LACUNARITY AS INDEX OF EVALUATION OF WIND DIRECTION IN PERNAMBUCO

 ABSTRACT: Wind direction data were recorded from January 2008 to February 2011, at nine weather stations (Recife, Caruaru, Arcoverde, Garanhuns, Surubim, Floresta, Ibimirim, Serra Talhada and, Cabrobó), in the state of Pernambuco, Brazil, operated by the Instituto Nacional de Meteorology (INMET). Lacunarity analysis was performed in order to evaluate wind direction on those weather stations. In Recife the predominance of the wind was in the east (E)-southeast (SE) direction. In the agreste of the state of Pernambuco the predominant direction of the wind is the same one of Recife. In the Sertão of Pernambuco the wind predominates in the east (E)-sul (S) direction. Small values of lacunarity are caused by the presence of local winds, while large values of lacunarity are caused by lack of wind in a certain direction.

found in their study that in most parts of the year there is predominance of winds from the southeast (SE), except for the period from December to February where the wind blows in the northwest (NW).According GALVANI et al. (1999) observed in their study that the predominant direction of winds in Maringa was northeast (NE) during the night period, in the southeast position (SW) in the period from January to May and from October to December and winds toward the east (E) in the months from July to September.SANTANA et al. (2008) study conducted in Cuiabá, MT noted that the prevailing wind direction in spring and summer is located in the north -northeast (N and NE), in the fall and winter are predominant in the south -southwest (S-SO).BUENO et al. (2011) in a study conducted in Lavras region, Minas Gerais, observed that the prevailing wind direction east (E) followed west (W) from February to November, already in January and December the predominant direction is inverted west (W) then east (E).SILVA et al. (1997) concluded in their study that the wind direction in Pelotas is east (E) for the period between spring and summer, in the fall is Southeast (SE) and in winter the prevailing direction is northeast (NE).SILVA et al. (2002) in their study of wind potential at 77 locations in Northeast Brazil observed that the wind predominates in the east (E).LUCENA et al. (2017) evaluated the direction of wind in the island of Fernando de Noronha through the analysis of lacunarity and concluded that greater the presence of wind smaller will be the lacunarity in a sector, the same number was verified by Lucena (2016) that evaluated the direction of wind through lacunarity in municipality of Serra Talhada.
Given the importance of wind energy, this study aims to characterize the wind direction in Pernambuco state using lacunarity analysis.

Data
Data of wind direction and speed observed from January 2008 to February 2011.Data are from nine stations (Recife, Caruaru, Arcoverde, Garanhuns, Surubim, Floresta, Ibimirim, Serra Talhada and Cabrobó) the climatological network in the state of Pernambuco operated by the National Institute of Meteorology (INMET).
Data were collected by the National Institute of Meteorology and measured through universal anemographs installed at 10 meters above the soil surface.The values of wind direction and speed were registered by anemograms hourly according to the following period's 0 to 1a.m., 1a.m. to 2a.m. and so on up to 24 hours.
The predominant wind directions were characterized by a frequency analysis of the daily observations, using the following expression where, f(x) is the frequency of occurrence of wind in a given direction, n is the number of occurrences of a given wind direction and N is the total number of observations.For better visualization of the frequency of wind directions was used to the compass rose.
The wind direction was registered hourly according to the following period's 0 to 1a.m., 1a.m. to 2a.m. and so on up to 24 hours.Each sector received the hourly record according to the following criteria described in Table 1.Thus for each sector of wind direction a time series of 0's and 1's was created, where 0 is represented by the absence of wind at a given hourly and 1 is represented by the presence of wind at a certain hourly.

Lacunarity
The concept of lacunarity was introduced by Mandelbrot (1982) as a measure of the distribution of gap sizes in the fractal object.Geometric objects with gap sizes distributed over a wide range have greater lacunarity than those with smaller and more uniform gaps.The lacunarity is a measure is very explored in climatic and ecological phenomena (MARTINEZ et al., 2007;MARTINEZ et al., 2010;LUCENA et al., 2017;LUCENA, 2016;LUCENA et al., 2015;LUCENA, 2015;LUCENA et al., 2014;LANA et al., 2014;LUCENA and CAMPOS, 2014;LUCENA and STOSIC, 2013).
Lacunarity is related to the deviation of a geometric object from translational invariance (GEFEN et al., 1983).Homogeneous and translationally invariant geometric objects have low lacunarity, while heterogeneous and not translationally invariant geometric objects have high lacunarity.Translational invariance (and lacunarity) is also scale dependent: the objects that are heterogeneous at small scale can appear homogeneous at higher scale and vice versa.Lacunarity was originally developed to describe properties of fractal objects, but it can be extended on general spatial patterns including those with fractal and multifractal properties and can be used with both binary and quantitative data in one, two and three dimensions (PLOTNICK et al., 1996).
Wind direction study using lacunarity analysis were reported by Lucena et al. (2017) where they found that the greatest lack of water was in the N direction (less presence of wind in the island) and smaller in direction SE (greater presence of wind in the island) by evaluating the wind in the island of Fernando de Noronha.In the municipality of Serra Talhada, Lucena (2016) found that the lowest lacunarity occurred in sector S and the largest in sector W.
To characterize the wind direction, the lacunarity is a measure of the distribution of segments, defined as a sequence of consecutive hours that the wind is found in a certain direction, and the gaps defined as sequence of consecutive hours that the wind is not in particular direction.Lacunarity analysis has been used in eight different wind directions.Quantitatively, we have n(s, r) is the number of windows mobile size r (2,4,8,16,32,64 and 128 hours) containing s segments (hours is not found in a particular direction).The probability p(s, r) is define as where, N(r)=L-r+1 is the total number of windows of size r, and L is the total length of record, including segments of occupied sites and gaps.
Lacunarity is now defined by where (  , ) are the first and second moment of p(s,r).
For scale free process lacunarity decreases with window size as power law given by where scaling exponent β can be determined as the slope of linear regression of log[L(r)] versus log(r) using least squares method and ε~N(0, δ 2 ) (MARTINEZ et al., 2007).If Y is values of log L(r) and X the values r and n is total of observations, soon  ̂=  ̅ −  ̂ ̅ and

Results and discussion
The wind in the city of Recife is very predominant in the eastern (E) regions with 21.0% and southeast (SE) reaching 34.2% of the winds in this region already in the northwest (NW) sector the frequency of winds does not reach 5.0%, Figure 1 and Table 2. Similar results were found in LUCENA et al. (2017) evaluating the direction of wind in the Island of Fernando de Noronha, Pernambuco, Brazil.Assessing the harsh region of Pernambuco is noted that in Caruaru the predominance of wind directions is in southeast (SE) with 30.8% of the winds and south (S) exceeding the 35.0%frequency, Figure 1 and Table 2. Investigation of the behavior of the wind in the city of Garanhuns we note that the wind is predominant in the eastward (E) with 37.0% and southeast (SE) with a frequency of 38.1% Figure 1 and Table 2.In Arcoverde noticed the predominance of air mass toward the southeast (SE) where 53.7% of the frequencies of the wind in this region, Figure 1 and Table 2.In the city of Surubim the predominant wind direction is from the east (E) with 38.0% and southeast (SE) reaching 36.0% of the frequencies, Figure 1 and Table 2.
Starting to evaluate the backcountry region of Pernambuco is observed that in the region of Ibimirim the predominance of wind reaches 35.6% frequency toward the southeast (SE) and 33.7% in the south (S), Figure 1 and Table 2.In the region of Floresta predominant wind is toward the southeast (SE) exceeding 47.0% frequency, Figure 1 and Table 2.In the city of Serra Talhada wind is predominant in the south (S) with 44.8% frequency, Figure 1 and Table 2.In the region of Cabrobó is observed that the predominant wind direction is located between the Southeast (SE) with 37.4% and South (S) exceeding the 40.0%frequency, Figure 1 and Table 2. Similar patterns of wind direction were found by LUCENA (2016), corroborating these findings.
In the analysis of lacunarity, it is observed through the scatterplot of the logarithm of lacunarity versus the logarithm of the size of the window to the city of Recife the smallest values of lacunarity are the eastward (E) and Southeast (SE) and higher values in north (N), Figure 2. In relation to harsh Pernambuco it is observed that lower values of lacunarity are found in the regions between the east (E) to the south (S) and higher values of lacunarity in the directions from the west (W) to north (N), Figure 3.In the backwoods of state of Pernambuco minors lacunarity values are found in the directions of Southeast (SE) and south (S) and larger from the southwest (SW) to the northwest (NW), Figure 4.It is observed in Table 3 that there are very significant differences for the estimates of alpha and beta parameters for each set, in different wind directions, in relation to municipalities.In directions where it has little wind presence, the values of the exponent beta lacunarity are studied independently of the smaller stations, while where the wind is more prevalent, have lower values of beta lacunarity exponent for all stations in the study, similar results are found such as those by Lucena (2016).
It is observed in Table 3 that there are quite significant differences between the estimates of alpha and beta parameters for each models fitted in different wind directions in relation to meteorological stations.In directions where wind predominance is small, the values of exponent beta of lacunarity are higher independent to the station.Where wind is more prevalent, the value exponent beta of lacunarity is small for all stations in the study as shown in Table 3 and Figure 5 (ratio between the beta exponent and wind frequency for all directions and stations).Figure 5 -Dispersion between exponent beta and wind direction ratio in region and direction.
Finally, we would like to point out that one may understand that there is a misconception when angular coefficient is used for the estimation of fractal dimension, once lacunarity computation does not involve the computation of an angular coefficient.Actually, there is a difference between the calculation of lacunarity and an estimate of the lacunarity.Lacunarity is calculated through a quotient taking into account the size of the boxes.Lacunarity can be estimated taking into account box sizes through a power model, where the variable response is the observed lacunarity itself, and box size as explanatory variable, as described by Martinez et al. (2007).The estimated angular coefficient was not calculated for lacunarity but for beta exponent of lacunarity.

Conclusions
In Recife the predominance of wind in the eastward (E) and Southeast (SE), to the harsh region of Pernambuco in which direction the wind is predominantly the same as Recife and in the backwoods of Pernambuco of the wind predominates in the east direction (E) to the south (S).

Figure 1 -
Figure 1 -Frequency distribution of the wind in relation to the areas of Recife, Caruaru Garanhuns Arcoverde Surubim, Ibimirim Floresta, Serra Talhada and Cabrobó from January 2008 to February 2011.

Figure 2 -
Figure 2 -Dispersion of the logarithm of lacunarity and the logarithm of the window size for different wind directions in Recife station.

Figure 3 -
Figure 3-Dispersion of the logarithm of lacunarity and the logarithm of the window size for different wind directions in Caruaru, Garanhuns, Arcoverde and Surubim stations.

Figure 4 -
Figure 4-Dispersion of the logarithm of lacunarity and the logarithm of the window size for different wind directions in Ibimirim, Floresta, Serra Talhada and Cabrobó stations.

Table 1 -
Hourly wind direction record in each sector

Table 2 -
Frequency distribution of wind direction in relation to the cities