On the applicability of temporal stability analysis to raingauge network design

Figures & data

Figure 1. General layout of the study area showing the selected watersheds and location of the operative rain gauges with identification numbers listed in Table 1. The study watersheds are sub-basins of the Tiber River basin, Italy.

Table 1. Main characteristics of the selected rain gauge network. The geographical location is in Universal Transverse Mercator (UTM) coordinates determined by the WGS84 ellipsoid model. Identification (ID) numbers are as referred to in Fig. 1.

ID number	Rain gauge station	UTM33 x (m)	UTM33 y (m)	Altitude (m a.s.l.)
1	Azzano	316615	4742431	235
2	Bastardo	300489	4748742	331
3	Bastia Umbra	301377	4769716	203
4	Bevagna	307370	4757320	212
5	Cannara	303078	4763180	191
6	Carestello	299943	4795529	517
7	Casa Castalda	309715	4783398	730
8	Casigliano	294947	4732331	273
9	Cerbara	275092	4821081	310
10	Città di Castello	277643	4815738	304
11	Collepepe	288292	4756269	167
12	Compignano	278394	4758593	240
13	Foligno	310678	4758225	220
14	Forsivo	337588	4740488	963
15	Gualdo Tadino	319870	4789953	599
16	Gubbio	302789	4802329	471
17	La Cima	266480	4790970	791
18	Lago di Corbara	273640	4731014	128
19	Massa Martana	297457	4738741	328
20	Monte Cucco	316046	4804934	1087
21	Narni Scalo	298381	4713916	109
22	Nocera Umbra	320281	4776405	534
23	Norcia	345052	4740189	691
24	Orvieto	263178	4733559	311
25	Orvieto Scalo	265508	4734213	109
26	Perugia Santa Giuliana	287387	4775762	417
27	Perugia Sede	286494	4775910	345
28	Petrelle	269830	4803553	342
29	Pianello	302003	4779669	233
30	Piediluco	316637	4711503	370
31	Pierantonio	287484	4792996	239
32	Ponte Felcino	291343	4778099	205
33	Ponte Nuovo di Torgiano	290491	4765144	174
34	Ponte Santa Maria	256802	4753550	240
35	Ponticelli	252657	4757685	245
36	Ripalvella	279329	4746964	453
37	San Benedetto Vecchio	294749	4812427	729
38	San Biagio della Valle	278380	4766281	257
39	San Gemini	298275	4720301	299
40	San Silvestro	309649	4736325	381
41	Spoleto	314952	4736162	353
42	Terni	307123	4714603	123
43	Todi	288089	4740319	329
44	Umbertide	284867	4798836	305
45	Vallo di Nera	324844	4736142	307

Table 2. Main characteristics (front typology and incoming direction of front deduced from the 700 mb wind) of the selected rainfall events. R_m: ensemble-averaged cumulative areal-average rainfall depth; σ_R: the corresponding standard deviation; CV_R: the coefficient of variation; R_min and R_max: minimum and maximum cumulative areal-average rainfall depth; D: average duration of rainfall event.

	Number of rainfall events	Rm	σR	CVR	Rmin	Rmax	D
		(mm)	(mm)	(−)	(mm)	(mm)	(h)
Areal rainfall amount >10 mm
All fronts	98	22.39	17.81	0.80	10.02	137.15	25.16
Warm	33	26.71	23.20	0.87	10.02	137.15	26.39
Cold	38	20.32	13.60	0.67	10.20	91.19	23.43
Other	27	20.02	14.91	0.74	10.19	85.88	26.07
North	5	34.85	32.43	0.93	11.87	91.19	29.20
Northwest	29	19.48	7.38	0.38	11.23	42.21	24.03
West	11	16.26	5.94	0.37	10.63	28.03	23.41
Southwest	35	23.28	15.84	0.68	10.02	85.88	24.40
South	11	20.05	9.35	0.47	11.41	44.28	30.41
Southeast	6	38.08	49.36	1.30	10.38	137.15	25.33
Northeast	1	12.26	–	–	–	–	24.50
Areal rainfall amount ≤10 mm
All fronts	53	6.86	2.18	0.32	0.85	9.72	16.54
Warm	15	6.49	1.86	0.29	3.00	9.14	15.10
Cold	17	6.67	2.23	0.34	2.30	9.41	18.21
Other	21	7.29	2.37	0.32	0.85	9.72	16.21
North	5	6.28	2.10	0.33	3.51	8.39	12.50
Northwest	9	7.39	2.97	0.40	0.85	9.68	16.39
West	7	6.15	2.25	0.37	2.30	8.65	18.14
Southwest	26	7.13	2.00	0.28	3.00	9.72	16.63
South	4	6.80	1.93	0.28	5.05	8.67	17.63
Southeast	1	4.38	–	–	–	–	21.00
Northeast	1	5.71	–	–	–	–	15.50

Table 3. Relative identification number associated with the rain gauges operative in each watershed. The corresponding identification number used in Fig. 1 is also given.

Watershed	Rain gauge station	ID number	Relative ID number
Alto Tevere	Cerbara	9	1
	Città di Castello	10	2
	La Cima	17	3
	Petrelle	28	4
	Pierantonio	31	5
	Ponte Felcino	32	6
	Ponte Nuovo di Torgiano	33	7
	San Benedetto Vecchio	37	8
	Umbertide	44	9
Chiascio	Bastia Umbra	3	1
	Carestello	6	2
	Casa Castalda	7	3
	Gualdo Tadino	15	4
	Gubbio	16	5
	Monte Cucco	20	6
	Pianello	29	7
Topino-	Azzano	1	1
Marroggia	Bevagna	4	2
	Cannara	5	3
	Foligno	13	4
	Nocera Umbra	22	5
	San Silvestro	40	6
	Spoleto	41	7
Paglia-Chiani	Orvieto	24	1
	Orvieto Scalo	25	2
	Ponte Santa Maria	34	3
	Ponticelli	35	4
Nestore	Compignano	12	1
	Perugia Santa Giuliana	26	2
	Perugia Sede	27	3
	Ripalvella	36	4
	San Biagio della Valle	38	5
Medio Tevere	Bastardo	2	1
	Casigliano	8	2
	Collepepe	11	3
	Lago di Corbara	18	4
	Massa Martana	19	5
	Todi	43	6
Nera	Forsivo	14	1
	Narni Scalo	21	2
	Norcia	23	3
	Piediluco	30	4
	San Gemini	39	5
	Terni	42	6
	Vallo di Nera	45	7

Figure 2. Comparison of the cumulative rainfall measured in the different stations of the Alto Tevere watershed for all the events that occurred between 29 November 2005 (event 1) and 7 December 2007 (event 30).

Figure 3. Comparison of the areal-average cumulative rainfall depth associated with the different selected watersheds for all the events that occurred between 4 May 2007(event 1) and 1 March 2009 (event 30).

Figure 4. Mean relative differences for each rain gauge (◊), ordered in each watershed by increasing value, with standard deviation (vertical bar): (a) Alto-Tevere, (b) Chiascio, (c) Topino-Marroggia, (d) Paglia-Chiani, (e) Nestore, (f) Medio Tevere, and (g) Nera. Labels refer to the relative identification numbers shown in Table 3 and circles identify stations with the lowest values of γ_i (see Table 5).

Figure 5. Map of the mean and standard deviation ${\bar{\delta }}_i$ and σ(δ_i), respectively, obtained by the relative differences approach for each rain gauge operative in the study area.

Figure 6. Spearman rank correlation coefficient (r_S, see text) for rainfall frontal systems observed during a 10-year period in the Alto Tevere watershed for events with spatially averaged cumulative rainfall: (a) less than 10 mm and (b) greater than 10 mm, and (c) for all the events.

Figure 7. Cumulative frequency of significant rainfall frontal systems (with specified front typology and incoming direction deduced from the 700 mb wind) characterized by the minimum and maximum values of the spatially averaged cumulative rainfall depth (>10 mm) observed in the watershed: (a) Alto Tevere (11–13 November 2012: warm, SW; 27–28 March 2008: other, S); (b) Chiascio (10–12 November 2013: other, SW; 24–25 December 2009: warm, SW); and (c) Topino-Marroggia (25–27 November 2005: cold, N; 9–10 March 2010: other, SW). The labels indicate the relative identification numbers of the rain gauges operative in the given watershed (see Table 3).

Table 4. Mean relative difference, ${\bar{\delta }}_i$, for each rain gauge operative in each watershed and the associated standard deviation, σ(δ_i), for events with cumulative areal-average rainfall, ${\bar{R}}_j$, greater and less than 10 mm.

Watershed	Rain gauge station	${\bar{R}}_j$ ≤ 10 mm		${\bar{R}}_j$ > 10 mm
		${\bar{\delta }}_i$	σ(δi)	${\bar{\delta }}_i$	σ(δi)
Alto Tevere	Cerbara	–0.164	0.463	–0.042	0.332
	Città di Castello	–0.102	0.390	0.013	0.261
	La Cima	0.005	0.582	0.050	0.452
	Petrelle	–0.016	0.443	0.090	0.314
	Pierantonio	0.051	0.393	0.117	0.305
	Ponte Felcino	0.265	0.629	–0.005	0.326
	Ponte Nuovo di Torgiano	0.138	0.815	–0.175	0.356
	San Benedetto Vecchio	–0.198	0.384	–0.169	0.283
	Umbertide	0.021	0.328	0.121	0.303
Chiascio	Bastia Umbra	–0.036	0.731	–0.082	0.584
	Carestello	0.150	0.548	0.077	0.575
	Casa Castalda	0.036	0.347	0.006	0.564
	Gualdo Tadino	–0.209	0.402	–0.069	0.595
	Gubbio	–0.027	0.681	0.051	0.573
	Monte Cucco	0.009	0.886	0.065	0.625
	Pianello	0.077	0.344	–0.048	0.529
Topino-	Azzano	–0.095	0.554	–0.040	0.529
Marroggia	Bevagna	0.011	0.572	–0.119	0.510
	Cannara	0.024	0.642	0.000	0.541
	Foligno	–0.015	0.569	0.029	0.561
	Nocera Umbra	0.046	0.673	–0.014	0.594
	San Silvestro	–0.010	0.692	0.049	0.544
	Spoleto	0.040	0.670	0.094	0.563
Paglia-Chiani	Orvieto	0.096	0.419	0.109	0.252
	Orvieto Scalo	–0.073	0.346	–0.021	0.231
	Ponte Santa Maria	0.005	0.420	–0.040	0.249
	Ponticelli	–0.028	0.436	–0.049	0.297
Nestore	Compignano	–0.150	0.513	–0.064	0.242
	Perugia Santa Giuliana	0.057	0.581	0.020	0.227
	Perugia Sede	0.023	0.626	–0.063	0.226
	Ripalvella	0.248	0.722	0.242	0.411
	San Biagio della Valle	–0.178	0.572	–0.136	0.160
Medio Tevere	Bastardo	0.046	0.752	0.008	0.627
	Casigliano	0.015	0.550	0.143	0.568
	Collepepe	0.019	0.610	0.001	0.542
	Lago di Corbara	0.039	0.842	–0.091	0.519
	Massa Martana	–0.017	0.560	–0.049	0.526
	Todi	–0.101	0.557	–0.012	0.501
Nera	Forsivo	–0.247	0.430	–0.216	0.287
	Narni Scalo	0.066	0.517	0.070	0.367
	Norcia	–0.269	0.416	–0.247	0.333
	Piediluco	0.289	0.576	0.239	0.323
	San Gemini	0.243	0.530	0.130	0.344
	Terni	–0.091	0.369	–0.029	0.270
	Vallo di Nera	0.010	0.549	0.053	0.321

Table 5. Index of accuracy, γ_i, expressed (see Equation (6)(6) ${\gamma }_i=\sqrt{{{\bar{\delta }}_i}^2+{\sigma }^2\left({\delta }_i\right)}$(6) ) through the mean relative difference and associated standard deviation, for each rain gauge operative in each watershed. All: all frontal systems considered (see Fig. 4).

		γi
Watershed	Rain gauge station	All	R≤ 10 mm	R> 10 mm
Alto Tevere	Cerbara	0.395	0.334	0.491
	Città di Castello	0.317	0.261	0.404
	La Cima	0.501	0.455	0.582
	Petrelle	0.371	0.327	0.444
	Pierantonio	0.351	0.327	0.397
	Ponte Felcino	0.479	0.326	0.683
	Ponte Nuovo di Torgiano	0.581	0.397	0.827
	San Benedetto Vecchio	0.368	0.329	0.432
	Umbertide	0.326	0.326	0.329
Chiascio	Bastia Umbra	0.541	0.589	0.732
	Carestello	0.476	0.580	0.568
	Casa Castalda	0.305	0.564	0.349
	Gualdo Tadino	0.463	0.599	0.454
	Gubbio	0.479	0.575	0.682
	Monte Cucco	0.665	0.628	0.886
	Pianello	0.317	0.531	0.352
Topino-	Azzano	0.542	0.531	0.562
Marroggia	Bevagna	0.542	0.523	0.573
	Cannara	0.582	0.541	0.643
	Foligno	0.564	0.562	0.569
	Nocera Umbra	0.625	0.594	0.675
	San Silvestro	0.606	0.546	0.692
	Spoleto	0.611	0.571	0.672
Paglia-Chiani	Orvieto	0.345	0.274	0.430
	Orvieto Scalo	0.287	0.232	0.354
	Ponte Santa Maria	0.330	0.252	0.420
	Ponticelli	0.361	0.301	0.436
Nestore	Compignano	0.510	0.251	0.534
	Perugia Santa Giuliana	0.509	0.228	0.584
	Perugia Sede	0.569	0.234	0.626
	Ripalvella	0.638	0.477	0.763
	San Biagio della Valle	0.528	0.210	0.599
Medio Tevere	Bastardo	0.671	0.627	0.753
	Casigliano	0.573	0.586	0.551
	Collepepe	0.565	0.542	0.611
	Lago di Corbara	0.649	0.527	0.842
	Massa Martana	0.538	0.528	0.560
	Todi	0.523	0.502	0.566
Nera	Forsivo	0.416	0.360	0.496
	Narni Scalo	0.434	0.374	0.521
	Norcia	0.447	0.414	0.496
	Piediluco	0.500	0.402	0.645
	San Gemini	0.458	0.368	0.583
	Terni	0.317	0.272	0.380
	Vallo di Nera	0.490	0.325	0.549

Figure 8. Cumulative rainfall depth represented by different combinations of measurements selected on the basis of the lowest values of the accuracy index given in Table 5 vs mean areal cumulative rainfall, for the events with depth (a) less than and (b) greater than 10 mm. Nera watershed.

Table 6. Coefficient of determination, R², associated with each watershed for the use of different combinations of representative stations defined through the temporal stability analysis. Combination 1 refers to the station with the lowest value of γ_i (see Table 5); combination 2 refers to the first and second lowest values of γ_i; and combination 3 refers to the first, second and third lowest values of γ_i. Three groups of events put together on the basis of the cumulative areal-average rainfall depth, ${\bar{R}}_j$, are considered.

Watershed	Combination 1	Combination 2	Combination 3
All rainfall events
Alto Tevere	0.947	0.977	0.984
Chiascio	0.917	0.921	0.967
Medio Tevere	0.943	0.958	0.976
Nera	0.812	0.957	0.965
Nestore	0.968	0.990	0.993
Paglia-Chiani	0.947	0.994	0.992
Topino-Marroggia	0.911	0.963	0.955
Average	0.921	0.966	0.976
${\bar{R}}_j$ ≤ 10 mm
Alto Tevere	0.527	0.761	0.837
Chiascio	0.594	0.662	0.723
Medio Tevere	0.559	0.704	0.838
Nera	0.440	0.637	0.745
Nestore	0.418	0.785	0.836
Paglia-Chiani	0.584	0.953	0.918
Topino-Marroggia	0.523	0.790	0.805
Average	0.521	0.756	0.815
${\bar{R}}_j$ > 10 mm
Alto Tevere	0.941	0.974	0.982
Chiascio	0.902	0.906	0.962
Medio Tevere	0.930	0.948	0.971
Nera	0.762	0.952	0.957
Nestore	0.967	0.989	0.992
Paglia-Chiani	0.943	0.994	0.991
Topino-Marroggia	0.887	0.952	0.944
Average	0.905	0.959	0.971

Figure 9. Mean areal cumulative rainfall depth approximated in each watershed by the more representative station vs its value derived from all the operative stations, for all observed frontal systems: (a) Alto-Tevere, (b) Chiascio, (c) Topino-Marroggia, (d) Paglia-Chiani, (e) Nestore, (f) Medio Tevere, and (g) Nera. The determination coefficients, R², are also reported.

Figure 10. Mean areal cumulative rainfall depth approximated in each watershed by the two more representative stations vs its value derived from all the operative stations, for all observed frontal systems: (a) Alto-Tevere, (b) Chiascio, (c) Topino-Marroggia, (d) Paglia-Chiani, (e) Nestore, (f) Medio Tevere, and (g) Nera. The determination coefficients, R², are also reported.

Figure 11. Map of the study area with the more representative rain gauge stations. Identification numbers are listed in Table 1.

Figure 12. Comparison of hyetographs deduced by all the gauges operative in a watershed and derived from the two best stations defined through the temporal stability analysis: (a) Alto Tevere, and (b) Chiascio.