Supplementary Material to
“Climate change and uncertainty assessment over a hydroclimatic transect
of Michigan”
Jongho Kim1 Valeriy Y. Ivanov1,2 and Simone Fatichi2
1
Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor,
MI
2
Institute of Environmental Engineering, ETH Zürich, Zürich, Switzerland
January 8, 2014
Corresponding author: Valeriy Y. Ivanov, Department of Civil and Environmental
Engineering, University of Michigan, Ann Arbor, MI 48103, tel: 734-763-5068, email:
ivanov@umich.edu.
S.1. Uncertainty bounds for cases assuming cross-correlation among 170 PDFs of FOC
Seven different cases with various cross-correlation assumptions among the 170
Factors of Change (FOCs) are tested to illustrate approximate dependence of the downscaling
uncertainty on the type assumption. The Case 1 refers to classifying the factors of change into
7 groups and assuming a specific cross-correlation structure: a complete dependence for the
factors of change within each group and a complete independence among FOC’s of different
groups [Fatichi et al., 2013]. The Case 2 refers to the asssumption of complete dependence
between all 170 FOCs, and choosing factors of change with varying percentiles from 1 to 99.
For example, as one sub-case, the median (the 50th percentile) in the distributions of all
FOCs can be selected, which was one particular scenario used in the study by Fatichi et al.,
[2011]. The cases 3 to 7 are similar to the case 2 except that FOC’s of temperature are fixed
as the 5, 25, 50, 75, and 95 percentiles and are independent from FOC’s of precipitation. For
a given fixed temperature FOC, all precipitation FOCs are assumped to be entirely correlated
with varying percentiles from 1 to 99. A description and the number of simulations used are
summarized in Table S.1.
All of the generated by AWE-GEN time-series corresponding to a given set of FOC’s
are 50-year long. The statistics of mean, variance, skewness, and non-precipitation frequency
are computed for each time-series. The specified number of simulations for each case is the
number of ensemble members used for calculating the uncertainty bounds in Fig. S.5. For
example, the Case 1 has 1,000 ensemble members, while the rest of the cases have 99
samples, from which the uncertainty bounds corresponding to the 5th and 95th percentiles are
computed.
Fig. S.5 presents the results of analysis of future time series of precipitation and
temperature, reflecting both the projection uncertainty of GCM results and the variability
introduced by the weather generator. Since the fully-independent case with 1,000 samples
(Case 1) is the most comprehensive, the rest of the cases are compared with the Case 1. For
all statistics of precipitation, the degree of variations among all of the cases is well preserved,
regardless of the employed assumptions of correlation. Even though the correlation structure
among 170 FOCs is unknown, the potential ranges of variation are well captured by any of
the cases. This implies that a certain number of simulations should be sufficient to describe
the related uncertainty. Furthermore, varying FOCs in temperature does not affect variations
in rainfall processes, while does influence the projection of temperature.
The source of uncertainty related to the unknown cross-correlation structure of factors
of change has not been fully explored (Fig. S.5) due to the difficulty of sampling multidimensional space. However, it appears to be negligible, at least for the mean statistical
properties.
Table S.1. Analyzed cases with different assumptions applied to cross-correlations among the
170 Factors of Change (FOCs).
Case
# of ensemble
members
1000
Descriptions of assumption for cross - correlation
1
FOCs among 7 groups are independent
2
All FOCs are correlated with FOCs varying from 1 to 99 %
99
3
All FOCs except for temperature are correlated; fixed temp. at 5 %
99
4
All FOCs except for temperature are correlated; fixed temp. at 25 %
99
5
All FOCs except for temperature are correlated; fixed temp. at 50 %
99
6
All FOCs except for temperature are correlated; fixed temp. at 75 %
99
7
All FOCs except for temperature are correlated; fixed temp. at 95 %
99
Table S.2. Projected changes of six indicators of Fd, GSL, DHI, DHE, HHI, and HHE for
mid- (the first row for each metric specified in the first column) and end- (the second row)
century. The changes are computed as the absolute difference between the historically
observed mean values (black dots in Fig. 9) and those corresponding to the median of each
box. The red colored numbers mean decreases while the black refer to increases.
TLD
DET
LAN
FLT
ALP
SSM
B1
A1B
A2
B1
A1B
A2
B1
A1B
A2
B1
A1B
A2
B1
A1B
A2
B1
A1B
A2
Fd
13.8
19.4
19.5
28.1
15.4
33.2
11.8
18.2
17.3
25.9
12.0
30.4
19.8
24.8
23.6
32.9
19.7
37.3
15.4
22.5
19.1
28.1
14.4
32.2
11.6
16.4
17.0
27.5
12.0
33.7
10.1
15.0
13.1
26.1
11.1
31.9
GSL
7.7
12.5
10.8
19.3
8.1
25.3
11.9
14.6
16.3
24.6
12.0
30.4
11.4
11.8
14.5
22.7
11.5
27.7
12.6
16.7
15.3
23.5
10.8
30.0
7.4
10.0
11.9
19.3
6.5
24.7
14.5
16.6
16.2
27.2
13.6
33.7
DHI
2.5
2.8
3.2
3.7
3.3
4.0
0.1
1.2
1.1
2.7
1.8
2.2
2.5
3.2
3.3
3.4
3.2
4.1
1.3
2.1
1.6
3.5
0.9
2.9
2.2
2.7
2.8
3.0
2.2
3.6
2.7
3.0
2.8
4.0
3.5
4.2
DHE
1.2
1.3
1.4
2.0
1.6
2.0
0.7
0.2
0.1
1.2
0.2
0.7
1.0
1.4
1.4
1.5
1.3
1.9
0.5
1.1
0.8
2.0
0.4
1.7
0.7
1.2
1.1
1.3
0.8
1.8
1.1
1.1
1.1
1.9
1.1
1.9
HHI
2.7
2.9
2.9
3.0
2.9
3.1
1.8
2.2
2.2
2.7
2.4
2.4
2.5
2.6
2.7
2.7
2.6
3.0
3.2
3.3
3.3
3.7
3.1
3.3
1.9
2.0
2.0
2.0
1.8
2.2
2.1
2.2
2.2
2.4
2.3
2.5
18.8
21.8
21.1
10.4
13.1
14.3
14.8
17.5
16.9
18.8
20.8
18.4
13.1
14.9
13.5
18.1
18.6
18.7
21.3
25.1
24.1
13.0
18.5
17.0
17.2
18.8
20.6
21.2
24.9
24.9
15.6
15.3
18.4
18.7
22.3
22.2
HHE
Fig. S.1. Sensitivity of precipitation extreme statistics to a threshold (top percent of unfiltered
series) defining the amount of extreme data to fit the generalized Pareto distribution (GPD)
for the location of Detroit. The 95th, 99th, and 99.9th percentiles are illustrated as blue,
black, and red colors, respectively.
Fig. S.2. The locations of meteorological stations along the SSE-NNW climatic gradient of
the state of Michigan.
Fig. S.3. Mean seasonality of precipitation and air temperature as inferred from observations
at 6 locations Toledo (TLD), Detroit (DET), Lansing (LAN), Flint (FLT), Alpena (ALP), and
Sault Ste. Marie (SSM). The same location abbreviation is used in subsequent figures.
Annual mean precipitation and temperature are specified in Table 3.
Fig. S.4. The ratio of the 5-95 percentile range for the five statistics computed from the time
series of ensemble of size N=6 (6 × 50), 10 (10 x50), 50 (50 × 50), 100 (100 × 50), and 300
(300 × 50) to the 5-95 percentile range estimated for the 500 × 50 = 25,000 year reference
simulation. The series were randomly selected from the reference simulation. The random
selection was replicated 1,000 times to provide “errorbars” that are standard deviations. The
results correspond to the A1B scenario, mid-century interval (2046-2065), and the location of
Detroit. The precipitation statistics were obtained for the 24-hour aggregation interval.
Fig. S.5. Seasonal distribution of uncertainty bounds corrsponding to the 5th and 95th
percentiles for the statistics of future time series generated by AWE-GEN for Cases detailed
in Table S.1, for the period of mid-century interval (2046-2065), the A1B scenario, and the
location of Detroit. The second subplot represents a zoomed in version of the first plot for a
better illustration. The aggregation interval of 24 hours was used for precipitation statistics.
The number of simulations specified in Table S.1 is the number of ensemble members used
for calculating the 5th and 95th percentiles.
Fig. S.6. as Fig. 4, for MID-century and B1 scenario.
Fig. S.7. as Fig. 4, for MID-century and A1B scenario.
Fig. S.8. as Fig. 4, for MID-century and A2 scenario.
Fig. S.9. as Fig. 4, for END-century and B1 scenario.
Fig. S.10. as Fig. 4, for END-century and A1B scenario.
Fig. S.11. as Fig. 4, for END-century and A2 scenario.
Fig. S.12. as Fig. 5, for MID-century and B1 scenario.
Fig. S.13. as Fig. 5, for MID-century and A1B scenario.
Fig. S.14. as Fig. 5, for MID-century and A2 scenario.
Fig. S.15. as Fig. 5, for END-century and B1 scenario.
Fig. S.16. as Fig. 5, for END-century and A1B scenario.
Fig. S.17. as Fig. 5, for END-century and A2 scenario.