HBV input, as vapor pressure and maximum temperature are included in the potential evapotranspiration calculation by Haude [31]. The window size specifies the number of previous timesteps included in the input data to model the streamflow. Window sizes of 365 days [6,40] and 270 days [45] have previously been applied. Whereas a window size of 250 resulted in a better model performance in this study during the hyperparameter tuning. The forcing data to simulate streamflow at timestep t has the vector-shape (5250) and contains the hydro–climatic drivers mentioned above of timestep t plus the 249 previous days. Zero-to-one scaling was applied to forcing data, given the range of the training data, before data were fed into the model.

#### 2.4.2. ML Model Architecture

#### 2.4.3. Routing

#### 2.5. Calibration and Validation

#### 2.6. Model Evaluation

^{2}. Each metric has its focus, either the overall water balance or a specific flow range [50]. These indices take the whole bandwidth of streamflow into account. Instead, the logNSE uses the logarithmic transformed values of observed and simulated streamflow. This adds more weight on lower values due to its mathematical formulation and, therefore, the logNSE is a recognized low-flow index. However, all these metrics only quantify the statistical characteristics of model residuals, which makes it necessary to include hydrologically based metrics [51]. Examples for hydrologically based metrics are signature indices derived from the flow duration curve [52]. The FDC is the cumulative distribution function where streamflow is plotted against its exceedance probability, showing the percentage of time when streamflow is equal or exceeds the given value [53]. Signature indices examine the influence of specific aspects of the hydrograph and are sensitive to detect differences in runoff generation, seasonality and reactivity [53,54]. Models which show similar NSE values might differ when the FDC is analyzed [54]. We, therefore, used a wide range of various indices, including performance indices and signature indices, to evaluate the model performances and to gain a deep understanding of the models’ strengths and weaknesses (Figure 3b). Due to the widespread application of the performance indices presented in Figure 3b, we refrain from the full explanation of all indices. Here, we explain the five less-common hydrological signature indices we used in this study. They were introduced by [52] in full detail (Figure 4).

- 1.
- Bias RR: bias of the mean values in percent (black circles in Figure 4)$$BiasRR=\frac{mean\left(FD{C}_{sim}\right)-mean\left(FD{C}_{obs}\right)}{mean\left(FD{C}_{obs}\right)}\times 100$$
- 2.
- Bias MM: bias of the median values in percent (black crosses Figure 4)$$BiasMM=\frac{median\left(FD{C}_{sim}\right)-median\left(FD{C}_{obs}\right)}{median\left(FD{C}_{obs}\right)}\times 100$$
- 3.
- Bias FDC midslope: bias of the mean slope in mid segment of FDC in percent (dashed lines in Figure 4)$$BiasFDC=\frac{\left(\mathrm{log}\left(FD{C}_{sim,0.2}\right)-\mathrm{log}\left(FD{C}_{sim,0.7}\right)\right)-\left(\mathrm{log}\left(FD{C}_{obs,0.2}\right)-\mathrm{log}\left(FD{C}_{obs,0.7}\right)\right)}{\left(\mathrm{log}\left(FD{C}_{obs,0.2}\right)-\mathrm{log}\left(FD{C}_{obs,0.7}\right)\right)}\times 100$$
- 4.
- Bias FLV: bias of the low segment of the FDC (orange and blue areas in Figure 4)$$BiasFLV=\frac{{{\displaystyle \int}}_{0.7}^{1}\left(\mathrm{log}\left(FD{C}_{sim,p}\right)-\mathrm{log}\left({Q}_{sim,min}\right)\right)dp-{{\displaystyle \int}}_{0.7}^{1}\left(\mathrm{log}\left(FD{C}_{ops,p}\right)-\mathrm{log}\left({Q}_{sim,min}\right)\right)dp}{{{\displaystyle \int}}_{0.7}^{1}\left(\mathrm{log}\left(FD{C}_{ops,p}\right)-\mathrm{log}\left({Q}_{obs,min}\right)\right)dp}\times 100$$
- 5.
- Bias FHV: bias of the high segment of the FDC (green area in Figure 4)$$BiasFHV=\frac{{{\displaystyle \int}}_{0}^{0.02}\left(FD{C}_{sim,p}\right)dp-{{\displaystyle \int}}_{0}^{0.02}\left(FD{C}_{ops,p}\right)dp}{{{\displaystyle \int}}_{0}^{0.02}\left(FD{C}_{ops,p}\right)dp}\times 100$$

## 3. Results

#### 3.1. Statistical Performance Indices

^{2}and Index of Agreement (IoA), with values of above 0.9, which can be considered “very good” [55]. Focusing on the logNSE, instead, exceptionally lower performance can be observed, particularly in sub-catchments 2 and 3. For the LSTM, the logNSE drops down to 0.68 in sub-catchment 2 and even down to 0.42 in sub-catchment 3, whereas the RNN only shows a decreased logNSE of −1.26 in sub-catchment 2. For the HBV model, the values of NSE, KGE, logNSE, R

^{2}and IoA usually are above 0.8. Exemptions occur in sub-catchment 2 with the KGE ranging slightly above 0.7 and in sub-catchment 5 when the logNSE falls below 0.8.

^{2}and IoA. The differences between the two ML models for each index usually range between 0.05 and 0.1. For the ML models in sub-catchments 1, 3, 4 and 5 during the validation period, the indices NSE, KGE, R

^{2}and IoA are all above 0.8. In sub-catchment 2, those indices only range above 0.7.

^{2}and IoA are all above 0.7 and up to 0.9. Lowest values also tend to occur in sub-catchment 2 and 3. The logNSE generally is also lower than the other indices and is down to 0.66 in sub-catchment 2, but it does not drop as significantly compared to the ML models. During the historical periods, the results are almost similar to the validation period. Larger differences occur only for the HBV model. In sub-catchments 1 and 2, the KGE is about 0.1 lower compared to the validation.

#### 3.2. FDC and Signature Indices

## 4. Discussion

#### 4.1. Statistical Performance Indices

#### 4.2. FDC and Signature Indices

#### 4.3. RNN Compared to LSTM

#### 4.4. Routing

## 5. Conclusions

^{2}and IoA). Signature indices, sampling specific parts of the FDC, reveal that the ML models have good representation of the overall water balance (low BiasRR), whereas the HBV model provides a better representation of streamflow dynamics (low BiasFDC). Further research is necessary to determine if this is a general model behavior or if this phenomenon only applies to catchments with certain characteristics.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Overview map of the study area location in Germany. (

**b**,

**c**) Maps of the river Ems showing the five individual sub-catchments delineated upstream from the last available streamflow gauge located 235 km downstream from the spring with its streamflow gauges, land use (

**b**) and elevation (

**c**). It appears that six separated catchments were delineated; yet, the smallest one actually has a connection to sub-catchment 2 on the bottom left side.

**Figure 2.**(

**a**) The architecture of a RNN cell is displayed on the left side with x being the input and h the internal state; t indicates the current timestep. (

**b**) Internals of the LSTM cell with h referring to the hidden state, c the cell state, f the forget gate, i the input gate and o the output gate at timestep t. Further details are explained by Equations (3)–(8).

**Figure 3.**(

**a**) Time ranges applied for the different model types during the modeling process. (

**b**) Overview of all goodness-of-fit criteria used to evaluate the model results.

**Figure 4.**Two exemplary flow duration curves with highlighted features to explain the signature indices after Casper et al. [53].

**Figure 5.**Hydrographs of all models and observed streamflow within sub-catchment 1 (

**a**) and sub-catchment 5 (

**b**) in the year 1998 to show annual variations in streamflow and the models’ performances exemplarily.

**Figure 6.**Radar plots showing the variety of statistical indices for the five different sub-catchments and the three different periods (historical (

**a**), calibration (

**b**) and validation (

**c**)). The minimal value visible in this figure is 0.5. Yet, in catchment 2 and 3, the logNSE value can be lower than 0.5, and, therefore, the logNSE values below 0.5 are mentioned in the following. Sub-catchment 2 (historical: LSTM: 0.45, RNN: −1.71; calibration: RNN: −1.26; validation: LSTM: 0.23, RNN: −1.80); Sub-catchment 3 (historical: LSTM: −0.47; calibration: LSTM: 0.42; validation: −0.11).

**Figure 7.**(

**a**) Flow duration curves of all three model types and observed streamflow. (

**b**) Bias of the five signature indices for the three model types compared to the observed streamflow.

**Table 1.**Number and data source of hydro–climatic stations used to derive forcing data for the models and to calculate the potential evapotranspiration.

Data | Number of Stations | Source | Required For |
---|---|---|---|

Precipitation (daily sum) | 142 | DWD | HBV, LSTM, RNN |

Rel. Humidity (hourly mean) | 10 | DWD | HBV (pET estimation after Haude [31]) |

Maximum Temperature (daily) | 25 | DWD | LSTM, RNN |

Minimum Temperature (daily) | 25 | DWD | LSTM, RNN |

Temperature (daily mean) | 33 | DWD | HBV |

Temperature (hourly mean) | 8 | DWD | HBV (pET estimation after Haude [31]) |

Vapor Pressure (daily mean) | 29 | DWD | LSTM, RNN |

Streamflow (daily mean) | 5 | GRDC | HBV, LSTM, RNN |

**Table 2.**Area (m

^{2}), flow length (km) and baseflow index (BFI) (baseflow/total streamflow) for each of the five individual sub-catchments. Sub-catchments are numbered systematically upstream-to-downstream.

Number of Sub-catchment | Area (km^{2}) | Flow Length (km) | BFI (%) |
---|---|---|---|

1 | 1448 | 87 | 55 |

2 | 1283 | 26 | 51 |

3 | 871 | 40 | 57 |

4 | 1150 | 59 | 61 |

5 | 3324 | 22 | 55 |

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**MDPI and ACS Style**

Ley, A.; Bormann, H.; Casper, M. Intercomparing LSTM and RNN to a Conceptual Hydrological Model for a Low-Land River with a Focus on the Flow Duration Curve. *Water* **2023**, *15*, 505.
https://doi.org/10.3390/w15030505

**AMA Style**

Ley A, Bormann H, Casper M. Intercomparing LSTM and RNN to a Conceptual Hydrological Model for a Low-Land River with a Focus on the Flow Duration Curve. *Water*. 2023; 15(3):505.
https://doi.org/10.3390/w15030505

**Chicago/Turabian Style**

Ley, Alexander, Helge Bormann, and Markus Casper. 2023. "Intercomparing LSTM and RNN to a Conceptual Hydrological Model for a Low-Land River with a Focus on the Flow Duration Curve" *Water* 15, no. 3: 505.
https://doi.org/10.3390/w15030505