Developing the Self-Calibrating Palmer Drought Severity Index

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Developing the Self-Calibrating Palmer Drought
Severity Index

• Is this computer science or climatology?

Steve Goddard
Computer Science Engineering, UNL
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Outline
1. What is Drought?
2. The PDSI
3. Self-Calibrating the PDSI
4. Summary
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What is Drought?
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What is the PDSI?

• The PDSI is a drought index that models the
  moisture content in the soil using a supply and
  demand model.
• Is an accumulating index
• Developed during the early 1960s by W. C.
  Palmer, published in 1965.
• Designed to allow for comparisons over time and
  space.

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Where is it used?
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How is it calculated?
Latitude
Temperature
Average Temp
Estimate Potential Evapotranspiration
Available Water Holding Capacity
Estimate Moisture Demand
Precipitation
Subtract
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How is it calculated?
Weighting process
Climatic Characteristic
Previous PDSI
Weighted Combination
Duration Factors
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Problems with the PDSI
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More Detail on PDSI Calculations

• Step 1 Supply ? Demand

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Moisture Departure d

• The moisture departure represents the excess or
  shortage of moisture.
• The same value of d may have a different effect
  at different places, as well as at different
  times.
• Examples
• A shortage of 1 will matter more during the
  growing season than during winter.
• An excess of 1 will be more important in a
  desert region than in a region that historically
  receives several inches of rain each month.

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Step 2 Adjustment

• The moisture departure, d, is adjusted according
  to the climate and time of year to produce what
  is called the Moisture Anomaly, which is
  symbolized as Z.
• Z is the significance of d relative to the
  climate of the location and time of year.
• Z is calculated by multiplying d by K, which is
  called the Climatic Characteristic.

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Climatic Characteristic K

• K is calculated as follows

where
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Step 3 Combine with Existing Trend

• The PDSI is calculated using the moisture anomaly
  as follows

The values of 0.897 and 1/3 are empirical
constants derived by Palmer, and are called the
Duration Factors. They affect the sensitivity of
the index to precipitation events.
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Self-Calibration

• Improving the spatial and temporal resolution of
  the index requires automatic calibration of

• Duration Factors
• Climatic Characteristic

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Duration Factors

• The Duration Factors are the values of 0.897 and
  1/3 that are used to calculate the PDSI.
• They affect the sensitivity of the index to
  precipitation as well as the lack of
  precipitation.

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Duration Factors - from Palmer
Palmer calculated his duration factors by
examining the relationship between the driest
periods of time and the SZ over those periods.
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Duration Factors - from Palmer

• The equation for this linear relationship is

Let b -10.764 and m -1.236. Then the
duration factors can be found as follows
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Duration Factors - Wet and Dry

• Most locations respond differently to a
  deficiency of moisture and an excess of moisture.
  
• Calculate separate duration factors for wet and
  dry periods by repeating Palmers process and
  examining extremely wet periods.

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Duration Factors - Automated
Example from Madrid, NE
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Climatic Characteristic

• The climatic characteristic adjusts d so that it
  is comparable between different time periods and
  different locations.
• The resulting value is the Moisture Anomaly, or
  the Z-index.
• This process can be broken up into two steps.

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Climatic Characteristic - Step 1

• The first step adjusts the moisture departure for
  comparisons between different time periods.

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Climatic Characteristic - Step 2

• The second step adjusts for comparisons between
  different regions.

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Climatic Characteristic - Redefinition

• All of the problems with the Climatic
  Characteristic come from Step 2.

What does this ratio really represent?
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Climatic Characteristic - Redefinition
Now what?
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Climatic Characteristic - Redefinition
Answer use the relationship between the ?Z and
the PDSI
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Climatic Characteristic - Redefinition
What is the expected average PDSI?
If there is one, it would be zero.
Now what?
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Climatic Characteristic - Redefinition

• From a users point of view, what are the
  expected characteristics of the PDSI?

• Besides zero, what other benchmarks does the PDSI
  have?

Answer A user would expect extreme values to
be extremely rare. The only other benchmarks
are the maximum and minimum of the range.
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Climatic Characteristic - Redefinition

• If extreme values are truly going to be
  considered extreme, they should occur at the same
  low frequency everywhere.
• What should this frequency be?
• There should be one extreme drought per
  generation.
• Frequency of extreme droughts about 2
• 12 months of extreme drought every 50 years.

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Climatic Characteristic - Redefinition

• Consider both extremely wet and dry periods
• To make the lowest 2 of the PDSI values fall
  below -4.00, map the 2nd percentile to -4.00.
• To make the highest 2 of the PDSI values fall
  above 4.00, map the 98th percentile to 4.00.

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Climatic Characteristic - Final Redefinition
Wait a second. Isnt K used to calculate the
PDSI? How can the PDSI be used to calculate K?
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Calibration Technique
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Calibration Technique - Summary

• Dynamically calculate the duration factors,
  following Palmers method and adjusting for poor
  correlation and abnormal precipitation.
• Redefine the climatic characteristic to achieve a
  regular frequency of extremely wet and dry
  readings by mapping the 2nd percentile to -4.00
  and the 98th to 4.00

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Calibration Technique

• Effects
• The index is now calibrated for both wet and dry
  periods.
• Almost all stations have about the same frequency
  of extreme values.
• The same basic algorithm can be used to calculate
  a PDSI over multiple time periods.

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Multiple Time Periods

• Why?
• To more easily correlate the PDSI with another
  type of climate data such as tree rings, or
  satellite data.
• Valid monthly periods are divisors of 12
• Single month, 2-month, 3-month, 4-month, 6-month.
• Valid weekly periods are divisors of 52
• Single week, 2-week, 4-week, 13-week, 26-week.

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Analysis

• How do we evaluate the Self-Calibrated PDSI?
• Best way
• Try to correlate the Self-Calibrated PDSI to
  actual conditions.
• Easy way
• Simply compare the Self-Calibrated PDSI to the
  original PDSI.
• Computer Science way
• Write a few number-crunching scripts to do the
  work performing any number of statistical
  examinations of the Self-Calibrated PDSI.

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Statistical Analysis

• What to look for in the statistical analysis.
• Frequency of extreme values
• Stations that are wet more often than dry and
  vice versa.
• Average range of PDSI values

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Statistical Analysis
Original Monthly Self-Calibrating Monthly Self-Calibrating Weekly
(max min) gt 1.0 The maximum PDSI value was significantly higher than the minimum was low. 35.90 16.03 16.67
(max min) lt -1.0 The minimum PDSI value was significantly lower than the maximum was high. 16.67 1.92 4.49
The frequency with which extremely wet PDSI values (above 4.00) was between 1 and 3 13.46 91.03 91.03
The frequency with which extremely dry PDSI values (below -4.00) was between 1 and 3 2.56 87.82 87.82
Range was greater than 16 17.31 0.00 0.00
Range was greater than 12 92.31 1.92 3.28
Range was greater than 10 100.00 52.56 65.38
Range was greater than 8 100.00 99.36 100.00
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Spatial Analysis
Percent of time the PDSI and SC-PDSI are at or
above 4.0
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Spatial Analysis
Percent of time the PDSI and SC-PDSI are at or
below -4.0
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Conclusion

• The SC-PDSI is now used throughout the world.
• Increased spatial and temporal resolution than
  feasible with PDSI.
• It is more spatially comparable than PDSI
• Performs the way we believe Palmer meant his
  drought index to perform, and the way he would
  have implemented it if computers were as readily
  available as they are today.
• Well, that is what we tell the climatologist
  anyway

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Questions