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Earlier work describes the correlation between utility revenues and a range of drought- related metrics (Baum et al. 2018). These included the Palmer Hydrologic Drought Index (PHDI) and cumulative measures of streamflow and precipitation (e.g. annual, seasonal, monthly) using data from 372 publicly owned, geographically diverse surface water utilities spread across 78 different climate divisions in the U.S. The PHDI is a measure of drought, normalized to conditions in each climate division with possible values between -12 and 12. Positive values indicate wetter conditions, while negative values indicate drier conditions, with values above an absolute value of 3 representing severe conditions. Thus, a PHDI of -4 indicates

very severe drought. It is also important to note that a similar PHDI level measured in two different climate divisions represents the same relative deviation from historic norms, not identical drought conditions as some regions are wetter/drier than others.

From a financial perspective, utility revenues are the primary metric of interest as utility costs are typically fixed and mostly a function of debt-service (Hanemann 1997; Hughes et al. 2014), making variations in revenue a reasonable surrogate for net revenues, the primary

measure of financial stability (Zeff and Characklis 2013; Zeff et al. 2014). In this analysis, utility revenues were most strongly correlated with PHDI, with 17% of the evaluated water utilities having a Pearson correlation value above 0.5 and an additional 19% having a Pearson correlation value between 0.25 and 0.5. This suggests that 36% of the examined water utilities experience some significant level of vulnerability to drought-related financial risks, as measured by PHDI. The assumption is that the other utilities are either less financially vulnerable to drought based on higher supply-to-demand ratios (which reduces the need to conserve), or employ alternative tools for managing drought-related financial risks (e.g. drought pricing, reserve funds) that are

sufficient to reduce their risk to acceptable levels. Index insurance is then used for managing financial risks from severe drought. For a more detailed comparison, and integration, of the tools used for managing drought-related financial risks for water utilities, see Zeff and Characklis 2013 and Zeff et al. 2014.

Data on PHDI are available for all 344 climate divisions in the U.S. over a 120-year period from 1895-2015. In order to better estimate drought-related financial risks for surface water utilities across the U.S. (beyond the 372 utilities in the 78 climate divisions considered in earlier research), synthetic PHDI data are produced to extend the PHDI record (based on historic data available from the NOAA Climatic Data Center (2017)).In creating the synthetic PHDI

data, it is important to preserve the spatial and temporal autocorrelation in the observed data across all 344 climate divisions. Autoregressive models that incorporate the spatial

autocorrelation of the historic data are used to generate 100,000 years of synthetic PHDI values for each climate division. The spatial autocorrelation matrix is calculated based on the historic PHDI data across all climate divisions and then broken down, using singular value

decomposition, to generate a matrix of random values (@₇) composed of a diagonal singular matrix (D), and left and right eigenvectors (U and V) from the historic PHDI data (Onorati et al. 2013), such that

@7 = √v ∗ V ∗ U (6)

where,

U = $a&ℎ$i$h"f ("&ack $z fbz& bcibh1b{&$as v = gc"i$h"f schi%f"a ("&ack $z bcibh1"f%bs | = $a&ℎ$i$h"f ("&ack $z aciℎ& bcibh1b{&$as

These random values (@₇) are then used in the autoregressive models to generate synthetic PHDI values, with models for each climate division determined based on the Akaike Information Criterion (AIC) and the temporal partial autocorrelation function (Akaike 1974; Shibata 1976) (Figure 12).

Figure 12. Number of autoregressive lags determined to be significant for each climate division The AIC aids in model selection by selecting a lag time that minimizes model

information loss by quantifying tradeoffs between the goodness-of-fit of the model and the total number of parameters.The partial autocorrelation function indicates the influence of each lagged year on overall temporal autocorrelation, controlling for the other years. The autoregressive models for PHDI in each climate divisions are best fit using either a one-year lag (AR(1)) (180 climate divisions) or a two-year lag (AR(2)) model (164 climate divisions).

The Thomas-Fiering model has frequently been used in generating synthetic hydrologic data as it maintains the spatial and temporal autocorrelation of the historic data (Stedinger and Taylor 1982). Synthetic PHDI values are calculated using the Thomas-Fiering Lag-1 Markov model (Equation 7) for AR(1) models, and a higher lag modification of the model (Equation 8) is used for AR(2) models (Reddy 1997; Singh and Yadava 2003).

AR(1) model: 6₇ = ; + }₉>6_7m9− ;? + @₇AB1 − }₉D ₍₇₎

AR(2) model: 67 = ; + }9>67m9− ;? + }D>67mD− ;? + @7A√1 − ~D (8)

AR Model

AR(1)

AR(2)

where,

67 = average monthly PHDI for the current year ; = mean of PHDI (based on historic data) α9= regression coefficient (for a one year lag) 6_7m9= PHDI for the previous year

@₇= random value from a multivariate normal distribution incorporating covariance A = standard deviation of PHDI (based on historic data)

}D= regression coefficient (for a two year lag) 6_7mD= PHDI for two years ago

~D_{= coefficient of multiple correlation}

~D ₌ ÑÖÜ8ÑÜÜmDÑÖÜÑÜ 9mÑÖÜ

where,

=₉ = autocorrelation (for a one year lag);

=_D = autocorrelation (for a two year lag);

Synthetic PHDI data exhibit a similar distribution to historic PHDI, with mean values being nearly identical (historic 0.1501, synthetic 0.1496) and the standard deviation differing only slightly (historic 2.051, synthetic 1.999) (Figure 13). Spatial autocorrelation of synthetic PHDI values also matches historic observations, fitting a linear model with an R2 _{of 0.9987.}

Figure 13. Annual PHDI data for all 344 climate divisions for historic data (from 1895-2015) and synthetic data (100,000 years)

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

PHDI

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

PHDI