Northern Prairie Wildlife Research Center

Drought Severity Index in the Central United States

The PDSI is defined as

(1) |

Here, subscripts *i* and *i* - 1 indicate current and previous
months at some arbitrary time, respectively, and *PDSI*_{0} =
0. The *Z*_{i} in Equation (1), called the monthly *Z*-index,
is defined as

(2) |

where *K* is a coefficient and

(3) |

In the above, *P* is actual monthly precipitation. The terms in the
parentheses on the right-hand-side (r.h.s.) of Equation (3) combine to yield
monthly 'climatologically appropriate rainfall'. In particular, *PE*
is potential evapotranspiration, *PR* potential water recharge to soil,
and *PRO* potential runoff. Palmer (1965) used a two-layer soil model
consisting of a surface layer, 'plow layer', and an underlying layer, 'root
zone', and defined *PL* as the sum of soil water of the two layers available
for evapotranspiration. He called this term 'potential loss of soil water
to evapotranspiration'. The four coefficients in Equation (3) are defined
as

(4) |

where the overbar denotes a long-term average of the parameter values for
the *i*th month. The numerators in the above four expressions are long-term
monthly averages of evapotranspiration, soil water recharge, runoff and available
soil water amount, respectively.

Using the recursive relation of *PDSI _{i}* and

(5) |

This new formula of PDSI indicates how soil dryness/wetness anomalies of
previous months affect the PDSI value of the *i*th month. Because *c*
= 0.897 < 1 and *m* is a positive integer, the effect of previous months
on the index gradually decreases as (*i - m*) increases. Soil wetness
anomaly in the *i*th month has a full impact on the index value for that
month. The simple form of Equation (5) for *Z* and the PDSI also indicates
that the task of examining separate effects of precipitation and temperature
anomalies on the PDSI can be carried out by examining the relationship between
*Z* and anomalies of precipitation and temperature.

Rearranging the terms in Equation (3), we have

(6) |

Here, we have combined the last three terms in the parentheses on the r.h.s
of Equation (3) into term *S*. Therefore, (*P - S*) yields *actual*
available soil water for evaporation and transpiration from rainfall events
in a specified time interval, e.g. one month. The term α*PE*
is total monthly loss of soil water by evapotranspiration. When this term
is larger than the first term, the monthly rainfall is inadequate to maintain
the 'climatologically required' soil water and the soil suffers a net loss
of water by an amount of α*PE*
- (*P - S*). In this situation, the available soil water decreases to
meet evapotranspiration demand and a drought condition develops. Because *S*
is determined by topography and physical and hydraulic properties of soils
at a site and is independent of surface air temperature, it can be represented
further as a fraction of total precipitation of the month, that is, *S*
= κ*P*. In this expression,
0 ≤ κ
≤ 1, and 0 ≤
κ came from the fact that κ
defines the fraction of monthly precipitation used in runoff, recharge to
soil and contribution to the storage of the available soil water, and these
contributions are zero when monthly rainfall is zero. Mathematically, this
is justified by the fact that the undefined expression *S/P* approaches
zero when both *P* and *S* are zero.

Only *PE* in Equation (6) is dependent on air temperature. The coefficient
of this term, α, satisfies 0 <
α < 1, and is nondimensional. Applying
Thornthwaite's (1948) calculation method for *PE*, which has been used
in calculation of the PDSI (Palmer, 1965; Karl, 1986), we have *PE* =
η*T*^{a}, where
*T* is monthly mean surface air temperature in degrees Celsius, η
= 1.6(10/*I*)^{a(I)}*T*^{a(I)}
with *I* being the heat index (Thornthwaite, 1948) and *a* a function
of *I* given in Thornthwaite (1948). Note that η
has a dimension [L][K]^{-1}, where L is length and K is Kelvin, and
η*T* has a length unit similar
to the precipitation unit. Combining these terms, we can rewrite Equation
(6) as

(7) |

Differentiating Equation (7) with *T* and *P* and writing the
result in a finite difference form yields

(8) |

This equation shows effects of temperature and precipitation anomalies on
changes of *d* and, hence, the *Z*-index. In Equation (8), change
of *d* is linearly dependent on precipitation anomaly with the coefficient
(1 - κ). The dependence of *d*
on temperature anomalies is a nonlinear function of temperature itself. According
to Thornthwaite (1948), η = 0.18 in
the central United States. In a temperature range of 0-40°C, *T*^{[a(I)-1]}
ranges from 0 to 5. In a more realistic temperature range of 20-40°C for summer,
*T*^{[a(I)-1]} varies between 3.5 and 5. With these
values, the coefficient of the temperature perturbation term in Equation (8)
varies between 0.5 and 0.8. Finally, because (1 - κ)
varies in a similar value range, the variation of *d* and, hence, monthly
*Z* is about equally dependent on air temperature anomaly and rainfall
anomaly. In other words, monthly contributions of temperature and precipitation
anomalies to total PDSI are nearly equal when they have similar magnitudes
of anomalies.

Equation (8) separates the effect of temperature and precipitation on the
PDSI. It shows that positive anomalies of monthly rainfall will yield a positive
*d* perturbation. If there coexists a negative anomaly in monthly temperature,
a temperature effect will enlarge this positive anomaly of *d*. Together
these will yield a positive value of *Z* and contribute to a positive
PDSI. On the other hand, a large positive anomaly of temperature could override
the effect of a smaller positive rainfall anomaly on variation of *d*
to result in a net negative value of *Z* and cause a negative PDSI. Similarly,
for a negative monthly rainfall anomaly, the *Z*-index of the month can
still be positive if temperature is significantly cooler than the 'appropriate'
temperature corresponding to 'appropriate rainfall' of the month.

The above analysis shows the effect of monthly temperature and precipitation
anomalies on monthly *Z*-index values. The linear relationship of Equation
(5) indicates that this effect will be linearly added over a dry period to
determine the PDSI for the period.

The relationship between the PDSI and temperature and precipitation anomalies is reasonable for a drought monitoring index, because droughts develop from combined effect of both precipitation and temperature anomalies on soil water availability to plants. However, because the PDSI variations can be significantly affected by temperature anomalies and are not necessarily consistent with precipitation fluctuations, using the PDSI to interpret precipitation variations in past climates could lead to ambiguous conclusions, especially when variations of either precipitation or temperature are unknown.

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