## Stochastic or Proababilistic Analysis

In the mathematical modeling of agricultural droughts, most often soil moisture records are taken as the basis where a time series of the soil moisture contents, X1, X2, X3,. . ., Xn is truncated at a threshold soil moisture value, X0, as shown in figure 4.1. An agricultural drought can be defined on the basis of some objective, random, probabilistic, or statistical properties or features: Figure 4.1 Variation in soil moisture content during a crop-growing season and its comparison with a threshold value, Xq. L and D refer to length and dry period, respectively.

1. A wet spell occurs when any time series value at ith instant is greater than the threshold level. The difference (Xi — Xq) is named as the soil moisture surplus (SMS) when it exceeds zero and is called a dry spell or soil moisture deficit (SMD) when it is less than zero.

2. A sequence of wet spells preceded and suceeded by a dry spell is referred to as the duration of wet period, during which there is no moisture stress and plant growth is normal. If the two successive dry spells that separate a wet period are Xi and Xj, then the duration of this wet period is equal to (j — i).

3. Similarly, if a sequence of dry spells is preceded and suceeded by a wet spell, it is then referred to as the duration of dry period, which might restrict the crop development during vegetative or reproductive phenological phases. If the two successive wet spells that separate a dry period are Xk and Xi, then the duration of this dry period is equal to (l — k).

4. If a dry spell (continuation of SMD) is followed by a wet spell, then there is a transition from the drought period to wet period (continuation of SMS) (i.e., Xi < Xj).

5. Similarly, if a wet spell is followed by a dry spell, then there is a transition from the wet period to drought period (i.e., Xi > Xj).

6. The maximum dry duration in the record of past soil moisture observations corresponds to the most critical agricultural drought period that has occurred in the history of the record site. Such a critical period is directly related to the critical phenological phases and is important to crop yield estimation.

7. The summation of water deficits during the whole drought period gives the total drought severity. This is equivalent to accumulation of soil moisture needed to offset the agricultural drought, which in turn is directly related to rainfall surplus.

8. Finally, the division of the accumulated soil moisture by drought duration can determine the average severity of agricultural drought.

It is possible to calculate almost all of the above objective drought features, provided measured soil moisture records are available. Accordingly, their statistical average, standard deviation, correlation coefficient, skew-ness, and probability distribution function (PDF) can also be determined.

In contrast, if the interest lies in drought frequency, then the probability statements can also be calculated from the same record. For instance, P(Xi > Xo) and P(Xi < X0) express simply the SMS and SMD probabilities, respectively. These basic probabilities help construct a probabilistic model that can be used to predict agricultural drought durations (Sen, 1976).

There is no procedure so far for accurately predicting the time of drought occurrence and durations or areal extent of drought. Although various subjective approaches were used in the past, they all failed. In modern times, drought estimations are sought on the basis of objective and systematic scientific procedures, and along this line the probability theory provides a convenient procedure for drought predictions. These techniques, in general, are used for depicting the quantitative relationships between the weather variables and the drought characteristics. For instance, multiple regression analysis or Monte Carlo simulation techniques are used to answer questions concerning regional and temporal drought frequencies.

The majority of drought analysis has concentrated on temporal assessments. The first classical approach to statistically analyzing droughts was evaluating the instantaneously smallest value in a measured sequence of basic variables such as soil moisture recorded at a single site (Gumbel, 1963). This method gives information on the maximum value of drought duration magnitude with a prescribed period of time such as 10, 25, 50, or 100 years. Yevjevich (1967) presented the first objective definition of temporal droughts. Applications of the above method have been performed by Downer et al. (1967), Llamas and Siddiqui (1969), Saldarriaga and Yevjevich (1970), Millan and Yevjevich (1971), Guerrero-Salazar (1973), Guerrero-Salazar and Yevjevich (1975), Sen (1976,1977,1980a) and brief descriptions have been presented by Dracup et al. (1980). Due to the analytical difficulties, regional droughts have been studied less. The first study of regional drought was by Tase (1976), who performed many computer simulations to explore various drought properties. Different analytical solutions of drought occurrences have been proposed by Sen (1980b) through random field concept. However, these studies are limited in the sense that they investigate regional drought patterns without temporal considerations.

Below, a systematic approach is presented for the calculation of temporal and regional drought occurrences by simple probability procedures. Recent improvements in statistical methods have tended to place a new emphasis on rainfall studies, particularly with respect to a better understanding of persistence (continuity of dry spells) effects (Sen, 1989, 1990).

### Temporal Drought Models

Statistical theory of runs provides a common basis for objectively defining and modeling critical drought given a time series (Feller, 1967). A constant soil moisture truncation level divides the whole series into two complementary parts: those greater than the truncation level, which are referred to as the positive run in statistics, a SMS period in the agricultural sense, and, similarly, a negative run or SMD period. Feller (1967) also gave a definition of runs based on recurrence theory and Bernoulli trials as follows.

A sequence of n events, S (success, SMS) and F (failure, SMD), contains as many S runs of length r as there are non-overlapping, uninterrupted blocks containing exactly r events S each. This definition is not convenient practically because it does not say anything about the start and end of the run (i.e., drought). In contrast, a definition of runs given by Feller (1967) seems to be most revealing for the analysis of various drought features because a run is defined as a succession of similar events preceded and succeeded by different events with the number of similar events in the run referred to as its length (figure 4.1).

Independent Bernoulli Model Truncation of a soil moisture series Xi (i = 1,2,... ,n) at a constant level yields two complementary and mutual distinct events—namely, SMS and SMD—with respective probabilities p and q (i.e., 1 — p). If the probability of the longest run-length, L (i.e., critical agricultural drought duration) in a sample size of i is equal to 1 and is denoted by Pi{L = 1}, then for sample size i = 1, one can simply deduce,

Since the occurrences of the elementary events are assumed independent from each other, the combined probabilities for i = 2 can be written as

Simply, P2{L = 0} indicates an SMD followed by another SMD. The first term on the right-hand side in P2{L = 1} represents the SMS followed by SMD, and the second term represents an SMD followed by an SMS. Finally, P2{L = 2} is the combination of SMS followed by another SMS event. It is possible to develop the same probability concepts for a soil moisture time series of length n (Sen, 1980a).

Dependent Bernoulli Model In the derivation of drought probabilities above, the occurrence of successive SMD and SMS is considered as independent from each other. However, in nature, there is a tendency of SMD to follow SMD, which implies dependence between successive occurrences. The simplest representation of dependence can be achieved by considering the relative situation of two successive time intervals. This leads to four possible outcomes as transitional probabilities, which are referred to also as conditional probability statements in the probability theory. For instance,

P(-/+) implies the probability of SMD ( —) at current time interval on the condition (given) that there is SMS (+) in the following time interval. In contrast, according to the joint probability definition, it is possible to state (1) transition from an SMS to an SMS with probability P(+/+); (2) transition from an SMD to an SMS with probability P(—/+); (3) transition from an SMS to an SMD with probability P(+/—) ; and, finally, (4) transition from a soil moisture to an SMD with probability P(—/—). Four soil moisture joint probability statements are P(+,+ ) = P(+/+)P(+); P(—,+) = P(—/+)P(+); P(+, —) = P(+/—)P( —) and P(—, —) = P(—/—)P( —). Similar to the independent Bernoulli case, there are two state probabilities: SMS P(+) and SMD P( —) probabilities. Because transition and state probabilities are independent of each other, the relationships between them can be written as:

where equation 4.7 expresses the probability of an SMS in the current time interval with its first right-hand side term as the probability of SMS P(+), in the previous time interval with its transition P(+/+) from SMS to SMS, and the second term on the right-hand side representing the transition P(+/—) from SMD in the previous time interval to SMS in the current time interval. Equation 4.8 has similar interpretations. Furthermore, due to the mutual exclusiveness of probabilities, the following sequences of probability statements are also valid. Any time interval may have either SMS or SMD cases with state probabilities P(+) or P( —), respectively, whereas transitional probabilities are valid between two successive time intervals, given that the state is in SMS in the previous interval. The derivation mechanism of agricultural drought probabilities are the same as independent Bernoulli case, but a slight change of notation is necessary due to the dependent nature of the successive events. Hence, the probability of the longest critical drought duration, L, being equal to an integer value, j, in a sample size of i with a surplus state at the final stage will be denoted by P+ {L = /}. Accordingly one can write

If two successive time intervals (i = 2) are considered, it is possible to obtain the following by enumeration:

P(+) = P(+/+)P(+) + P(+/-)P(-) P(-) = P(-/+)P(+) + P(-/-)P(-)

P-{L = 0} = P-{L = 0} P(-/-) P+{L = 1} = Pf{L = 0} P(+/-) P-{L = 1} = P+{L = 1} P(-/+) P2+{L = 2} = P+{L = 1} P(+/+)

Equation 4.11 refers to the transition from SMD to SMD, whereas Equation 4.13 shows transition from SMS to SMD. Similarly, probabilities can be determined for sample size i. The numerical solutions of these equations for different sample sizes can be obtained using computer porgrams and are presented in figure 4.2, on the basis of a given SMS probability (p = 0.7). Using this graph, it is possible to read probability of critical agricultural drought of a given duration for given number of samples.

### Markov Model

Although dependence between successive SMS or SMD events is accounted for simply by the dependent Bernoulli model, in nature dependences are more persistent. To model critical agricultural droughts more realistically, the second-order Markov process is presented. This process requires three-interval basic transitional probabilities in addition to two-interval probabilities. The SMS and SMD probabilities remain as they were in the previous models. The complete description of the second-order Markov model for critical drought probability predictions is presented by Sen (1990). The new set of transitional probabilities can be defined as 8 (i = 23 = 8), which are mutually exclusive and collectively exhaustive alternatives. For example, one of the the eight alternatives will be:

P(+/ +-) = P(X, > Xo, X— > Xo, Xi-2 > Xo)

Here, P(+/+-) refers to the probability of an SMS at current interval, given that two successive intervals included an SMD and SMS, respectively. In contrast, mutual exclusiveness implies that P(+/++) + P(-/++) = 1; P(-/-+) + P(-/+-) = 1; P(+/-+) + P(-/-+) = 1 and P(+/--) + Length

Figure 4.2 Cumulative probabilities of critical agricultural drought of given duration (length in years) and return period n (in years).

Length

Figure 4.2 Cumulative probabilities of critical agricultural drought of given duration (length in years) and return period n (in years).

P( —/--) = 1. The critical agricultural drought durations for the first two samples are the same as in the dependent Bernoulli case. However, when the sample size is greater than two, the relevant drought probabilities differ. Numerical solution of these equations are achieved through the use of digital computers, and some of examplary results are shown in figures 4.3 and 4.4.

Spatio-Temporal Drought Models Almost all the studies in the literature are confined to temporal drought assessment, with few studies concerning areal coverage. But there is a significant regional dimension to agricultural drought that occurs at regional scales. Most often, droughts affect not only one country but many countries, in different regional proportions. In this section we include models that represent both spatial (regional) and temporal drought behaviors simultaneously.

In regional studies, clustering of dry spells in a region will be referred to Length

Figure 4.3 Critical drought duration distribution for different sample sizes at P(+/+ +) = 0.40.

Length

Figure 4.3 Critical drought duration distribution for different sample sizes at P(+/+ +) = 0.40. Figure 4.4 Expectation of the critical drought duration for different sample sizes and P(+/+ +) = 0.8.

Figure 4.4 Expectation of the critical drought duration for different sample sizes and P(+/+ +) = 0.8.

as "drought area" and clustering of the wet spell as "wet area." Here we explain two different regional drought models. The first model relies on probabilities of dry and wet areas, spell probabilities, pr and qr, respectively. Because these two events are mutually exclusive, pr + qr = 1. This model assumes that once a subarea of agricultural land is hit by a dry spell, it remains under this state in the subsequent time instances. Therefore, as time passes, the number of dry spells hitting the subareas increases steadily until the whole region comes under the influence of drought. Such a regional model has been referred to as regional persistence model (Sen, 1980b). The application of this model is convenient for agricultural droughts in arid and semiarid regions where long drought periods occur.

The second model takes into account the regional as well as the temporal occurrence probabilities of wet and dry spells. The probabilities of temporal SMS (pt) and SMD (qt) are mutually exclusive, and therefore, pt + qt = 1. In this model, in an already drought-stricken area, subareas are subject to temporal drought effects in the next time interval. This model is also known as multiseasonal model because it can be applied for a duration that may include several dry and wet periods. Since agriculture is a seasonal activity, this seasonal model is suitable for agricultural drought modeling.

Regional Drought Modeling Let an agricultural land be divided into m mutually exclusive subareas, each with the same chance of spatial and temporal drought. The Bernoulli distribution theory can be used to find the extent of drought area, Ad, during a time interval, At. The probability of n1 subareas affected by drought can be written according to Bernoulli distribution as (Feller, 1967)

jpn1 qm-n1 pr + qr = 1.0 [4.16] This implies that out of m possible drought-prone subareas, n1 have SMD, and hence the areal coverage of drought is equal to n1 or n\lm. For the subsequent time interval, At, there are (m — ni) drought-prone subareas. Assuming that the evolution of possible SMD and SMS spells along the time axis is independent over mutually exclusive subareas, similar to the concept in equation 4.16, it is possible to write for the second time interval (Tase, 1976; Sen, 1980b). The PDFs of areal agricultural droughts for this model are shown in figure 4.5 with parameters m = 10, pr = 0.3, pt = 0.2 and i = 1, 2, 3, 4, and 5. The probability functions exhibit almost symmetrical forms regardless of time intervals, although they have very small positive skewness.

Another version of the multiseasonal model is interesting when the number of continuously SMD subareas appear along the whole observation period. In such a case, the probability of drought area in the first time interval can be calculated using equation 4.16. At the end of the second time interval, the probability of j subareas with two successive SMDs given that already n1 subareas had SMD in the previous interval can be expressed as

This expression computes the probability of having n1 subareas to have SMD, out of which j subareas are hit by two SMDs; in other words, there are (n1 — j) subareas with one SMD. Hence, the marginal probability of continuous SMD subarea numbers is m—i f h 4-

P2At (Ad = j) = J2 PAt (Ad = k + j) ( h + M pjt qh [4.18] k=0 \ ) /

In general, for the ith time interval, it is possible to write m—j / h A

PiAt (Ad = j) = J2 P(i—1)At (Ad = k + j)(k + M pt qk [4.19]

The numerical solutions of this expression are presented in figure 4.6 for m = 10, pr = 0.3, and pt = 0.5. The probability distribution function is positively skewed. AREA Cn)

Figure 4.5 Probability of drought area for multiseasonal model (m = 10; pr = 0.3; pt = 0.2).

AREA Cn)

Figure 4.5 Probability of drought area for multiseasonal model (m = 10; pr = 0.3; pt = 0.2). Drought Parameters The global assessment of model performances can be achieved on the basis of drought parameters such as averages (i.e., expectations and variances), but for drought predictions, the PDF expressions as derived above are significant. The expected (i.e., average) number of SMDs, Ei (Ad), over a region of m subareas during time interval, ¡At, is defined as m

Similarly, the variance, Vi(Ad), of a drought-affected area is given by definition as m

The drought-stricken average area within the whole region can be determined as:

or, succinctly,

Furthermore, the percentage of agricultural drought area, P'A, can be calculated by dividing both sides by the total number of subareas, m, leading to

Figure 4.7 shows the variation in a drought-affected area with the number of SMD subareas, i, for given SMD probability, qr. Spatio-temporal drought behaviors were investigated by Sirdas and Sen (2003) for Turkey, where the drought period, magnitude, and standardized precipitation index (SPI) values were presented to depict the relationships between drought duration and magnitude.