# Spatiotemporal Scaling Effect on Rainfall Network Design Using Entropy

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}per station on mountainous islands with irregular precipitation, one of which is Taiwan, the selected case in this study.

## 2. Methodology

#### 2.1. Spatiotemporal Scale

^{2}. A total of 346, 45, and 20 grids were created, respectively. The center of each grid was assigned the location of a candidate rain gauge station. At each of the three different spatial scales, hourly records for typhoon events, monthly, six dry and wet months and annual rainfall were individually analyzed. Hourly data are used to investigate fluctuations of short duration for extreme events while the monthly, six dry and wet months and annual data depict the possible seasonal and annual trends or variations. Therefore, a total of fifteen combinations for different conditions are evaluated.

#### 2.2. Kriging

#### 2.3. Entropy

_{i}is probability.

_{i}is the probability of event x

_{i}. Equation (3) refers only to the information state before receiving data. Thus, H(x

_{1}) measures the average amount of information. H(x

_{1}) = 0 when the event is certain (p

_{i}= 0 or 1) and there is no surprise. Because of the uniformity resulting in an inability to believe any outcome being more likely than any other, uniform distribution results correspond to maximum ignorance. Maximum entropy can be seen as a generalization of the classical principle of indifference and can be used to obtain unbiased probability assessments.

_{1}and x

_{2}. Corresponding to Equation (3), the joint entropy of two variables is [24]:

_{ijk}is the joint probability of x

_{1}, x

_{2}, and x

_{3}.

_{1}rain gauge station is examined to record rainfall data, the remaining uncertainty of the x

_{2}rain gauge station will be exhibited by the conditional entropy. The probability of x

_{2}under the influence of x

_{1}’s condition can be shown as below [24]:

_{2}|x

_{1}) is the conditional entropy of x

_{2}given x

_{1}. To find out the amount of mutual or overlapped information of the two stations, a transferable information calculation can be utilized to do so, as if using the x

_{1}rain gauge station to forecast information from the x

_{2}rain gauge station.

#### 2.4. Optimization of Network Design

_{n}|x

_{1},x

_{2},···,x

_{n−1})] can be selected to arrange the order of data overlap of all the stations, in which the station with minimum overlap is the first to be added to the network, and the station with maximum overlap is the last to be added.

_{n}|x

_{1},x

_{2},···,x

_{n−1})] to detect the critical data volume and the supposed number of stations. The coefficient k

_{m}denotes the specific value of the m-th entropy value, as compared with all entropy values of the study area; it is assumed to be used as the reveal data volume of the m-th station. Assuming n stations in the study area and a number of basic stations have been selected, and the addition of new candidate rain gauge stations is prioritized on the basis of the entropy value, the definition of k

_{m}in this study can be expressed as [24]:

_{1}, k

_{2},..., k

_{m},..., k

_{n}

_{−1}, k

_{n}< 1. Hence, in determining the number of stations in a catchment area, a threshold value ${k}_{m}^{*}$ must be set, and by setting a limit such as ${k}_{m}>{k}_{m}^{*}$, the number can be secured.

_{m}. In this study, k

_{m}is set to 0.95, which is 95% of the information. Hence, if the number of rain gauge stations in the existing network is greater than that in the candidate network, those existing stations sequenced behind the candidate station are to be eliminated; otherwise, more stations are added.

_{m}, we define the percentage of required stations reaching 95% information PRS (%) as:

#### 2.5. Study Area and Data

^{2}(Figure 1). The catchment area extends from the Mt. Jade (elevation 3952 m, the highest peak in Taiwan) to Gueitsuto (220 m). Coursing through major forestland within the Chenyulan catchment, the river flows over 41.4 km, with an average slope of 2.7%. Most of the catchment is covered by mature forests, and the geological features of this region include a complex suite of rocks, such as granite, gneiss, schist, sandstone, conglomerate, and marl. Because of differences in altitude, the climate is divided into subtropical, warm temperate, cold temperate, subfrigid, and frigid zones. The mean annual temperature ranges between a low of 4 °C (Mt. Jade) and a high of 23 °C (Jushan). The NTUEF area usually experiences enhanced rainfall, and it receives an average (1992–2012) rainfall of 2408 mm; however, the rainfall is unevenly distributed, with more than 70% of it occurring between May and September. The annual rainfall increases from north to south and east to west, and consists of several centers (Figure 2a). During the period 1992 to 2012, the most severe rainfall event, Typhoon Morakot, which occurred 5 August to 9 August 2009, poured almost the entire average annual rainfall within a few days (Figure 2b). The largest rainfall recorded by the Alishan weather station (bottom left near the boundary in Figure 1) is 2884 mm, which caused a large-scale landslide and debris flow.

## 3. Result and Discussion

#### 3.1. Validation of Kriging Estimates

#### 3.2. Uncertainty Distributed in Space

#### 3.3 Spatial Scale Effect

#### 3.4. Temporal Scale Effect

#### 3.5. Optimal Rain Gauge Station Network of the NTUEF Area

^{2}is equivalent to 13.1 gauges, very close to the number analyzed for hourly and monthly rainfall in the 5-km grid in Table 4. More rainfall information can be obtained as the required rain gauges increased for 3- and 1-km scale. However, for efficiency, 13 and 14 candidate stations at monthly and hourly at the 5-km grid, equivalent to one-fourth of existing rain gauge stations, are enough for general use, respectively; for hydrologic design and using the prioritized network at 3- or 1-km grid, the number will double or even more. Compromising the accuracy and network density, 13 candidate stations were identified as the optimal network according to the prioritized and overlapped gauge stations across all spatiotemporal scales in Figure 8.

^{2}, rain gauge number 27, with respect to 328 km

^{2}and 50 in this study.

## 4. Conclusions

- (1)
- It exhibits different locations for first prioritized candidate rain gauges between spatiotemporal scales.
- (2)
- The effect of spatial scales is insignificant in comparison to temporal scales for network design. From the joint entropy value, the difference between hourly and monthly scales is more significant than the six dry, wet months and annual rainfall. However, the difference is significant across the spatial scale.
- (3)
- A smaller number and a lower percentage of required stations (PRS) are needed to reach stable joint entropy of long duration (six months or year) at finer spatial scale. Compromising the accuracy and network density, we suggest the optimal network design comprising of 13 candidate stations be suitable across all spatiotemporal scales.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Hackett, O.M. National water data program. J. Am. Water Work Assoc
**1966**, 58, 786–792. [Google Scholar] - Campbell, S.A. Sampling and Analysis of Rain; American Society of Testing and Materials: West Conshohocken, PA, USA, 1983. [Google Scholar]
- World Meteorological Organisation (WMO), Guide to Hydrometeorological Practices; WMO Technical Paper 82; WMO: Geneva, Switzerland, 1970.
- Markus, M.H.; Knapp, V.; Tasker, G.D. Entropy and generalized least square methods in assessment of the regional value of streamgauges. J. Hydrol
**2003**, 283, 107–121. [Google Scholar] - Cheng, K.S.; Lin, Y.C.; Liou, J.J. Rain-gauge network evaluation and augmentation using geostatistics. Hydrol. Process
**2008**, 22, 2554–2564. [Google Scholar] - Harmancioglu, N. Measuring the information content of hydrological processes by the entropy concept. J. Civ. Eng. Facul. Ege Univ
**1981**, 13–40. [Google Scholar] - Awumah, K.; Goulter, I. Assessment of reliability in water distribution networks using entropy based measures. Stoch. Hydrol. Hydraul
**1990**, 4, 309–320. [Google Scholar] - Awumah, K.; Goulter, I.; Bhatt, S.K. Entropy-based redundancy measures in water-distribution networks. J. Hydraul. Eng
**1991**, 117, 595–614. [Google Scholar] - Krstanovic, P.F.; Singh, V.P. Evaluation of rainfall network using entropy II: Application. Water Resour. Manag
**1992**, 6, 295–314. [Google Scholar] - Al-Zahrani, M.; Husain, T. An algorithm for designing a precipitation network in the southwestern region of Saudi Arabia. J. Hydrol
**1998**, 205, 205–216. [Google Scholar] - Ozkul, D.S.; Harmancioglu, N.B.; Singh, V.P. Entropy-based assessment of water quality monitoring networks. J. Hydraul. Eng
**2000**, 5, 90–100. [Google Scholar] - Mogheir, Y.; Singh, V.P. Application of information theory to groundwater quality monitoring networks. Water Resour. Manag
**2002**, 16, 37–49. [Google Scholar] - Mogheir, Y.; de Lima, J.L.M.P.; Singh, V.P. Characterizing the spatial variability of groundwater quality using the entropy theory: I. Synthetic data. Hydrol. Process
**2004**, 18, 2165–2179. [Google Scholar] - Nunes, L.M.; Cunha, M.C.; Ribeiro, L. Groundwater Monitoring Network Optimization with Redundancy Reduction. J. Water Resour. Plann. Manag
**2004**, 130, 33–43. [Google Scholar] - Masoumi, F.; Kerachian, R. Assessment of the groundwater salinity monitoring network of the Tehran region: Application of the discrete entropy theory. Water Sci. Technol
**2008**, 58, 765–771. [Google Scholar] - Mogheir, Y.; Singh, V.P.; de Lima, J.L.M.P. Spatial assessment and redesign of a groundwater quality monitoring network quality using entropy theory, Gaza Strip, Palestine. Hydrogeol. J
**2006**, 14, 700–712. [Google Scholar] - Yoo, C.; Jung, K.; Lee, J. Evaluation of rain gauge network using entropy theory: Comparison of mixed and continuous distribution function applications. J. Hydrol. Eng
**2008**, 13, 226–235. [Google Scholar] - Mogheir, Y.; de Lima, J.L.M.P.; Singh, V.P. Entropy and multi-objective based approach for groundwater quality monitoring network assessment and redesign. Water Resour. Manag
**2009**, 23, 1603–1620. [Google Scholar] - Alfonso, L.; Lobbreche, A.; Price, R. Information theory-based approach for location of monitoring water level gauges in polder. Water Resour. Res
**2010**, 46, W03528. [Google Scholar] - Alfonso, L.; Lobbrecht, A.; Price, R. Optimization of water level monitoring network in polder systems using information theory. Water Resour. Res
**2010**, 46, W12553. [Google Scholar] - Li, C.; Singh, V.P.; Mishra, A.K. Entropy theory-based criterion for hydrometric network evaluation and design: Maximum information minimum redundancy. Water Resour. Res
**2012**, 48, W5521. [Google Scholar] - Gong, W.; Gupta, H.V.; Yang, D.W.; Sricharan, K.; Hero, A.O. Estimating epistemic and aleatory uncertainty during hydrologic modeling: An information theoretic approach. Water Resour. Res
**2013**, 49, 2253–2273. [Google Scholar] - Chen, Y.C.; Wei, C.; Yeh, H.C. Rainfall network design using kriging and entropy. Hydrol. Process
**2008**, 22, 340–346. [Google Scholar] - Yeh, H.C.; Chen, Y.C.; Wei, C.; Chen, R.H. Entropy and Kriging Approach to Rainfall Network Design. Paddy Water Environ
**2011**, 9, 343–355. [Google Scholar] - Awadallah, A.G. Selecting optimum locations of rainfall stations using kriging and entropy. Int. J. Civil Environ. Eng
**2012**, 12, 36–41. [Google Scholar] - Amorocho, J.; Espildora, B. Entropy in the assessment of uncertainty in hydrologic systems and models. Water Resour. Res
**1973**, 9, 1511–1522. [Google Scholar] - Chapman, T.G. Entropy as a measure of hydrologic data uncertainty and model performance. J. Hydrol
**1986**, 85, 111–126. [Google Scholar] - Harmancioglu, N.; Yevjevich, V. Transfer of hydrologic information among river points. J. Hydrol
**1987**, 91, 103–118. [Google Scholar] - Yang, Y.; Burn, D.H. An entropy approach to data collection network design. J. Hydrol
**1994**, 157, 307–324. [Google Scholar] - Singh, V.P. The Use of Entropy in Hydrology and Water Resources. Hydrol. Process
**1997**, 11, 587–626. [Google Scholar] - Kawachi, T.; Maruyama, T.; Singh, V.P. Rainfall entropy for delineation of water resources zones in Japan. J. Hydrol
**2001**, 246, 36–44. [Google Scholar] - Mishra, A.K.; Coulibaly, P. Hydrometric network evaluation for Canadian watersheds. J. Hydrol
**2010**, 380, 420–437. [Google Scholar] - Memarzadeh, M.; Mahjouri, N.; Kerachian, R. Evaluating sampling locations in river water quality monitoring networks: Application of dynamic factor analysis and discrete entropy theory. Environ. Earth Sci
**2013**, 70, 2577–2585. [Google Scholar] - Sivapalan, M.; Grayson, R.; Woods, R. Preface Scale and scaling in hydrology. Hydrol. Process
**2004**, 18, 1369–1371. [Google Scholar] - Viney, N.R.; Sivapala, M. A framework for scaling of hydrologic conceptualizations based on a disaggregation–aggregation approach. Hydrol. Process
**2004**, 18, 1395–1408. [Google Scholar] - Serra, Y.L.; Mcphaden, M.J. Multiple Time- and Space-Scale Comparisons of ATLAS Buoy Rain Gauge Measurements with TRMM Satellite Precipitation Measurements. J. Appl. Meteorol
**2003**, 42, 1045–1059. [Google Scholar] - Singh, V.P. Entropy Theory and Its Application in Environmental and Water Engineering; John Wiley & Sons: Hoboken, NJ, USA, 2013. [Google Scholar]
- Stewart, J.B.; Engman, E.T.; Feddes, R.A.; Kerr, Y. Scaling up in Hydrology Using Remote Sensing; John Wiley & Sons: Hoboken, NJ, USA, 1996. [Google Scholar]
- Matheron, G. Traite De Geostatistique Appliqué, Tome 1; Editions Technip: Paris, France, 1962. (In French) [Google Scholar]
- Rodriguez-Iturbe, I.; Sanabria, M.G.; Bras, R.L. A geomorphoclimatic theory of the instantaneous unit hydrograph. Water Resour. Res
**1982**, 18, 877–886. [Google Scholar] - Chiles, J.-P.D.P. Geostatistics-Modeling Spatial Uncertainty; Willey: New York, NY, USA, 1999. [Google Scholar]
- Wackernagel, H. Multivariate Geostatistics; Springer-Verlag: Berlin, Germany, 2003. [Google Scholar]
- Kebaili, B.Z.; Chebbi, A. Comparison of two kriging interpolation methods applied to spatiotemporal rainfall. J. Hydrol
**2009**, 365, 56–73. [Google Scholar] - Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J
**1948**, 27, 623–656. [Google Scholar] - Shannon, C.E.; Weaver, W. Mathematical Theory of Communication; University of Illinois Press: Champaign, IL, USA, 1949. [Google Scholar]
- Kay, P.A.; Kutiel, H. Some remarks on climatic maps of precipitation. Clim. Res
**1994**, 4, 233–241. [Google Scholar] - Kutiel, H.; Kay, P.A. Effects of network design on climatic maps of precipitation. Clim. Res
**1996**, 7, 1–10. [Google Scholar]

**Figure 2.**Contour maps of (

**a**) average annual rainfall between 1992 and 2012; and (

**b**) Typhoon Morakot rainfall of 5–9 August 2009 at the NTUEF area.

**Figure 3.**Validation of monthly rainfall at three stations in Xitou Tract by Ordinary Kriging; (

**a**) Phoenix 3.8 K Station (January 2004 to September 2005); (

**b**) Liu Long Gully Station (January 2004 to September 2005); and (

**c**) Upper Station of University Gully (December 2004 to September 2005).

**Figure 4.**Entropy contour maps at (

**a**) hourly; (

**b**) monthly; (

**c**) six dry monthly; (

**d**) six wet monthly; and (

**e**) annual scale.

**Figure 5.**Variation of joint entropy for first 20 prioritized candidate gauges at (

**a**) hourly, (

**b**) monthly; (

**c**) six dry monthly; (

**d**) six wet monthly; and (

**e**) annual scale.

**Figure 6.**Variation of joint entropy vs. candidate gauge number for (

**a**) 1-km scale; (

**b**) 3-km scale; (

**c**) 5-km scale.

**Figure 7.**First 10 prioritized candidate gauges in the study area at (

**a**) 1-km scale; (

**b**) 3-km scale; and (

**c**) 5-km scale at different temporal scales.

No. | Rain Gauge Station | Elevation (m) | TM2 (m) | Hourly Rainfall for Typhoon Events (mm) | Monthly Rainfall (mm) | Annual Rainfall (mm) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Easting | Northing | Maximum | Minimum | Mean | Standard Deviation | Maximum | Minimum | Mean | Standard Deviation | Maximum | Minimum | Mean | Standard Deviation | |||

1 | AliShan | 2,413 | 230,043 | 2,600,476 | 123.0 | 0.0 | 25.8 | 25.6 | 3,346.0 | 0.0 | 336.6 | 438.1 | 5,886.7 | 2,196.5 | 4,039.2 | 1,140.9 |

2 | Mt. Jade | 3,845 | 245,030 | 2,598,435 | 64.0 | 0.0 | 14.7 | 13.4 | 2,189.9 | 0.0 | 254.8 | 299.0 | 4,705.2 | 1,702.7 | 3,058.2 | 830.3 |

3 | Xitou Nursery | 1,169 | 228,583 | 2,618,722 | 110.0 | 0.0 | 17.0 | 20.6 | 1,770.0 | 0.0 | 202.3 | 246.1 | 4,053.0 | 1,291.0 | 2,455.3 | 673.5 |

4 | Jushan-NTU | 156 | 216,693 | 2,628,383 | 145.0 | 0.0 | 10.2 | 18.1 | 1,173.0 | 0.0 | 181.0 | 214.8 | 2,821.6 | 1,355.1 | 2,221.0 | 449.4 |

5 | Shueli-NTU | 295 | 234,893 | 2,633,571 | 123.5 | 0.0 | 10.5 | 17.2 | 1,512.5 | 0.0 | 150.7 | 200.4 | 2,816.0 | 2,12.5 | 1,835.9 | 724.6 |

6 | Nemoupu-NTU | 509 | 233,987 | 2,620,868 | 125.5 | 0.0 | 11.2 | 15.8 | 1,008.0 | 0.0 | 153.3 | 175.4 | 2,805.0 | 946.0 | 1,820.9 | 491.0 |

7 | Heshe-NTU | 780 | 237,830 | 2,609,920 | 74.0 | 0.0 | 11.2 | 14.3 | 1,258.0 | 0.0 | 154.2 | 185.0 | 2,688.5 | 1,062.0 | 1,855.9 | 498.8 |

8 | Chinshueigao-NTU | 520 | 227,576 | 2,629,098 | 100.0 | 0.0 | 9.2 | 14.5 | 1,271.6 | 0.0 | 187.6 | 222.1 | 4,275.0 | 680.5 | 2,234.6 | 852.4 |

9 | Hsingouko | 2,540 | 236,749 | 2,597,543 | 112.5 | 0.0 | 16.8 | 15.9 | 2,203.0 | 0.0 | 241.7 | 294.0 | 4,524.5 | 787.0 | 2,828.7 | 1,068.2 |

10 | Dann | 1,528 | 224,672 | 2,619,646 | 75.5 | 0.0 | 8.6 | 11.2 | 945.0 | 0.0 | 180.7 | 197.9 | 3,154.0 | 773.0 | 2,088.8 | 627.1 |

11 | Jushan | 151 | 217,157 | 2,629,012 | 170.0 | 0.0 | 8.9 | 16.9 | 1,133.5 | 0.0 | 177.6 | 207.3 | 3,205.0 | 613.0 | 2,047.8 | 611.7 |

12 | Wanshian | 2,403 | 240,080 | 2,613,075 | 85.0 | 0.0 | 12.8 | 15.0 | 1,633.5 | 0.0 | 208.3 | 247.9 | 3,642.0 | 924.0 | 2,421.6 | 832.5 |

13 | Phoenix Garden | 878 | 227,485 | 2,625,117 | 141.0 | 0.0 | 11.9 | 18.4 | 1,292.0 | 0.0 | 218.2 | 235.6 | 3,671.0 | 948.0 | 2,522.5 | 741.5 |

14 | Xitou Observation | 1,771 | 229,514 | 2,617,731 | 61.0 | 0.0 | 9.2 | 9.6 | 1,053.5 | 0.0 | 192.8 | 203.2 | 3,139.0 | 909.5 | 2,219.5 | 629.0 |

15 | Long-Shen Bridge | 339 | 236,100 | 2,630,858 | 130.5 | 0.0 | 9.0 | 15.0 | 900.0 | 0.0 | 164.5 | 179.4 | 2,812.5 | 1,133.5 | 1,921.2 | 653.3 |

16 | Ji-Ji | 235 | 226,257 | 2,636,039 | 103.5 | 0.0 | 8.3 | 13.2 | 975.0 | 0.0 | 188.7 | 210.0 | 3,100.5 | 1,504.5 | 2,256.7 | 923.4 |

17 | GuanShan | 1,780 | 240,135 | 2,601,472 | 81.5 | 0.0 | 14.8 | 15.4 | 1,171.5 | 0.0 | 227.7 | 225.2 | 3,695.9 | 1,296.5 | 2,444.9 | 820.9 |

18 | Pasture | 2,677 | 237,860 | 2,597,660 | 136.0 | 0.0 | 18.3 | 17.4 | 2,383.5 | 0.0 | 304.1 | 387.7 | 5,218.8 | 1,653.5 | 3,719.5 | 1,025.3 |

19 | Shenmu Village | 1,595 | 233,125 | 2,603,668 | 91.5 | 0.0 | 16.4 | 17.0 | 2,141.5 | 0.0 | 260.1 | 330.5 | 4,649.5 | 1,653.5 | 3,114.4 | 1,664.2 |

20 | Chungshinlun | 661 | 219,839 | 2,625,192 | 63.5 | 0.0 | 9.4 | 12.6 | 1,075.0 | 0.0 | 231.0 | 264.0 | 3,682.0 | 1,554.5 | 2,731.8 | 1,468.3 |

21 | Shueli | 593 | 234,295 | 2,636,644 | 110.0 | 0.0 | 10.2 | 17.5 | 911.0 | 0.0 | 193.9 | 218.3 | 3,094.0 | 1,451.0 | 2,341.8 | 1,276.9 |

22 | Fongchiou | 1,151 | 237,647 | 2,618,491 | 84.5 | 0.0 | 11.1 | 14.3 | 1,211.0 | 0.0 | 166.9 | 210.1 | 2,938.0 | 1,088.0 | 2,021.5 | 1,114.5 |

23 | ShangAn | 781 | 236,321 | 2,625,167 | 66.0 | 0.0 | 8.6 | 12.2 | 804.5 | 0.0 | 162.0 | 190.1 | 2,914.0 | 1,193.0 | 1,973.3 | 1,074.3 |

24 | Hsin-shin Bridge | 897 | 235,680 | 2,606,957 | 96.5 | 0.0 | 14.2 | 17.2 | 1,751.5 | 0.0 | 193.9 | 266.1 | 3,277.5 | 1,291.0 | 2,425.1 | 1,297.9 |

25 | Dongpu | 887 | 241,493 | 2,606,091 | 67.0 | 0.0 | 10.2 | 11.8 | 1,307.0 | 0.0 | 169.6 | 219.2 | 2,917.0 | 1,107.0 | 2,092.9 | 1,138.7 |

26 | Siluang | 1,001 | 237,315 | 2,628,058 | 78.5 | 0.0 | 10.7 | 15.5 | 963.5 | 0.0 | 186.7 | 217.3 | 3,061.0 | 1,313.5 | 2,193.8 | 1,218.5 |

27 | Xitou office | 1,156 | 228,453 | 2,619,028 | 56.0 | 0.0 | 13.5 | 12.9 | 1,218.5 | 0.0 | 223.2 | 277.4 | 4,005.5 | 1,125.5 | 3,053.3 | 1,369.3 |

28 | TienDi | 787 | 230,728 | 2,624,199 | 53.5 | 0.0 | 11.3 | 12.6 | 1,360.5 | 0.0 | 140.3 | 266.4 | 3,122.5 | 137.0 | 2,326.5 | 973.3 |

29 | GuangHsin | 645 | 225,917 | 2,625,831 | 49.5 | 0.0 | 9.8 | 12.2 | 1,190.5 | 0.0 | 238.9 | 286.9 | 3,628.5 | 329.0 | 3,124.5 | 1,299.3 |

30 | No.3 Gully | 1,185 | 228,811 | 2,619,174 | 25.0 | 0.0 | 4.2 | 5.7 | 776.0 | 0.0 | 167.2 | 180.2 | 3,339.5 | 712.5 | 1,937.0 | 907.2 |

31 | Neihu elementary school | 772 | 227,181 | 2,623,316 | 52.0 | 0.0 | 9.8 | 11.7 | 1,214.5 | 0.0 | 201.6 | 258.7 | 3,560.5 | 901.0 | 3,090.3 | 1,157.6 |

32 | Lower University Gully | 1,197 | 227,492 | 2,618,456 | 106.0 | 0.0 | 15.1 | 16.8 | 1,663.0 | 0.5 | 255.6 | 402.7 | 3,956.0 | 222.5 | 3,334.2 | 1,207.7 |

33 | Wushio | 1,495 | 225,450 | 2,620,064 | 32.0 | 0.0 | 7.3 | 7.5 | 2,296.5 | 0.0 | 212.4 | 394.3 | 3,941.0 | 610.0 | 3,400.8 | 1,221.4 |

34 | Yashanpin | 1,390 | 233,383 | 2,611,144 | 86.0 | 0.0 | 17.5 | 19.4 | 1,872.5 | 0.0 | 269.3 | 367.1 | 4,221.0 | 2,264.5 | 3,534.0 | 1,478.1 |

35 | Alibudon | 1,208 | 235,227 | 2,609,712 | 31.5 | 0.0 | 4.8 | 7.9 | 823.5 | 0.0 | 135.8 | 162.1 | 2,876.0 | 530.5 | 1,549.0 | 780.0 |

36 | Salishian | 1,216 | 241,259 | 2,602,664 | 61.5 | 0.0 | 10.3 | 12.3 | 1,211.5 | 0.0 | 208.0 | 274.3 | 3,284.5 | 824.5 | 1,941.2 | 799.8 |

37 | Neuchangpin | 1,306 | 237,549 | 2,606,292 | 83.0 | 0.0 | 13.8 | 16.5 | 1,500.0 | 0.0 | 185.3 | 247.9 | 2,672.5 | 1,768.5 | 2,302.8 | 1,008.6 |

38 | Shenmu | 1,315 | 235,142 | 2,602,259 | 75.0 | 0.0 | 12.9 | 15.5 | 1,490.5 | 0.5 | 297.5 | 413.5 | 3,885.0 | 425.0 | 2,185.3 | 955.8 |

39 | 32-compartment | 1,823 | 240,123 | 2,602,231 | 59.0 | 0.0 | 15.6 | 13.5 | 1,714.5 | 0.0 | 223.5 | 298.9 | 3,073.0 | 2,050.5 | 2,318.0 | 1,157.2 |

40 | 30-compartment | 2,097 | 238,588 | 2,603,814 | 66.0 | 0.0 | 14.8 | 14.4 | 1,725.0 | 0.0 | 227.2 | 341.3 | 4,294.0 | 1,134.5 | 2,903.7 | 1,244.7 |

41 | 29-compartment | 2,298 | 233,408 | 2,596,924 | 80.5 | 0.0 | 20.2 | 20.7 | 2,307.5 | 0.0 | 347.0 | 466.6 | 5,450.5 | 974.5 | 3,453.3 | 1,774.8 |

42 | 20-compartment | 967 | 233,765 | 2,615,241 | 72.0 | 0.0 | 12.8 | 15.1 | 1,372.5 | 15.0 | 174.8 | 282.1 | 2,010.0 | 603.0 | 1,456.5 | 573.6 |

43 | 21-compartment | 1,280 | 231,832 | 2,618,174 | 99.5 | 0.0 | 22.3 | 24.4 | 2,243.5 | 7.0 | 296.3 | 433.4 | 2,946.0 | 2,048.5 | 2,518.5 | 1,023.4 |

44 | 22-compartment | 892 | 230,636 | 2,618,475 | 79.0 | 0.0 | 13.7 | 17.1 | 1,403.0 | 3.0 | 198.1 | 259.8 | 2,859.5 | 1,859.5 | 2,129.6 | 877.4 |

45 | 24-compartment | 1,278 | 231,635 | 2,621,701 | 107.0 | 0.0 | 18.9 | 21.2 | 2,042.0 | 0.0 | 171.1 | 339.8 | 2,820.5 | 442.0 | 1,582.8 | 797.0 |

46 | 13-compartment | 454 | 231,953 | 2,629,686 | 51.5 | 0.0 | 7.9 | 11.0 | 708.5 | 8.5 | 178.2 | 193.9 | 2,728.5 | 1,003.5 | 1,870.9 | 804.6 |

47 | 16-compartment | 1,002 | 232,038 | 2,630,932 | 71.0 | 0.0 | 10.6 | 14.9 | 303.0 | 2.5 | 114.4 | 203.5 | 1,473.5 | 508.5 | 1,058.4 | 458.3 |

48 | 17-compartment | 454 | 230,194 | 2,632,283 | 51.0 | 0.0 | 8.9 | 10.4 | 343.0 | 0.0 | 166.6 | 259.3 | 1,403.0 | 409.5 | 1,110.8 | 300.3 |

49 | 11-compartment | 1,228 | 230,931 | 2,626,757 | 60.0 | 0.0 | 12.9 | 14.5 | 915.0 | 21.0 | 216.6 | 207.2 | 2,998.0 | 1,402.0 | 2,219.9 | 929.7 |

50 | 9-compartment | 1,213 | 232,127 | 2,628,823 | 57.0 | 0.0 | 10.4 | 12.3 | 755.0 | 21.5 | 216.4 | 189.2 | 2,391.5 | 1,301.5 | 1,839.6 | 762.0 |

Typhoon | Date | Maximum Wind (m/s) | Rainfall Duration (h) | Damage (Billion, NT) |
---|---|---|---|---|

Herb | 29 July–1 August 1996 | 53 | 44 | 39.3 |

Toraji | 28–31 July 2001 | 38 | 24 | 14.7 |

Mindulle | 28 June–3 July 2004 | 45 | 72 | 6.5 |

Kalmeigi | 16–18 July 2008 | 33 | 32 | 3.4 |

Silaku | 11–16 September 2008 | 51 | 75 | 5.6 |

Marakot | 5–10 August 2009 | 40 | 96 | 47.7 |

Saola | 30 July–3 August 2012 | 38 | 42 | 16.2 |

Temporal Scale | b (Sill, mm^{2}) | a (Range Parameter, m) | Kriging Variance (mm^{2}) |
---|---|---|---|

Hour | 165 ± 292 | 40,243 ± 25,538 | 21 ± 62 |

Month | 23,529 ± 67,316 | 39,481 ± 67,316 | 3,154 ± 7,760 |

Dry six months | 43,209 ± 52,813 | 50,926 ± 22,708 | 3,198 ± 4,184 |

Wet six months | 583,324 ± 560,410 | 39,499 ± 27,171 | 64,250 ± 58,808 |

Annual | 645,623 ± 654,175 | 31,337 ± 29,685 | 104,080 ± 107,613 |

Scale | Candidate Station Number | Hour | Month | Six Dry Months | Six Wet Months | Year |
---|---|---|---|---|---|---|

1-km | 346 | 126 (36.4%) | 143 (41.3%) | 3(0.9%) | 2(0.6%) | 4 (1.1%) |

3-km | 45 | 26 (57.8%) | 28 (62.2%) | 5(11.1%) | 3(6.7%) | 4 (8.9%) |

5-km | 20 | 14 (70.0%) | 13 (65.0%) | 6(30%) | 3(15%) | 3 (15.0%) |

© 2014 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Wei, C.; Yeh, H.-C.; Chen, Y.-C.
Spatiotemporal Scaling Effect on Rainfall Network Design Using Entropy. *Entropy* **2014**, *16*, 4626-4647.
https://doi.org/10.3390/e16084626

**AMA Style**

Wei C, Yeh H-C, Chen Y-C.
Spatiotemporal Scaling Effect on Rainfall Network Design Using Entropy. *Entropy*. 2014; 16(8):4626-4647.
https://doi.org/10.3390/e16084626

**Chicago/Turabian Style**

Wei, Chiang, Hui-Chung Yeh, and Yen-Chang Chen.
2014. "Spatiotemporal Scaling Effect on Rainfall Network Design Using Entropy" *Entropy* 16, no. 8: 4626-4647.
https://doi.org/10.3390/e16084626