# Improvement of Hydrological Simulations by Applying Daily Precipitation Interpolation Schemes in Meso-Scale Catchments

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

^{2}) sub-catchments lying in the Sulejów reservoir catchment in central Poland. Four methods were tested: the default SWAT method (Def) based on the Nearest Neighbour technique, Thiessen Polygons (TP), Inverse Distance Weighted (IDW) and Ordinary Kriging (OK). =The evaluation of methods was performed using a semi-automated calibration program SUFI-2 (Sequential Uncertainty Fitting Procedure Version 2) with two objective functions: Nash-Sutcliffe Efficiency (NSE) and the adjusted R

^{2}coefficient (bR

^{2}). The results show that: (1) the most complex OK method outperformed other methods in terms of NSE; and (2) OK, IDW, and TP outperformed Def in terms of bR

^{2}. The median difference in daily/monthly NSE between OK and Def/TP/IDW calculated across all catchments ranged between 0.05 and 0.15, while the median difference between TP/IDW/OK and Def ranged between 0.05 and 0.07. The differences between pairs of interpolation methods were, however, spatially variable and a part of this variability was attributed to catchment properties: catchments characterised by low station density and low coefficient of variation of daily flows experienced more pronounced improvement resulting from using interpolation methods. Methods providing higher precipitation estimates often resulted in a better model performance. The implication from this study is that appropriate consideration of spatial precipitation variability (often neglected by model users) that can be achieved using relatively simple interpolation methods can significantly improve the reliability of model simulations.

## 1. Introduction

^{2}. For larger catchments (above 5000 km

^{2}), no significant difference in performance between studied methods was reported.

^{3}km

^{2}[24]. The secondary goal is to analyse the effect of certain catchment properties in order to explain the between-catchment differences in results. Even though there exists a body of literature on this topic (e.g., [22,23,25,26,27,28]), our study brings certain improvements in methodological design, by combining simultaneously the following features:

- -
- We perform interpolation of daily precipitation data for a climate normal period (in contrast to many other studies that used a monthly or annual time step, e.g., [19,29,30,31,32,33], or a daily time step for a much shorter period, e.g., [20]). Long simulation periods are recommended for model application for climate change impact assessment [34];
- -
- -
- We focus on the effect of different interpolation methods on hydrology (in contrast to some interesting papers that limit their attention to the comparison of performance of interpolators, e.g., [17,20,35]) and evaluate this effect using a semi-automated SUFI-2 (Sequential Uncertainty Fitting) algorithmin 11 catchments spanning in size between 119 and 3935 km
^{2}. Taking advantage of the relatively large number of studied catchments (compared to other studies that usually focused on one catchment), we investigate the influence of certain catchment characteristics on evaluation results, which to our knowledge has not been done to date; - -

## 2. Materials and Methods

#### 2.1. Study Area

^{6}m

^{3}of total capacity) but its effect on river flow is outside the scope of this paper, because all catchments studied in this paper (i.e., sub-catchments of the SRC) are situated upstream of it. The SWAT model is however set up for the whole SRC, whose total area equals 4928 km

^{2}, of which the Pilica catchment contributes nearly 80% and the Luciąża catchment nearly 16% [36].

^{6}m

^{3}. Such a quasi-natural character of the SRC makes it a suitable study area for hydrological modelling.

#### 2.2. SWAT Model

#### 2.2.1. General Features

#### 2.2.2. Model Setup

- -
- DEM based on the ASTER satellite data, 1:25,000 topographic map and Regional Water Management Authority (RZGW) water cadastre.
- -
- Land cover data derived from reclassified Corine Land Cover 2006 (CLC2006) available from General Directorate of Environmental Protection (GDOŚ).
- -
- Soil map composed of a 1:100,000 digital map from Institute of Soil Science and Plant Cultivation (IUNG) and 1:25,000 soil map available from Regional Directorate of State Forests (RDLP).

#### 2.3. Precipitation Data and Interpolation Methods

#### 2.4. Precipitation Station Density Factor

**Figure 3.**A surface of kernel density function (

**A**) applied for estimation of station density within each catchment (

**B**) expressed in units (number of stations) per km

^{2}.

#### 2.5. Strategy for Evaluation of Hydrological Simulations

#### 2.5.1. SWAT-CUP and SUFI-2

^{2}which is the coefficient of determination R

^{2}multiplied by the coefficient of the regression line b. This modified coefficient of determination allows accounting for the discrepancy in the magnitude of two signals (depicted by b) as well as their dynamics (depicted by R

^{2}) [50]. It can range between 0 and 1, where 1 is the optimal value. Equation for bR

^{2}is [50]:

Name | Lower Limit | Upper Limit | Definition |
---|---|---|---|

ESCO.hru ^{2} | 0.7 | 1 | Soil evaporation compensation factor |

EPCO.hru ^{2} | 0 | 1 | Plant uptake compensation factor |

SOL_Z().sol ^{1} | −0.4 | 0.4 | Depth from soil surface to the bottom of layer |

SOL_AWC().sol ^{1} | −0.4 | 0.4 | Available water capacity of the soil layer |

SOL_BD().sol ^{1} | −0.4 | 0.4 | Moist bulk density |

SOL_K().sol ^{1} | −0.9 | 2 | Saturated hydraulic conductivity |

HRU_SLP.hru ^{1} | −0.3 | 0.3 | Average slope steepness |

ALPHA_BF.gw ^{2} | −0.9 | 2 | Baseflow alpha factor |

GW_DELAY.gw ^{2} | 50 | 400 | Groundwater delay time |

GWQMN.gw ^{2} | 0 | 1000 | Threshold depth of water in the shallow aquifer required for return flow to occur |

GW_REVAP.gw ^{2} | 0.02 | 0.2 | Groundwater “revap” coefficient |

RCHRG_DP.gw ^{2} | 0 | 0.3 | Deep aquifer percolation fraction |

CN2.mgt ^{1} | −0.15 | 0.15 | Initial SCS (Soil Conservation Service) runoff curve nr for moisture condition II |

SURLAG.bsn ^{2} | 0.3 | 3 | Surface runoff lag coefficient |

SLSUBBSN.hru ^{1} | −0.3 | 0.3 | Average slope length (m) |

CH_N2.rte ^{2} | 0.01 | 0.1 | Manning's “n” value for the main channel |

CH_N1.sub ^{2} | 0.01 | 0.1 | Manning's “n” value for the tributary channel (-) |

SMTMP.bsn ^{2} | −2 | 2 | Snow melt base temperature |

TIMP.bsn ^{2} | 0 | 1 | Snow pack temperature lag factor |

SNOCOVMX.bsn ^{2} | 0 | 40 | Minimum snow water content that corresponds to 100% snow cover |

^{1}parameter multiplied by 1 + r, where r is a number between lower and upper limits;

^{2}parameter replaced by the new value from the range.

^{2}refer to one single parameter set that produces the best value of the objective function, p- and r-factors refer to the whole body of simulations resulting from the 95PPU, not one simulation.

#### 2.5.2. The Observed Data and Catchment Properties

^{3}/s) required for model testing in SUFI-2 were used in two temporal aggregations: daily and monthly, most widely used in hydrological modelling. They were obtained from 11 IMGW-PIB flow gauges for the period of 28 hydrological years from 1984–2011 (the hydrological year in Poland begins on 1 November). Names and codes of the gauges, corresponding rivers and available data periods as well as some basic catchment characteristics are presented in Table 2, whereas their location is shown in Figure 2.

**Table 2.**List of IMGW-PIB flow gauges in the Sulejów reservoir catchment used in this study with data availability and selected catchment properties.

No | Gauge Name | River Name | Code | A (km^{2}) | Period of Available Data | Years | Flow (m^{3}/s) | q_{m} (m^{3}/s/km^{2}) | c_{v} (-) | |
---|---|---|---|---|---|---|---|---|---|---|

Mean | St. Dev. | |||||||||

1 | Sulejów | Pilica | Pil-SUL | 3934 | 11/1/1983–10/31/2011 | 28 | 21.9 | 14.9 | 5.5 | 0.68 |

2 | Przedbórz | Pilica | Pil-PRZ | 2491 | 11/1/1983–10/31/2011 | 28 | 13.5 | 9.6 | 5.3 | 0.71 |

3 | Wąsosz | Pilica | Pil-WAS | 974 | 11/1/2005–10/31/2011 | 6 | 5.7 | 4.7 | 6.5 | 0.82 |

4 | Szczeko-ciny | Pilica | Pil-SZC | 360 | 11/1/1983–10/31/2009 | 26 | 1.8 | 1.3 | 5.0 | 0.69 |

5 | Kłudzice | Luciąża | Luc-KLU | 507 | 11/1/1983–10/31/2011 | 28 | 2.6 | 2.5 | 5.2 | 0.95 |

6 | Dąbrowa | Czarna Maleniecka | CzM-DAB | 946 | 11/1/1983–10/31/2008; 11/1/2009–10/21/2011 | 27 | 5.6 | 5.6 | 5.8 | 1.00 |

7 | Wąsosz-Stara Wieś | Czarna Maleniecka | Cza-WSW | 119 | 11/1/1991–10/31/2003 | 12 | 0.91 | 1.2 | 7.6 | 1.09 |

8 | Wąsosz-Stara Wieś | Krasna | Kra-WSW | 120 | 11/1/1991–10/31/2011 | 20 | 0.77 | 1.2 | 6.3 | 1.51 |

9 | Janusze-wice | Czarna Włoszczowska | CzW-JAN | 598 | 11/1/1983–10/31/2011 | 28 | 3.3 | 4.0 | 5.5 | 1.20 |

10 | Bonowice | Żebrówka | Zeb-BON | 128 | 11/1/1983–10/31/2009 | 26 | 0.51 | 0.47 | 6.5 | 0.82 |

11 | Bonowice | Krztynia | Krz-BON | 256 | 11/1/1990–10/31/2009 | 19 | 1.3 | 0.68 | 5.0 | 0.54 |

_{m}is the area-specific runoff and c

_{v}is coefficient of variation in daily flows. St. Dev. is abbreviation of standard deviation

^{2}, mean area-specific runoff (q

_{m}) from 4.1–7.6 m

^{3}/s∙km

^{2}and the coefficient of variation of daily flows (c

_{v}) from 0.54 to 1.51. The two last hydrological characteristics are indices that accumulate the complex interplay of physiographic (e.g., elevation, slope, land cover, soil permeability), climatic (e.g., evapotranspiration, precipitation) and water management (e.g., abstractions, reservoirs) properties. They will be used along with KD index illustrated in Figure 3 in an attempt to explain the differences in results (e.g., objective function values) between different catchments.

#### 2.5.3. Study Design

^{2}) and for each of the 11 catchments (gauging stations) separately. More precisely, when the observed data input file contains data from multiple gauging stations, SUFI-2 enables assigning weights to different stations and thus calculating weighted objective function values:

^{2}) for the interpolation method $X$ (Def, TP, IDW or OK) and temporal aggregation $t$ ($d$ for day or $m$ for month). In the first step, since the sample size is small ($n=11$) and the population cannot be assumed to be normally distributed, we applied a Wilcoxon signed-rank test that is a nonparametric analogue to the paired t-test [54]. Eleven catchments and three pairs of interpolation methods (Def vs. TP, Def vs. IDW and Def vs. OK) served as nominal variables, while $O{F}_{X,t}$ served as measurement variable. The null hypotheses is that the median difference in $O{F}_{X,t}$ between two given interpolation methods is zero. We applied this test at two significance levels: $p=0.05$ and $p=0.1$. Additionally, the $O{F}_{X,t}$ values will be illustrated as box plots showing the median, interquartile range and minimum/maximum values.

## 3. Results and Discussion

#### 3.1. Evaluation of Interpolation Results

**Figure 4.**Mean annual precipitation for the period 1982–2011 calculated for 11 studied catchments for the Default method (

**A**); The difference in mean annual precipitation between three interpolation methods (TP, IDW, OK) and Def (

**B**).

**Figure 5.**Spatial distribution of precipitation on four selected wet days (4 January 1983); 17 April 1989; 9 July 1994; 7 November 1998) estimated using Ordinary Kriging (

**A**,

**C**,

**E**,

**G**) and respective prediction standard errors for the same days (

**B**,

**D**,

**F**,

**H**).

#### 3.2. Effect of Interpolation Methods on Model Performance

#### 3.2.1. Statistical Summary for All Catchments

^{2}all three interpolation methods led to a significant improvement over the default method, however the difference between each of them individually was very small (Figure 6b,d). The median difference of $\u2206O{F}_{X,Def,d}$ was equal to 0.05 for $X$ being TP, IDW or OK. The median difference of $\u2206O{F}_{X,Def,m}$ was equal to 0.07 for $X$ being TP or OK and 0.06 for $X$ being IDW. A comparison of results between daily and monthly aggregations leads to a conclusion that the effect is broadly similar and its magnitude is only a bit stronger for monthly aggregation.

^{2}as the objective functions. It can be noted that the positive values of PBIAS largely predominate, which shows that the model generally underestimated the discharge. This underestimation was always higher for bR

^{2}than for NSE. For NSE, it was significantly lower for the monthly temporal aggregation than for the daily aggregation. For bR

^{2}no such trend was observed. As regards comparison between methods, the results were quite variable. For monthly NSE, clearly OK produced the values of PBIAS closest to zero as compared to all other methods. For daily NSE, OK was characterised by lower variability of PBIAS than other methods, particularly lower than IDW. For monthly bR

^{2}, the PBIAS box plot statistics were slightly lower for Def than for other methods, whereas for daily bR

^{2}no clear trend could be found, even though the variability for IDW and OK was slightly higher than for Def and TP.

**Figure 6.**Box plots of selected objective functions across all 11 flow gauging stations for different interpolation methods (Def—Default version, TP—Thiessen Polygons, IDW—Inverse Distance Weighted, OK—Ordinary Kriging) and different temporal aggregations (

**a**,

**b**: daily;

**c**,

**d**: monthly).

**Figure 7.**Box plots of PBIAS across all 11 flow gauging stations for different interpolation methods (Def—Default version, TP—Thiessen Polygons, IDW—Inverse Distance Weighted, OK—Ordinary Kriging) and different objective function/temporal aggregation combinations (

**a**—$NS{E}_{d}$;

**b**—$b{R}_{d}^{2}$;

**c**—$NS{E}_{m}$;

**d**—$b{R}_{m}^{2}$).

^{2}as objective functions. In general, the values of RMSE were always lower for NSE than for bR

^{2}, and always lower for the monthly temporal aggregation than for daily aggregation. When it comes to the comparison between methods, since RMSE is not dimensionless, it cannot be compared between different catchments. Hence, we have calculated percent changes in RMSE for each pair of methods (Figure 8). As with PBIAS, the results are different for different objective functions. For NSE, RMSE for OK is significantly lower than for all other methods for both temporal aggregations, which is in agreement with observations from Figure 6a,c. For daily bR

^{2}all interpolation methods are characterised by lower RMSE than Def and OK has slightly lower RMSE than IDW. For monthly bR

^{2}the results are highly variable between catchments and no clear relationship can be distinguished.

**Figure 8.**Box plots of percent changes in RMSE across all 11 flow gauging stations for different interpolation methods (Def—Default version, TP—Thiessen Polygons, IDW—Inverse Distance Weighted, OK—Ordinary Kriging) and different objective function/temporal aggregation combinations (

**a**—$NS{E}_{d}$;

**b**—$b{R}_{d}^{2}$;

**c**—$NS{E}_{m}$;

**d**—$b{R}_{m}^{2}$).

- (1)
- p-factor for method A is higher than p-factor for method B and r-factor for method A is not higher than r-factor for method B.
- (2)
- p-factor for method A is not lower than p-factor for method B and r-factor for method A is lower than r-factor for method B.

**Figure 9.**Box plots of uncertainty coefficients across all 11 flow gauging stations for different interpolation methods (Def—Default version, TP—Thiessen Polygons, IDW—Inverse Distance Weighted, OK—Ordinary Kriging) and different temporal aggregations (

**a**,

**b**: daily;

**c**,

**d**: monthly).

#### 3.2.2. Relationship between the Objective Functions and Catchment Characteristics

**Table 3.**Pearson correlation matrix between selected catchment properties (precipitation station density indicator KD (km

^{−2}), upstream catchment area $A$, (km

^{2}), coefficient of variation of daily/monthly flows ${c}_{\text{v}}$ (-) and the difference in mean annual precipitation between methods $X$ and $Y$$\u2206{P}_{X,Y}$ (mm) and the differences in objective functions $\u2206O{F}_{X,Y,t}$.

Catchment Properties | $TP\u2013Def$ | $IDW\u2013Def$ | $OK\u2013Def$ | $IDW\u2013TP$ | $OK\u2013TP$ | $OK\u2013IDW$ |
---|---|---|---|---|---|---|

$\u2206NS{E}_{X,Y,d}$ | ||||||

$\text{KD}$ | −0.15 | −0.36 | −0.14 | −0.36 | 0.00 | 0.28 |

$A$ | −0.25 | −0.49 | 0.14 | −0.44 | 0.60 ^{‡} | 0.73 ^{‡} |

${c}_{\text{v}}$ | 0.19 | 0.28 | −0.10 | 0.18 | −0.46 | −0.44 |

$\u2206{P}_{X,Y}$ | 0.03 | 0.10 | −0.04 | 0.14 | 0.07 | −0.88 ^{‡} |

$\u2206b{R}_{X,Y,d}^{2}$ | ||||||

$\text{KD}$ | −0.54 ^{†} | −0.56 ^{†} | −0.60 ^{†} | −0.07 | −0.32 | 0.17 |

$A$ | −0.13 | −0.13 | −0.16 | −0.09 | −0.22 | 0.08 |

${c}_{\text{v}}$ | −0.58 ^{†} | −0.60 ^{‡} | −0.59 ^{†} | 0.14 | −0.05 | 0.20 |

$\u2206{P}_{X,Y}$ | 0.81 ^{‡} | 0.78 ^{‡} | 0.79 ^{‡} | 0.19 | 0.51 | 0.24 |

$\u2206NS{E}_{X,Y,m}$ | ||||||

$\text{KD}$ | 0.20 | −0.29 | −0.43 | −0.69 ^{‡} | −0.50 | −0.10 |

$A$ | 0.27 | 0.03 | 0.35 | −0.40 | −0.04 | 0.34 |

${c}_{\text{v}}$ | 0.51 | 0.49 | −0.27 | −0.23 | −0.71 ^{‡} | −0.88 ^{‡} |

$\u2206{P}_{X,Y}$ | −0.60 ^{†} | −0.25 | 0.32 | 0.34 | 0.74 ^{‡} | −0.37 |

$\u2206b{R}_{X,Y,m}^{2}$ | ||||||

$\text{KD}$ | −0.54 ^{†} | −0.59 ^{†} | −0.70 ^{‡} | −0.52 ^{†} | −0.80 ^{‡} | −0.56 ^{†} |

$A$ | −0.27 | −0.29 | −0.26 | −0.20 | −0.13 | −0.03 |

${c}_{\text{v}}$ | −0.25 | −0.16 | −0.33 | 0.26 | −0.38 | −0.55 ^{†} |

$\u2206{P}_{X,Y}$ | 0.63 ^{‡} | 0.53 ^{†} | 0.70 ^{‡} | 0.32 | 0.64 ^{‡} | 0.46 |

^{†}Significant at significance level $p=0.1$.

^{‡}Significant at significance level $p=0.05$.

- (1)
- $\u2206NS{E}_{X,Y,d}$ (Figure 10A,B): OK is superior over IDW and TP in catchments with larger drainage areas; OK is superior over IDW in catchments with small mean precipitation difference between these two methods.
- (2)
- $\u2206b{R}_{X,Y,d}^{2}$ (Figure 10C,D): IDW is superior over Def in catchments with lower daily ${c}_{\text{v}}$ (more stable flow regime); TP, IDW and OK are superior over Def in catchments with high positive difference in mean precipitation.
- (3)
- $\u2206NS{E}_{X,Y,m}$ (Figure 10E–G): TP is superior over IDW in catchments with higher station densities; OK is superior over IDW and TP in catchments with lower monthly ${c}_{\text{v}}$ (more stable flow regime); OK is superior over TP in catchments for which the difference in mean precipitation between OK and TP is positive.
- (4)
- $\u2206b{R}_{X,Y,m}^{2}$ (Figure 10H,I): OK is superior over Def (more apparently) and TP (less apparently) in catchments with low station density. TP and OK are superior over Def in catchments with high positive difference in mean precipitation.

**Figure 10.**Scatter plots of $\u2206O{F}_{X,Y,t}$ and various catchment descriptors for relationships with significant correlation (at significance level $p=0.05)$ from Table 3. Note: Dbr_X_Y = $\u2206N{S}_{X,Y}$, Dbr_X_Y = $\u2206b{R}_{X,Y}^{2}$.

#### 3.3. Discussion

**Table 4.**Literature review of studies evaluating precipitation interpolation methods for hydrological modelling.

ID | Publication Code and Material | Catchment (Country) | Area (km^{2}) | Number of Precipitation Gauges | Station Density (Stations/1000 km^{2}) | Model Name | Simulation Period (Years) | Number of Flow Gauges | Analysis Time Step | Interpolation Methods | Evaluation Criterion | Main Conclusion |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | This paper; Figure 6 and Figure 8 | Pilica (PL) | 4,928 | 46 | 9.3 | SWAT | 30 | 11 | d, m | Def (NN), TP,IDW, OK | NSE, bR^{2} | OK, IDW, TP outperformed Def for bR^{2} for both daily and monthly time step; OK slightly better than others for NSE (daily and monthly) |

2 | Haberlandt1998 [22]; Figure 7A | Mackenzie (CA) | 1,800,000 | 81 | 0.05 | SLURP | 16 | 29 | m | NN, OK | Relative standard error | OK superior over NN but mainly in smaller subbasins (below 50,000 km^{2}) |

3 | Hwang2012 [26]; Table 7, Figures 17 and 18 | Animas (CO, USA) | 1,792 | 37 | 20.6 | PRMS (distributed) | 26 | 1 | d, s, a | IDW, MLR, CMLR, LWP | RMSE, NSE, Flow statistics | All methods similar in terms of NSE and RMSE; all methods provide accurate timing of flood events but the magnitude is underestimated |

4 | Hwang2012 [26]; Tables 6 and 7, Figures 17 and 18 | Alapaha (GA, USA) | 3,626 | 28 | 7.7 | PRMS (distributed) | 22 | 1 | d, s, a | IDW, MLR, CMLR, LWP | RMSE, NSE, Flow statistics | LWP and MLR superior over CMLR in terms of NSE and RMSE; all methods provide accurate timing of flood events but the magnitude is underestimated |

5 | Masih2011 [23] Table 3, Figures 5–7 | Karkheh (IR) | 4,2620 | 41 | 0.96 | SWAT | 15 | 15 | d, m | Def (NN), IDEW | R^{2}, NSE, Flow statistics | Little difference between two methods for R^{2}, but IDEW superior over Def for NSE, especially in smaller subbasins (below 2500 km^{2}) |

6 | Ruelland2008 [25]; Table 5, Figure 10 | Bani (ML, CI, BF) | 100,000 | 13 | 0.13 | Hydrostrahler | 6 | 7 | 10d | TP, IDW, Spline, OK | NSE, VE, PE | The best results in terms of selected criteria were obtained for IDW, intermediate for TP and OK and the worst for Spline; all methods underestimated flood peaks |

7 | Shen2013 [27]; Tables 2 and 3, Figure 3a,b | Daning (CN) | 4,426 | 19 | 4.3 | SWAT | 7 | 3 | m | Def (NN), TP, IDW, Dis-Kriging, CoKriging | NSE, flow statistics | All methods showed an improvement over the Default method in terms of NSE (the highest for CoKriging); all methods underestimate most of flow characteristics |

8 | Wagner2012 [28]; Tables 4 and 5, Figure 8 | Mula and Mutha (IN) | 2,036 | 16 | 7.9 | SWAT | 21 | 4 | d | RIDW_{x}, RIDW_{trmm}, RK_{x}, RK_{trmm} | NSE, PBIAS, flow statistics | RIDWTrmm and RKTrmm outperform RIDWX and RKX in terms of NSE and PBIAS; RKX overestimates runoff and does not reproduce right timing of floods in contrast to RKTrmm |

^{2}—coefficient of determination, RMSE—root mean squared error; VE—volume error, PE—relative peak error; PBIAS—percentage bias. Methods: Def—Default, NN—Nearest Neighbour, TP—Thiessen Polygons, IDW—Inverse Distance Weighted, OK—Ordinary Kriging, MLR—Multiple Linear Regression, CMLR—Climatological Multiple Linear Regression LWP—Locally Weighted Polynomial Regression, IDEW—Inverse Distance and Elevation Weighting, RIDW

_{x}, RIDW

_{trmm}—Regression Inverse Distance, RK

_{x}, RK

_{trmm}—Regression Kriging methods.

^{2}, whereas three others have drainage areas in the range 42,620–1,800,000 km

^{2}. The first group, containing our study, can thus be classified as meso-scale applications, and the second one as macro-scale applications. We used these values as well as the numbers of precipitation stations used for interpolation in order to estimate the values of mean station density indicator for each case: these varied considerably between studies, from 0.05 stations per 1000 km

^{2}[22] to 20.6 stations per km

^{2}[26]. In this respect, our study (9.3 stations per km

^{2}) belongs to the group of higher station densities together with [26,27,28]. SWAT was the most frequently used hydrological model (four cases), whereas other models were: semi-distributed SLURP (Semi-distributed Land Use-based Runoff Processes), Hydrostrahler, and fully-distributed PRMS (Precipitation Runoff Modeling System). The length of simulation period was the longest in our study (30 years) and exceeded almost by a factor of two the mean value across all other studies. Since calibrated parameter values are very sensitive to climatic conditions and those calibrated for dry and short periods might not be suitable for simulating the opposite conditions [34,56], we conclude that study designs for this type of assessments should contain longer simulation periods (20–30 years) rather than only several years. The number of flow gauging stations varied between 1 and 29. Our study used 11 stations, however when referring the number of stations to catchment unit area, our study had the largest station density. Temporal aggregations were diversified, however daily and monthly aggregation used in our study were the most frequent. The range of applied interpolation schemes was also wide and included also regression-based methods [26,28] and more sophisticated than OK geostatistical methods (Dis-kriging, Cokriging; [27]). Only three studies including ours have applied the model default (usually NN) method that served as a reference for other more complex methods. The assessment criteria for method evaluation were non-uniform as well. However, all studies apart from Haberlandt et al. [22] used NSE as one of objective functions. In five cases, various flow statistics (e.g., mean flow and extreme flows) were reported and compared to measured values. None of the studies apart from ours used bR

^{2}.

^{2}and 50,000 km

^{2}, respectively. Figure 10 A for daily NSE shows a relationship in the opposite direction: OK outperforms TP and IDW mainly in catchments with larger drainage areas. This discrepancy can be explained by a considerable scale and station density differences between our study and aforementioned studies (cf. Table 4), i.e., the relationships found by Haberlandt et al. [22] and Masih et al. [23] are not necessarily valid for smaller spatial scales and/or for catchments with significantly higher station densities.

## 4. Conclusions

^{2}, regardless of temporal aggregation. The difference between these three interpolation methods and Def was, however, spatially variable and a part of this variability was attributed to catchment properties: catchments characterised by low station density, low coefficient of variation of daily/monthly flows and a higher interpolated precipitation estimation experienced more pronounced improvement as a result of using interpolation methods. The implication of this study is that appropriate consideration of spatial precipitation variability (often neglected by model users) that can be achieved applying interpolation methods can significantly improve the reliability of model simulations across different scales. Ordinary Kriging should be considered as the optimal method, however we recommend testing various methods, as the results tend to be catchment-specific. From the practical point of view, this study identified certain circumstances (sparse precipitation gauge networks, catchments with stable flow regimes, higher precipitation estimation) under which one can expect a larger improvement in model efficiency criteria by applying precipitation interpolation schemes.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Szcześniak, M.; Piniewski, M.
Improvement of Hydrological Simulations by Applying Daily Precipitation Interpolation Schemes in Meso-Scale Catchments. *Water* **2015**, *7*, 747-779.
https://doi.org/10.3390/w7020747

**AMA Style**

Szcześniak M, Piniewski M.
Improvement of Hydrological Simulations by Applying Daily Precipitation Interpolation Schemes in Meso-Scale Catchments. *Water*. 2015; 7(2):747-779.
https://doi.org/10.3390/w7020747

**Chicago/Turabian Style**

Szcześniak, Mateusz, and Mikołaj Piniewski.
2015. "Improvement of Hydrological Simulations by Applying Daily Precipitation Interpolation Schemes in Meso-Scale Catchments" *Water* 7, no. 2: 747-779.
https://doi.org/10.3390/w7020747