developing the self-calibrating palmer drought severity index

Developing the Self-Calibrating Palmer Drought Severity Index

Is this computer science or climatology?

Steve Goddard

Computer Science & Engineering, UNL

Oct. 26th, 2007 Computer Science & Engineering, UNL

Outline

1. What is Drought? 1. What is Drought?

2. The PDSI 2. The PDSI

3. Self-Calibrating the PDSI 3. Self-Calibrating the PDSI

4. Summary4. Summary


What is Drought?


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

1960’s by W. C. Palmer, published in 1965.

• Designed to allow for comparisons over time and space.


Where is it used?


How is it calculated?

Latitude Temperature Average Temp

Estimate Moisture Demand

Moisture Departure

Estimate Potential Evapotranspiration

Available Water Holding Capacity

Precipitation

Subtract


How is it calculated?

Moisture Departure

Weighting process

Weighted Combination

Moisture Anomaly Previous PDSI

Duration Factors

Climatic Characteristic

Current PDSI


Problems with the PDSI


• Step 1: Supply Demand

More Detail on PDSI Calculations

P̂-P d =

.P as symbolized is whichlevel, moisture soil normal

a maintain to needed ionprecipitat the Calculate

ˆ

ion.precipitat actual the and

P between difference the is Departure Moisture The ˆ


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.


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.

Kd Z ⋅=


Climatic Characteristic: K

• K is calculated as follows:

i

jjj

i KKD

K ′′

=∑=

12

1

67.17

5.08.2

log5.1 10 +

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ ++

++

⋅=′i

ii

iii

i DLPRORPE

K

where


Step 3: Combine with Existing Trend

• The PDSI is calculated using the moisture anomaly as follows:

Zii ⋅+⋅=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

− 31PDSI897.0PDSI 1

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.


Self-Calibration

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

• Duration Factors

• Climatic Characteristic


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.


Duration Factors - from Palmer

Palmer calculated his duration factors by examining the relationship between the driest periods of time and the ΣZ over those periods.


Duration Factors - from Palmer

The equation for this linear relationship is:

∑=

−−=t

tii tZ 764.10236.1

31

1244

897.01

==+

=⎟⎠⎞

⎜⎝⎛

+−

bm

bmm

Let b = -10.764 and m = -1.236.

Then the duration factors can be found as follows:


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 Palmer’s process and examining extremely wet periods.


Duration Factors - Automated

Example from Madrid, NE


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.


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

Climatic Characteristic - Step 1

5.08.2

log5.1 10 +

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ ++

++

⋅=′i

ii

iii

i DLPRORPE

K

KdZ ′=′


Climatic Characteristic - Step 2

The second step adjusts for comparisons between different regions.

Z

KD

Z

iii

′′

=∑=

12

1

67.17

locations.

different nine fromaverage the is 17.67 value The 12

1∑=

′i

iiKD

• Edwards Plateau, Texas• Southern Texas• Western Kansas• Texas High Plains

• Western Tennessee• West Central Ohio• Central Iowa• Scranton, Pennsylvania

• Northwestern North Dakota


Climatic Characteristic - Redefinition

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

∑=

′12

1

67.17

iiiKD

What does this ratio really represent?


∑∑

′′KDKD

Observed Expected

∑∑

′′Kd

Kd

Average Observed

Average Expected

∑∑

′′Z

Z

Average Observed

Average Expected

Now what?



Answer: use the relationship between the ∑Z and the PDSI



PDSI Average ObservedPDSI Average Expected

What is the “expected average” PDSI?

If there is one, it would be zero.

Now what?



• 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.


• From a user’s point of view, what are the expected characteristics of the PDSI?


• 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.



• 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.



⎪⎪

⎩

⎪⎪

⎨

⎧

≥⎟⎟⎠

⎞⎜⎜⎝

⎛⋅′

<⎟⎟⎠

⎞⎜⎜⎝

⎛⋅′

=

0 if(PDSI)percentile 98

4.00

0 if(PDSI)percentile 2

4.00-

th

nd

dK

dK

K

Climatic Characteristic - Final Redefinition

Wait a second…. Isn’t K used to calculate the PDSI?

How can the PDSI be used to calculate K?


Calibration Technique

KdZ ′=′ using PDSI the Calculate 1.

PDSI. the of spercentile 98th and 2nd the using Calculate 2.

K

.K index with-Z the eRecalculat 3.

index.-Znew the using PDSI final the Calculate 4.


Calibration Technique - Summary

– Dynamically calculate the duration factors, following Palmer’s 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


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.


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.


Analysis

• How do we evaluate the Self-Calibrated PDSI?– Best way

Try to correlate the Self-Calibrated PDSI to actual conditions.

– Easy waySimply 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.


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


Statistical Analysis

Original Monthly

Self-Calibrating

Monthly

Self-Calibrating

Weekly

(max + min) > 1.0The maximum PDSI value was significantly higher than the minimum was low.

35.90% 16.03% 16.67%

(max + min) < -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%





Spatial Analysis

Percent of time the PDSI and SC-PDSI are at or above 4.0


Spatial Analysis

Percent of time the PDSI and SC-PDSI are at or below -4.0


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…


Questions

developing the self-calibrating palmer drought severity index

Documents

duration factors wet

separate duration factors

excess of moisture

moisture anomaly

shortage of moisture

moisture content

deficiency of moisture

different time periods