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Development and validation of a leaf wetness duration model using a fuzzy logic system

Development and validation of a leaf wetness duration model using a fuzzy logic system

Agricultural and Forest Meteorology 127 (2004) 53–64 www.elsevier.com/locate/agrformet Development and validation of a leaf wetness duration model us...

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Agricultural and Forest Meteorology 127 (2004) 53–64 www.elsevier.com/locate/agrformet

Development and validation of a leaf wetness duration model using a fuzzy logic system K.S. Kima,*, S.E. Taylorb, M.L. Gleasona a

Department of Plant Pathology, Iowa State University, Ames, IA 50011, USA b Department of Agronomy, Iowa State University, Ames, IA 50011, USA Received 20 January 2004; accepted 29 July 2004

Abstract A model to estimate leaf wetness duration (LWD) based on fuzzy logic was developed and validated using hourly weather measurements at 15 sites in the midwestern U.S. from 1997–1999. The Fuzzy model, which required relatively few input variables and simplified calculations compared with physical models, was comparable to sensor measurements in overall accuracy of LWD estimation (<1 h/day error). The Fuzzy model was also more portable than the CART/SLD/Wind model, an empirical model that used the same weather variables, in that the Fuzzy model classified presence or absence of wetness more accurately than the CART/SLD/Wind model at most sites. This suggests that incorporating energy balance principles and empirical computation methods in a fuzzy logic system makes it possible to accurately estimate LWD with a relatively small number of input variables. The accuracy of the Fuzzy model may be improved using correction factors when additional relevant inputs, e.g., solar radiation, become available. Therefore, the Fuzzy model may merit further study to verify this adaptability. # 2004 Elsevier B.V. All rights reserved. Keywords: Energy balance model; Empirical model; Fuzzy logic; Leaf wetness; Dew

1. Introduction Leaf wetness duration (LWD) is an important input to many disease-warning systems (Campbell and Madden, 1990), but is seldom measured at conventional weather stations. Even when LWD measurements are available, they frequently fail to represent

* Corresponding author. E-mail address: [email protected] (K.S. Kim).

LWD with acceptable accuracy at sites distant from a weather station because of the spatial variability of wetness occurrence (Rao et al., 1998). On the other hand, existence of wetness is dependent on energy exchange at a surface and can be simulated using physical or empirical approaches. Models have therefore been developed to estimate LWD using conventional weather variables such as air temperature, water vapor pressure, and wind speed, which are relatively invariant over space (Pedro and Gillespie, 1982a,b; Gleason et al., 1994; Francl and Panigrahi,

0168-1923/$ – see front matter # 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.agrformet.2004.07.006

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1997; Chtioui et al., 1999; Madeira et al., 2002; Kim et al., 2002). Models based on energy balance principles estimate LWD by calculating latent heat flux (e.g., Pedro and Gillespie, 1982a,b; Anderson et al., 2001; Madeira et al., 2002). Since these models are mathematical representations of physical principles, they can be utilized wherever the required input weather data are available. Accuracy of their outputs, however, is quite sensitive to accuracy of the weather data inputs (Magarey, 1999). This sensitivity makes it difficult to estimate LWD accurately in practice, especially on a micrometeorological scale, because input data must be measured at every site for which LWD estimation is intended (Rao et al., 1998). Another limitation of physical models is that net radiation, a key input to these models, is difficult to calculate accurately during nights without cloud cover data, which are unavailable at most weather stations. It is possible, however, to establish empirical relationships as surrogates for certain variables used in physical models. For example, the flag leaf temperature of maize was estimated for a grower’s production field using air temperature measured at a nearby weather station (Pedro and Gillespie, 1982b; Gillespie and Barr, 1984; Rao et al., 1998). Such empirical estimates, however, could compromise spatial portability of the physical models since the empirical relationships may differ from site to site. Empirical LWD models have been developed and validated as alternatives to physical models (Gleason et al., 1994; Francl and Panigrahi, 1997; Chtioui et al., 1999; Kim et al., 2002). Francl and Panigrahi (1997) reported that a neural network model estimated presence and absence of wetness on a surface with >90% accuracy using weather data gathered from a standard weather station. The CART/SLD model, which estimated LWD more accurately than the neural network model, was developed using a statistical approach called classification and regression tree (CART) analysis that derived empirical relationships between LWD and weather variables (Breiman et al., 1984; Gleason et al., 1994). The CART/SLD/Wind model proposed by Kim et al. (2002) improved accuracy of the CART/SLD model by extrapolating wind speed to the height of a wetness sensor. The CART/SLD/Wind model requires relatively few input variables, which makes it easier to implement than

more complex models. However, the empirical nature of the CART/SLD/Wind model may limit its spatial and temporal portability. A fuzzy logic system may provide a computational framework to develop an empirical LWD model that complies with energy balance principles. According to Zadeh (1992), outcomes of fuzzy logic need not be precise, and its conclusions tend to be dispositional. Because the presence of wetness, especially dew, is estimated by identifying whether or not water vapor will condense on a surface, it is possible to estimate wetness occurrence by calculating the direction of latent heat flux rather than the scalar amount of the flux. It is therefore reasonable to utilize fuzzy logic for the empirical formulation of relationships between latent heat flux and other energy fluxes, e.g., sensible heat and net radiation, to identify wetness occurrence. The objectives of this study were to develop and validate a LWD model using a fuzzy logic system and to compare its accuracy with that of an empirical model, the CART/ SLD/Wind model.

2. Materials and methods 2.1. Weather data Hourly measurements of air temperature, relative humidity (RH), and wind speed were obtained from 15 sites in Iowa (IA), Illinois (IL), and Nebraska (NE) during May to September of 1997, 1998, and 1999 (Fig. 1). Weather data gathered in 1997 and 1998– 1999 were used as training and validation sets, respectively. Air temperature and RH were measured at 1.5-m height. Wind speed was measured at 3-m height in IA and NE and 10-m height in IL. Flat, printed-circuit, electronic wetness sensors (Model 237, Campbell Scientific, Logan, UT) were deployed at a 458 angle, facing north, at level unobstructed sites on managed turfgrass, 0.3 m above the ground. Wetness sensors were coated with latex paint to enhance sensitivity to small water droplets and to approximate the emissivity of plant leaves (Davis and Hughes, 1970; Gillespie and Kidd, 1978; Lau et al., 2000). When electrical impedance <1000 kV was detected for 30 min in an hour, the hour was counted as wet (1); otherwise, the hour was scored as dry (0).

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Fig. 1. Locations of weather stations where leaf wetness duration, air temperature, wind speed, and relative humidity were measured during May to September of 1997, 1998, and 1999.

2.2. Development of a fuzzy logic system to estimate LWD 2.2.1. Identification of variables using energy balance principles An LWD model based on fuzzy logic (Fuzzy model) consisted of variables, membership functions of each variable, and rules to determine whether or not wetness existed on a surface. Variables of the Fuzzy model were selected using the Penman equation: LE ¼

rCp VPD=ra þ DðRn  GÞ Dþg

(1)

where LE = latent heat flux (W m2), r = density of air (1.204 kg m3), Cp = specific heat of air at constant pressure (1004.67 J kg1 K1), VPD = vapor pressure deficit (kPa), D = slope of the saturated vapor pressure versus temperature curve (kPa K1), Rn = net radiation (W m2), G = ground heat flux (W m2), g = psychrometric constant (0.066 kPa K1), and ra = aerodynamic resistance for heat and water vapor (s m1). The ra can be estimated from wind speed. From the Penman equation, vapor pressure deficit (VPD), Rn, and wind speed were chosen as variables of the Fuzzy model since they are influential in determining occurrence of wetness. VPD was obtained from saturated and actual vapor pressure of the air calculated using the Clausius–Clapeyron

equation:    1 1 1  es ¼ e0 exp R 273 T þ 273

(2)

and ea ¼ RH es =100

(3)

where e0 = 0.611 kPa, R = the gas constant for water vapor (461 J kg1 K1), ea = the partial pressure of water vapor in the air (kPa), and es = the saturated vapor pressure (kPa) at T, the air temperature (8C). The Rn term of the Penman equation requires measurements of solar radiation, leaf surface temperature, and cloud cover fraction for accurate estimation of net radiation. In order to minimize the number of input variables, however, a para-Rn (pRn) variable was used to estimate net radiation as follows: pRn ¼ ea sðT þ 273Þ4  ½es sðTdew þ 273Þ4 þ ð1  es Þea sðT þ 273Þ4 

(4)

where ea and es = emissivity of the atmosphere and of the sensor surface, respectively, s = Stefan–Boltzmann constant (5.67 108 W m2 K4), and Tdew = dew point temperature (8C). The atmospheric emissivity, ea, was calculated by using the Idso–Jackson formula under the assumption of a clear sky condition

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as follows: ea ¼ 1  0:261 expð7:77 104 T 2 Þ

(5)

and es was assumed to be 0.98 (Idso and Jackson, 1969). Solar radiation was excluded in calculating pRn since most hours of wetness in the midwestern USA occur during the night, as a result of dew, and rainfall during the daytime coincides with cloud cover, which reduces the amount of incident solar radiation. To estimate outgoing radiation, surface temperature was assumed to be Tdew. Under neutral conditions, ra can be calculated as ra ¼

ln½ðz  dÞ=z0 2 k2 uðzÞ

(6)

where z = the height of wetness sensor (0.3 m), d = the zero plane displacement (0.19 m), z0 = the roughness length (0.39 m), k = von Karman’s constant (0.41), and u(z) = the wind speed at z. Since ra is a function of wind speed, wind speed was used as a surrogate for ra. Wind speed was extrapolated to the height of the sensor surface using a log profile on the assumption of neutral conditions: uðzÞ lnðz  dÞ  lnz0 ¼ uðzm Þ lnðzm  dÞ  lnz0

(7)

where z and zm = the height of wetness sensors and anemometers, respectively. The extrapolation of wind speed was performed only during the night. 2.2.2. Fuzzy logic and a fuzzy logic system Fuzzy logic originated as an extension of classic or Boolean logic (Zadeh, 1992). Classic logic assigns binary values, such as 1 for true and 0 for false, to a variable, which makes logical reasoning exact; i.e., either true or false. In fuzzy logic, by contrast, truth is represented in relative terms, referred to as degrees, which allow an approximated inference rather than an exact conclusion. In fuzzy logic, a variable can be expressed in natural language, such as ‘‘slow’’ or ‘‘fast,’’ to simulate physical processes. For example, net radiation can be represented as ‘‘low’’, ‘‘moderate’’, or ‘‘high.’’ These linguistic terms constitute a fuzzy set, which is defined by a membership function (m). A membership function assigns a degree of membership between 0 and 1 to the value of a variable (Fig. 2). Membership functions can be represented as

geometric shapes such as triangles and trapezoids (Yen and Langari, 1999). There are various ways to define a domain of a membership function, which is the set of input values that gives a valid degree of membership (>0). For instance, its domain can be determined from opinion or knowledge of an expert (Klir and Yuan, 1995). The membership function may also have different domains in different contexts. For example, the domain of a membership function of ‘‘high’’ for surface temperature will differ between a leaf and the sun. Using fuzzy logic, it is possible to express relationships between meteorological variables and occurrence of wetness in the form of rules that describe specific conditions under which wetness is likely to be present or absent. In order to infer occurrence of wetness on a surface, two groups of fuzzy rules were devised to encompass net radiation (pRn) and sensible heat flux (VPD and wind speed). For example, fuzzy rules to identify wetness occurrence included the propositions, ‘‘if VPD is high and wind speed is fast, then wetness is absent’’, and ‘‘if pRn is very low, then wetness is present.’’ Once a set of rules was annunciated in accordance with energy balance principles, the membership functions for each variable were defined heuristically. In this study, the domains of membership functions were determined from our experience to associate them with a linguistic term, e.g., high, moderate, or low. Membership functions were also defined in the form of geometric shapes (Fig. 2). The rules, their weight values, and parameters of membership functions were subsequently adjusted to yield the least root mean square error (RMSE) using an iterative training set, which included weather data measured at 15 sites in 1997. In this study, MATLAB ver. 6 (The Mathworks, Inc.; Natick, MA) was used to develop the fuzzy logic system to estimate LWD. The fuzzy rules and membership functions for each variable are shown in Table 1 and Fig. 2, respectively. To deduce occurrence or absence of wetness at a given time, each rule in the fuzzy logic system was evaluated using hourly weather data. Weather data were ‘‘crisp’’ values (e.g., 0.2 kPa of VPD), whereas a fuzzy rule consisted of a linguistic term, e.g., ‘‘VPD is moderate,’’ or ‘‘VPD is low.’’ The crisp values were converted to the linguistic term by using a membership function. The procedure that assigns the degree of

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Fig. 2. Membership functions (m) of variables included in a fuzzy logic system to estimate leaf wetness duration: (A) vapor pressure deficit (VPD; kPa); (B) wind speed (m/s); (C) pRn (W/m2), which was estimated net radiation; (D), occurrence of wetness.

each term to the crisp value of a variable is called fuzzification. The fuzzified value can vary according to the terms of each rule. For example, 0.2 kPa of VPD is fuzzified as ‘‘VPD is moderate’’ at the degree of 0.63 for the rule: ‘‘If VPD is moderate and wind speed Table 1 Fuzzy rules to infer occurrence of wetness Antecedenta c

VPD

High Moderate Moderate Low Low Low Low

a

Consequencea Wind speed

pRnc

Wetness

High Moderate Moderate Low Very low

Absent Likely absent Likely absent Likely present Likely absent Likely present Likely absent Absent Likely absent Likely present Likely present Likely present

Slow Not slow Slow Fast Moderate Moderate

Weightb

0.55 0.50 0.95 0.90 0.85 0.65 0.30 0.65 0.60 0.20 0.30 0.45

A fuzzy statement includes an antecedent, p, and a consequence, q, in form of ‘‘If p, then q;’’ p may consist of a combination of variables, e.g., VPD and Wind connected with AND operand. b Weight values corresponding to each logic statement were multiplied to calculate fuzzy numbers for the statements. c VPD = vapor pressure deficit and pRn = para-net radiation calculated from Eq. (4).

is not slow, then wetness is likely absent’’ (Table 1 and Fig. 2). In contrast, the same value of VPD is fuzzified as ‘‘VPD is low’’ at the degree of 0.37 to evaluate the rule: ‘‘If VPD is low and wind speed is slow, then wetness is likely present.’’ When an antecedent in a fuzzy rule consisted of multiple statements, combining the statements by means of logical operation was necessary to obtain a single degree of membership. In this study, the AND operator was used as follows: mA ðxÞ AND mB ðyÞ ¼ mA ðxÞmB ðyÞ

(8)

and for the NOT operator, NOT mA ðxÞ ¼ 1  mA ðxÞ

(9)

where mA(x) is a membership function of fuzzy set A for x 2 [1,1]. For example, to evaluate a statement, ‘‘VPD is moderate and wind speed is not slow,’’ the crisp values of VPD (0.2 kPa) and wind speed (2 m/s) were fuzzified as ‘‘VPD is moderate’’ at the degree of 0.63 and ‘‘wind speed is slow’’ at the degree of 0.37, respectively (Fig. 2). To obtain the degree of ‘‘wind speed is not slow’’ in our example, the NOT operator was used to yield a degree of 0.63. After performing an AND operation between two sentences in our example, the degree of our statement became 0.40. In addition, a weight value may be used to adjust a

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degree of antecedent in a fuzzy rule. In our example, the statement was associated with a rule weight (0.95), which yielded 0.38 as the degree of antecedent (Table 1). Once the degree of an antecedent was determined to have a value between 0 and 1, a conclusion of the fuzzy rule was obtained in the form of a fuzzy set using an implication operator, IF-Then. The implication operator in this study was given by: IF  Then ða; mC ðxÞÞ ¼ min ða; mC ðxÞÞ

(10)

where a = the degree of antecedent and mC(x) = membership function of consequence over x 2 [0,1]. For example, to determine a fuzzy set that represents the conclusion of the rule, ‘‘if VPD is moderate and wind speed is not slow, then wetness is likely absent,’’ the value of membership corresponding to the output term, ‘‘likely absent’’ and 0.38, which was the degree of antecedent from our previous example, were subjected to an implication operation. Taking the lower of these two values yielded a fuzzy set, as illustrated in Fig. 3A. An aggregation operation was followed to combine multiple fuzzy sets that represented conclusions of each rule. In this study, an algebraic sum, termed PROBOR, was used in the aggregation operation: PROBOR ðm1 ðxÞ; m2 ðxÞÞ ¼ m1 ðxÞ þ m2 ðxÞ  m1 ðxÞm2 ðxÞ

(11)

where m1(x) and m2(x) are the membership functions of the resulting fuzzy sets and x 2 [0,1]. As a result, a single fuzzy set was obtained as a final conclusion, which represented presence or absence of wetness during a given hour in the form of a fuzzy set. For example, two fuzzy sets that represented conclusions of the rules, ‘‘if VPD is moderate and wind speed is not slow, then wetness is likely absent,’’ and ‘‘if pRn is moderate, then wetness is likely present,’’ were combined into a single fuzzy set, as illustrated in Fig. 3C. The fuzzy set obtained from the aggregation operation was converted back to a crisp value through a defuzzification process. The crisp value of inference outcome was determined using a weighted mean given by: R m ðxÞxdx Centroid ðRÞ ¼ R R mR ðxÞdx

where mR(x) is a membership function of the aggregated fuzzy set R and x 2 [0,1]. When the defuzzified value was <0.5, the hour was classified as dry (0); otherwise, the hour was identified as wet (1). Further description of fuzzy logic and fuzzy logic systems can be found in Klir and Yuan (1995), Cox (1999), Yen and Langari (1999), and Nelles (2000). 2.3. Analysis of LWD estimates 2.3.1. The CART/SLD/Wind model Accuracy of LWD estimation by the Fuzzy model was compared with that of the CART/SLD/Wind (CART) model. Using the CART model, wetness occurrence at a given hour was determined through a series of decision points, or nodes, of the model as follows (Gleason et al., 1994): after potential dry hours were excluded by discarding hours with DPD >3.7 8C, the hours with wind speed <2.5 m/s were subjected to the following inequality: ð1:6064

Fig. 3. An example of operations of implication and aggregation with 0.2 kPa of VPD, 2 m/s of wind speed, and 50 W/m2 of pRn: (A) and (B) indicate fuzzy sets derived from fuzzy rules after an implication operation and (C) indicates the result of an aggregation operation using fuzzy sets illustrated in (A) and (B). Membership functions of each fuzzy set are indicated by m.

(12)

pffiffiffiffiffi T þ 0:0036T 2 þ 0:1531 RH

 0:4599uðzÞ DPD  0:0035T RHÞ > 14:4674

(13)

Next, after hours with RH <87.8% were discarded as potential dry hours, the remaining hours were sub-

K.S. Kim et al. / Agricultural and Forest Meteorology 127 (2004) 53–64

jected to another inequality: pffiffiffiffi ð0:7921 T þ 0:0046 RH  2:3889uðzÞ

approximately equivalent to midnight to sunrise, daytime, and sunset to midnight, respectively.

 0:0390T uðzÞ þ 1:0613uðzÞ DPDÞ > 37:0

3. Results (14)

When the relevant inequality was met, wetness was considered to be present during that hour. 2.3.2. Accuracy in LWD estimation For quantification of model accuracy in daily LWD estimation, mean error (ME) was calculated by averaging differences between measured and modelestimated LWD for 24-h periods that began at 12:00 and ended at 11:00 the next day. Mean absolute error (MAE) was also computed by averaging the absolute values of hourly errors during each 24-h period. Mean error provided a measure of the tendency to over- or underestimate LWD, whereas MAE assessed overall accuracy. In order to determine the pattern of model accuracy when wetness was associated with rainfall, 24-h periods during which rainfall was measured (0.25 mm) were classified as rain days and days with no measured rainfall as dew-only days. Accuracy in hourly prediction of wetness was compared between the CART model and the Fuzzy model. An ANOVA table was obtained using a statistical procedure proposed by Looney (1998). The values of F were computed and evaluated against the F distribution to test the null hypothesis using S-plus (Insightful Corp., Seattle). Hours classified as wet and dry were grouped and subjected to the F tests. Sunset and sunrise are transitional events in energy exchange on a surface in terms of short-wave radiation. Therefore, para-elevation of the sun (bp), which ranged from 908 to 2708, was defined to identify time periods of sunrise and sunset using the solar elevation b. For time periods during which solar elevation was increasing (morning), bp was given by: bp ¼ b

(15)

For hours at which solar elevation was declining (after solar noon), bp ¼ 180  b

59

(16)

Therefore, the time periods during which 908 < bp < 08, 08  bp  1808, and 1808 < bp < 2708 were

Overall, both Fuzzy and CART models estimated LWD with a ME of <1 h/day (Table 2). The magnitude of ME for the Fuzzy model was similar to that of the CART model, but variability (SEM) of the Fuzzy model was slightly lower. The Fuzzy model also exhibited a narrower range of ME across sites (1.4 to 2.7 h/day) than did the CART model (2.1 to 3.3 h/ day). The Fuzzy model, furthermore, had smaller MAE than did the CART model at 14 of 15 sites. During dew-only days, the ME of the Fuzzy model was slightly smaller than that of the CART model (Table 3). During rain days, in contrast, the Fuzzy model estimated LWD with an ME of 1.3 h/day whereas the CART model had an ME of <1 h/day. The Fuzzy model and the CART model estimated LWD within 1 h/day during 44.6 and 40.9% of dew-only days, respectively. During the same period, the Fuzzy model estimated LWD within 1 h/day during a higher percentage of days than did the CART model at 13 of 15 sites (Fig. 4A). The number of days on which the Fuzzy model estimated LWD within 3 or 5 h/day also exceeded that of the CART model at a majority of sites. On rain days, however, the superior accuracy of the Fuzzy model was less pronounced than on dewonly days (Fig. 4B). Nevertheless, the Fuzzy model estimated LWD within 1 h/day during a higher percentage of rain days (38.3%) than did the CART model (32.7%). The Fuzzy model identified presence and absence of hourly wetness more accurately than the CART model (Table 4). The Fuzzy model was significantly (P < 0.05) more accurate than the CART model in identifying wet or dry hours at 14 of 15 sites. The rate of misclassification of wetness occurrence peaked near sunrise and sunset for both the Fuzzy model and the CART model (Fig. 5). The Fuzzy model, however, had a smaller absolute error rate than did the CART model during those transition periods (Fig. 5A). Immediately after sunrise, the Fuzzy model and the CART model over- and underestimated wetness occurrence, respectively (Fig. 5B). Immediately after sunset, in contrast, the Fuzzy model and the CART

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Table 2 Mean error (M.E.) and mean absolute error (M.A.E.) for estimation of wetness duration (h/day) in 1998 and 1999 Na

Sites

Ames, IA Lewis, IA Nashua, IA Sutherland, IA Crawfordsville, IA Belleville, IL Bondville, IL Dixon Springs, IL Monmouth, IL St. Charles, IL Red Cloud, NE Gordon, NE O’Neill, NE Sidney, NE West Point, NE All 15 sites

MWDb (h/day)

303 307 301 306 303 196 204 225 231 218 317 322 270 322 167 3992

8.7 7.6 8.0 8.0 8.1 7.6 10.0 9.2 7.5 8.6 7.8 8.5 6.9 6.5 10.5 8.1

M.E. (h/day) (S.E.M.)c

M.A.E.d

CARTe

Fuzzye

CART

Fuzzy

0.3 1.2 3.3 1.4 1.9 2.9 1.6 0.4 0.8 0.6 1.9 2.1 2.4 1.0 1.0 0.6

0.2 1.4 2.7 1.8 0.9 2.2 1.4 1.3 0.8 0.0 1.6 1.3 2.7 0.2 1.1 0.6

3.7 4.3 4.4 4.2 3.5 4.7 3.8 3.0 4.0 3.5 4.6 3.9 5.1 3.0 2.9 3.9

3.5 3.8 3.8 3.9 2.7 3.4 3.3 2.8 3.4 2.7 3.8 3.7 4.7 2.7 3.1 3.4

(0.22) (0.25) (0.23) (0.24) (0.18) (0.24) (0.27) (0.22) (0.27) (0.22) (0.25) (0.20) (0.30) (0.20) (0.23) (0.07)

(0.22) (0.24) (0.23) (0.25) (0.17) (0.20) (0.24) (0.22) (0.24) (0.21) (0.23) (0.22) (0.29) (0.19) (0.25) (0.06)

a

Number of 24-h periods included in the analysis. MWD = Wetness duration measured during study periods. c M.E. = Mean error (S(estimated  measured)/h) and S.E.M. = standard error of the mean difference. d M.A.E. = Mean absolute error (S(estimated  measured)/h). e CART = The CART/SLD/Wind model (Kim et al., 2002); fuzzy = Fuzzy model. Wind speed used in both models was corrected to the level of wetness sensors during night. b

Table 3 Mean error (M.E.) of model-estimated wetness duration during days on which wetness >0 h/day was measured in 1998 and 1999 Sites

Dew-only daya Nb

Ames, IA Lewis, IA Nashua, IA Sutherland, IA Crawfordsville, IA Belleville, IL Bondville, IL Dixon Springs, IL Monmouth, IL St. Charles, IL Red Cloud, NE Gordon, NE O’Neill, NE Sidney, NE West Point, NE All 15 sites a

180 180 177 162 175 132 127 163 140 153 195 203 158 166 119 2430

MWDc (h/day)

8.9 8.0 8.4 8.5 8.5 8.5 10.7 9.4 8.6 8.8 7.7 9.0 6.8 7.7 10.7 8.6

Rain day M.E. (h/day) (S.E.M.)d CARTe

Fuzzye

0.3 0.9 3.0 1.2 1.7 2.4 1.8 0.3 1.2 0.4 1.9 2.7 2.2 1.6 0.9 0.4

0.0 1.1 3.8 0.4 2.0 2.6 2.2 0.1 1.3 1.6 0.9 2.3 1.8 2.3 1.5 0.1

(0.28) (0.29) (0.29) (0.32) (0.24) (0.28) (0.35) (0.23) (0.36) (0.25) (0.28) (0.25) (0.38) (0.26) (0.26) (0.08)

(0.27) (0.27) (0.26) (0.30) (0.22) (0.21) (0.33) (0.23) (0.34) (0.21) (0.26) (0.27) (0.37) (0.24) (0.24) (0.08)

Nb

107 93 92 100 102 42 60 48 55 54 94 81 77 78 42 1125

MWD (h/day)

9.8 9.5 10.2 10.7 9.6 8.6 11.3 11.3 9.6 9.6 10.4 11.2 10.0 10.3 11.5 10.2

M.E. (h/day) (S.E.M.) CART

Fuzzy

0.0 1.1 3.8 0.4 2.0 2.6 2.2 0.1 1.3 1.6 0.9 2.3 1.8 2.3 1.5 0.3

0.6 2.8 3.5 1.9 1.5 2.8 0.8 2.0 0.3 0.7 2.2 0.4 3.4 0.1 0.1 1.3

(0.37) (0.52) (0.42) (0.38) (0.32) (0.45) (0.44) (0.61) (0.48) (0.49) (0.48) (0.41) (0.60) (0.40) (0.50) (0.13)

(0.40) (0.51) (0.48) (0.47) (0.30) (0.47) (0.38) (0.59) (0.38) (0.47) (0.47) (0.43) (0.58) (0.39) (0.67) (0.13)

Dew-only day was defined as days on which rainfall <0.25 mm was measured; otherwise, rain day. Number of days corresponding to dew-only day or rain day. c Measured wetness duration during either dew-only day or rain day. d M.E. = Mean error (S(estimated  measured)/h) and S.E.M. = standard error of the mean difference. e CART = The CART/SLD/Wind model (Kim et al., 2002); fuzzy = Fuzzy model. Wind speed used in both models was corrected to the level of wetness sensors during night. b

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Fig. 4. The percentage of days on which LWD was estimated by a model within various error ranges (0, 1, 3, and 5 h). E indicates mean estimation error (estimation  measurement) during a 24-h period. Each point represents the frequency of days at corresponding sites over 2 years (1998–1999) of the validation set: (A) dew-only days defined as days on which no rainfall was measured within a 24-h period; (B) rain days defined as days on which rainfall (0.25 mm/ day) was measured. Data points above the 1:1 line indicate that the Fuzzy model estimated LWD within the specified error range for a higher percentage of days than did the CART model.

model under- and overestimated wetness occurrence, respectively. On the other hand, both models overestimated wetness occurrence during most of the night as well as for several hours after sunrise.

4. Discussion and conclusions Our results indicated that a fuzzy logic system could be used to develop an LWD model whose accuracy

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benefits from incorporating both energy balance principles and empirical computation methods. Estimation of LWD using the Fuzzy model was comparable to sensor measurements in terms of overall accuracy (<1 h/day). The Fuzzy model tended to be more accurate than the CART model in terms of MAE and hourly wetness classification across the study sites, which suggested that the Fuzzy model could be more spatially portable (Fig. 4 and Table 4). The Fuzzy model therefore appeared to be an advance over earlier empirical models to estimate LWD, since the CART model was reported to estimate LWD more accurately than previous empirical models (Gleason et al., 1994; Francl and Panigrahi, 1997; Kim et al., 2002). Accuracy of LWD estimation by the CART model varied among sites in Iowa, Illinois, and Nebraska. The CART model overestimated or underestimated LWD at several of the same locations at which LWD was over- or under-estimated by the CART/SLD model in an earlier study (Gleason et al., 1994). For example, the CART model overestimated LWD within 2 h/day at Crawfordsville and Sutherland, IA, in both studies. On the other hand, the magnitude of ME for the CART model was consistently less than 1 h/day at Ames, IA, Monmouth, IL, and Sidney, NE, suggesting that the CART model is subject to location-based errors. Variation in accuracy of the Fuzzy model among sites was roughly parallel to that of the CART model (Table 2). Nonetheless, the magnitude of MAE for the Fuzzy model was smaller than the CART model at every site, suggesting that the Fuzzy model possesses higher spatial portability than does the CART model. A physically oriented model is apt to estimate LWD caused by dew more accurately than wetness caused by rainfall because dew onset and dew duration on a surface are governed by energy balance, whereas wetness occurrence by rainfall is much less affected by latent heat on the surface. Although ME for the CART model was similar during both dew-only days and rain days, ME for the Fuzzy model was <1 h/day during dew-only days but >1 h/day during rain days. Nevertheless, it is important to note that the Fuzzy model estimated LWD within 1 h/day of ME more frequently than did the CART model during rain days (Fig. 4). A ME within 1 h/day is considered to be the limit of LWD measurement accuracy (Pedro and Gillespie, 1982b).

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Table 4 The accuracy in hourly estimation of wetness by models in 1998 and 1999 Sites

Wet hoursa

All hours Nb

Accuracyc CART

Ames, IA Lewis, IA Nashua, IA Sutherland, IA Crawfordsville, IA Belleville, IL Bondville, IL Dixon Springs, IL Monmouth, IL St. Charles, IL Red Cloud, NE Gordon, NE O’Neill, NE Sidney, NE West Point, NE All 15 sites

7398 7456 7359 7452 7376 4747 5107 5570 5768 5393 7718 7814 6687 7789 4082 97716

0.846 0.823 0.819 0.823 0.852 0.804 0.842 0.873 0.831 0.857 0.810 0.838 0.788 0.876 0.880 0.836

d

N

Accuracy CART

Fuzzy

2697 2373 2455 2474 2513 1482 2111 2125 1812 1905 2529 2775 1914 2099 1786 33050

0.806 0.803 0.933 0.819 0.898 0.883 0.733 0.852 0.681 0.759 0.828 0.645 0.802 0.689 0.816 0.796

0.790 0.848** 0.933 0.866** 0.893 0.922** 0.764 0.776** 0.722* 0.844** 0.864** 0.706** 0.857** 0.800** 0.800 0.825**

d

Fuzzy

0.853 0.843** 0.843** 0.837 0.888** 0.858** 0.863* 0.886 0.857** 0.888** 0.843** 0.847 0.806 0.885 0.871 0.857**

Dry hoursa N

Accuracy CART

Fuzzy

4701 5083 4904 4978 4863 3265 2996 3445 3956 3488 5189 5039 4773 5690 2296 64,666

0.869 0.833 0.761 0.825 0.829 0.768 0.920 0.886 0.899 0.910 0.801 0.944 0.783 0.945 0.929 0.857

0.890*e 0.841 0.798** 0.822 0.885** 0.829** 0.933 0.953** 0.919* 0.912 0.833** 0.925** 0.785 0.917** 0.927 0.873**

a

Hours at which wetness was detected >30 min was defined as wet hours; otherwise, dry hours. Number of hours. c Accuracy = 1  S(estimated  measured)/N. d CART = The CART/SLD/Wind model (Kim et al., 2002); fuzzy = Fuzzy model. Wind speed used in both models was corrected to the level of wetness sensors during night. e * and ** indicated that accuracy of models was significantly different at p < 0.05 and p < 0.01, respectively. b

The Fuzzy model simulates changes in latent heat flux on a surface, i.e., condensation and evaporation at a surface using estimated net radiation (pRn). To calculate pRn, the Fuzzy model assumed that the surface temperature of a leaf was at the dew point temperature at all times. Therefore, it was likely that the Fuzzy model estimated net radiation relatively accurately during hours in which dew occurred. When wetness results from rainfall, the surface temperature should be close to the temperature of rain drops, which also justifies our assumption to some extent. The use of VPD may also contribute to a relatively accurate estimation of LWD during rain days. Under high RH associated with rainfall, it is likely that the value of VPD is very small, which would favor wetness. Therefore, it appears that the Fuzzy model is fundamentally preferable to the CART model for estimating LWD caused by either rain or dew. The Fuzzy model was more accurate in classifying presence or absence of wetness at all sites than the CART model (Table 4). Since a physical principle is portable to any location, the spatial portability of the

Fuzzy model was ascribed to its incorporation of energy balance principles. In another study, the Fuzzy model estimated LWD substantially more accurately than did the CART model in northwestern Costa Rica during the wet season, providing additional evidence of its spatial portability (Kim et al., 2003). Both Fuzzy and CART models exhibited lower accuracy near sunset and sunrise than at other times (Fig. 5). LWD errors from the CART model also peaked during onset of leaf wetness (Rao et al., 1998; Madeira et al., 2002), which generally occurs in the hours soon after sunset. The physical process component of both models assumes steady state conditions, but the steady state assumption is least valid at sunrise and sunset. The CART model underestimated wetness occurrence immediately after sunrise, which may result from the empirical process that it uses to identify wet hours. In contrast, the Fuzzy model misclassified hours at which wetness dried off after sunrise as wet hours, which is attributable to the fact that the Fuzzy model used no measured solar radiation data.

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Fig. 5. Error pattern in LWD estimation with respect to progress of a day in terms of solar elevation that identifies time periods of sunset and sunrise: (A) absolute error rate (S(estimation  measurement)/ N, where N = the total number of hours that fall within each 108 range of relative solar elevation); (B) pattern of the mean error rate, which was calculated as S(estimation  measurement)/N.

Where solar radiation measurements are available, it is possible to modify the Fuzzy model to use them. At sites lacking a pyranometer, however, it is likely that the modified model may not have improved accuracy unless solar radiation is estimated with reasonable accuracy. Instead of changing an input term of the Fuzzy model, e.g., pRn, it is possible to adjust the output of the Fuzzy model using a correction factor. For example, after multiplying a correction factor, 0.975, to the defuzzified output of the Fuzzy model, percentage of absolute error for time periods during which bp was within 158–258 and 258–358 decreased from 20.2 to 17.9% and from 12.0 to 9.9%, respectively (Kim, unpublished data). Net radiation of a leaf surface depends on various factors including free water on the leaf. To simulate wetness occurrence accurately, calculation of energy balance using a conventional physical approach requires calculation of these factors within the

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computation framework. Fuzzy logic allows imprecise inputs, which lessens the burden of computing each detail within the model. By using the pRn term instead of an Rn term, therefore, it was possible to circumvent the complex computation process needed to obtain exact values of Rn without sacrificing accuracy of LWD estimation. The small number of input variables and simplicity in calculation are major advantages of the Fuzzy model compared with physical models. The Fuzzy model used the same input variables as the CART model, yet the estimation error was consistently lower over space and time. These patterns suggested that application of a fuzzy logic system makes it possible to develop a physically oriented model that retains temporal and spatial extendibility without sophisticated calculations, such as solution of nonlinear equations, that require iteration or numerical analysis. Iterative processes pose a potential risk when input data are not accurate, since a small error can be amplified in repeated calculations. Therefore, a physical model that includes an iterative procedure may not be able to estimate LWD accurately when relatively inaccurate weather data such as site-specific weather estimates are used (Magarey, 1999). Furthermore, because the outcome of the Fuzzy model represents an approximated possibility of wetness occurrence, it is possible to adjust the outcome under specific sets of climatic or geographic conditions instead of adding weather variables or input parameters. This adaptability may give the Fuzzy model an additional advantage over other models. The Fuzzy model, therefore, merits further validation as a successor to current LWD modeling efforts using physical and empirical approaches.

5. Acknowledgements We thank K.J. Koehler, Department of Statistics, Iowa State University, who provided advice and help with data handling and analysis and T.J. Gillespie, University of Guelph, for evaluation of this manuscript; K. Hubbard, High Plains Climate Center, Lincoln, NE, S. Hollinger, Midwest Regional Climate Center, Champaign, IL, and K. Berns, Department of Agronomy, Iowa State University, Ames, IA, facilitated field data collection.

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