Probabilistic Assessment of Agricultural Droughts using

Graphical Models

69th SWCS International Annual ConferenceLombard, IL | July 29, 2014

Presented by

Meenu RamadasPhD StudentLyles School of Civil Engineering Purdue University

Co-authors

Dr. Rao S GovindarajuDr. Indrajeet Chaubey

Dr. Dev NiyogiDr. C X Song

Introduction

• Agricultural drought conditions are, by and large, determined by soil moisture rather than by precipitation.

• Since water needs vary with crops, agricultural droughts in a region can be assessed better, if crop responses to soil water deficits are also accounted for in the drought index.

• A probabilistic assessment would convey the uncertainty in agricultural drought classification that popular indices (Standardized Precipitation Index, SPI; Palmer Drought Severity Index, PDSI; Standardized Precipitation Evapotranspiration Index, SPEI) do not provide.

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Objectives of the Study

To investigate agricultural drought events in Indiana in a probabilistic framework using graphical models (hidden Markov models), and compare the modeling results at different spatial and temporal resolutions.

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Study Area and Data Used

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Data Used:• Cropland Data Layer (CDL) for 2000-2010 hosted on

CropScape (Han et al., 2012) developed by the National Agricultural Statistics Service (NASS) of the United States Department of Agriculture (USDA).

• Soil moisture data, the Climate Prediction Center’s (CPC) 0.5° resolution global monthly datasets (Fan and van den Dool, 2004)- data from 1948-2010.

• Soil moisture data, daily values at 4 km spatial resolution from NASA-LIS model from 1981-2010.

• Major Crops:Corn, soybean, winter wheat, alfalfa, sorghum, pasture grass• Cultivation practices:Fallow land, crop rotation, double cropping

Source: http://www.cpc.ncep.noaa.gov/products/analysis_monitoring/regional_monitoring/CLIM_DIVS/indiana.gif

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Methodology

Estimation of Crop Moisture Stress Function ζ

Proposed by Rodriguez-Iturbe et al. (1999)5

**

*

*

1 for

- ( )( ) for ( )-

0 for

wq

ww

s s

s s tt s s t ss s

s s

Soil moisture

q=2ζ

1 

0 sw s*

Crop water stress

DroughtStatet=i-1

DroughtStatet=i

Drought Statet=i+1

ζi-1 ζi ζi+1

where,s* - soil moisture content at the level of incipient stomatal closuresw - soil moisture content at wilting pointq - measure of nonlinearity

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Methodology

Graphical model • A family of distributions that can be efficiently represented by directed or

undirected graph

Hidden Markov Models (HMMs)• The graph structure in HMMs comprises of hidden nodes in addition to

the observed nodes, such that dependencies exist between these hidden nodes.

• Outputs of the system are assumed to be dependent on a sequence of hidden states.

• In this study, the hidden nodes are the latent drought states, while the observations could be the drought indicator variable (crop water stress).

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Source: Fig.1, Sudderth et al. 2010. “Nonparametric Belief Propagation”

Methodology

Methodology

Basic Elements of HMM

Transition probabilities Emission densities-a suitable probability distribution with

parameters Initial state probabilities Constraints:

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1 2 1 1( | , , , ) ( | )t t t t tP q q q q P q q

( | )t tP q { , }B

1{ },s.t. ( ), 1, 2, ,i i P q i i K

1

1

1

1, 1 i

K

iiK

ijj

a K

{ }ijA a

Methodology

• Parameter estimation consists of determining of the HMM from the crop water stress time series.

• Continuous Gaussian HMMs are easily modeled. Closed-form expressions derived for Expectation Maximization (EM) algorithm [Rabiner, 1989].

• Crop stress values are however bounded in the range of [0,1]. Beta emission distribution was found suitable for the emission model. Derivation of new parameter estimation formula was performed from first principles of EM algorithm.

• These expressions are obtained by treating parameter estimation as a constrained optimization of Probability of Observations given the Model parameters or P(O|model), subject to constraints, and estimation formula are developed using Lagrange multipliers technique.

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{ , , }A B

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Results

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Mutual Information Statistic

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Investigating temporal dependence between crop drought states at monthly time scale using Mutual Information (MI) statistic

• The crop stress function values are standardized and categorized into drought categories, similar to SPI-based drought classification.

• The drought categories were then combined into 2, 4 and 6 bins.

• In this example, temporal dependence exists between January and February values; requiring use of hidden Markov models over mixture models.

,,

( , )( , ) ( , ) log

( ) ( )x y

x yx y x y

p x yMI X Y p x y

p x p y

11Results

12Results

Statistical interpretation of results is possible.Drought state classification uncertainties vary from location to location.

Loc. id 7

Loc. id 35

Emission PDFs for the monthly drought model at two locations, using HMM and data at(a) 4 km spatial resolution, and (b) 0.5 degree resolution

Initial probabilities at loc. id 7(a) (b)

State 1 2 3 4 State 1 2 3 4

π 1 0 0 0 π 1 0 0 0

Transition probability matrices for loc. id 7 (a) (b)

State 1 2 3 4 State 1 2 3 4

1 0.80 0.20 0 0 1 0.80 0.20 0 0

2 0.20 0.65 0.15 0 2 0.19 0.68 0.13 0

3 0 0.47 0.26 0.27 3 0 0.40 0.59 0.01

4 0 0 0.97 0.03 4 0 0 0.34 0.66

Initial and Transition State Probabilities using data at(a) 4 km spatial resolution, and (b) 0.5 degree resolution

13Results

A tridiagonal transition probability matrix was assumed

14Results

A tridiagonal transition probability matrix was assumed, and stable results obtained

Initial probabilities at loc. id 35(a) (b)

State 1 2 3 4 State 1 2 3 4

π 1 0 0 0 π 1 0 0 0

Transition probability matrices for loc. id 35 (a) (b)

State 1 2 3 4 State 1 2 3 4

1 0.79 0.21 0 0 1 0.80 0.20 0 0

2 0.21 0.61 0.18 0 2 0.19 0.70 0.11 0

3 0 0.44 0.42 0.14 3 0 0.38 0.58 0.04

4 0 0 0.48 0.52 4 0 0 0.67 0.33

Initial and Transition State Probabilities using data at(a) 4 km spatial resolution, and (b) 0.5 degree resolution

Drought state probabilities at monthly time scale, using HMM and soil moisture data at(a) 4 km spatial resolution, and (b) 0.5 degree resolution, for the period 2001-2010 at loc# 7

15Results

• At the two locations, HMM results were found to be different, transition probabilities as well as the emission pdfs varied from place to place.

• The emission pdfs for drought states evolved differently while using data of different resolutions at the same grid point. The drought classification ,as a result, changes with the data used, especially between the severe and extreme categories.

• Hence, a large number of extreme drought events are captured by the high resolution model, and not by the CPC data-based model.

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HMMs at different locations and spatial resolutions

16Results

• Comparison of drought events identified by the proposed index and indices such as SPEI, PDSI and SPI was performed.

• Given that proper definitions of corresponding SPI, SPEI and PDSI index values for each drought state in the proposed HMM framework—near normal, moderate, severe and extreme droughts—are not available, direct comparisons could not be made.

• Using the graphical model-based index, modeling of uncertainty in crop water stress-based drought analysis is possible.

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Comparison of Drought Indices

17Results

18Loc. id 7 :41.25° N, 87.25°W

18Results

Comparison of Drought Indices

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Results

Loc. id 35 :39.25° N, 85.75° W

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Comparison of Drought Indices

20Results

Drought state probabilities for Loc. id 7 at weekly time scale using HMM and data at 4 km resolution at (a) loc id 7 and (b) loc id 35 respectively, for the period Jun 1987-May 1989

JUN 1987

JUN 1987

MAY 1989

MAY 1989

JAN 1988

JAN 1988

MAY 1988

MAY 1988

JAN 1989

JAN 1989

Temporal dependencies at weekly time scales exhibited by soil moisture do not follow first order Markov property at one time step. A more complex HMM framework is needed. However, with finer temporal resolutions, information derived can be useful for planning irrigation, drought monitoring and effective management of available soil water.

Conclusions

• A probabilistic agricultural drought index based on crop water stress was formulated within a graphical model framework, with hidden states representing different drought categories

• Crop water stress was modeled using HMMs with a tridiagonal transition matrix and beta emission densities to develop a probabilistic model based on a bounded stress function.

• Drought severity category is defined differently for each location by the HMM, and hence, an averaged or aggregated assessment for a region cannot be considered accurate.

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Conclusions

• The transition trend among drought states is not similar at all places in Indiana. Results tend to be site-specific, suggesting the need for advanced regionalization studies for regional agricultural drought outlook.

• Comparison of indices indicated that many of drought events during dominant crop growing season (May-October) that were not identified by the SPI, SPEI and SC-PDSI, were revealed by the proposed index.

• Drought analysis based on weekly time series data are more useful in irrigation scheduling and soil water management. However, temporal dependencies at weekly time scale require more complex HMMs.

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THANK YOU

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