Goes-13 IR Images for Rainfall Forecasting in Hurricane Storms

Home , Hurricane Ernesto (2012), Hurricane Manuel, Hurricane Patricia

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Technical Note Goes-13 IR Images for Rainfall Forecasting in Hurricane Storms

Marilu Meza-Ruiz 1 and Alfonso Gutierrez-Lopez 2,*

1 Water Research Center, Centro de Investigaciones del Agua-Queretaro (CIAQ), Young Water Professionals, International Water Association-Mexico, Universidad Autonoma de Queretaro, Queretaro 76010, Mexico; [email protected] 2 Water Research Center, Centro de Investigaciones del Agua-Queretaro (CIAQ), International Flood Initiative, Latin-American and the Caribbean Region (IFI-LAC), Intergovernmental Hydrological Programme (IHP-UNESCO), Universidad Autonoma de Queretaro, Queretaro 76010, Mexico * Correspondence: [email protected]

Received: 16 September 2019; Accepted: 29 April 2020; Published: 30 April 2020

Abstract: Currently, it is possible to access a large amount of satellite weather information from monitoring and forecasting severe storms. However, there are no methods of employing satellite images that can improve real-time early warning systems in different regions of Mexico. The auto-estimator is the most commonly used technique that was developed for specific locations in the United States of America (32–49◦ latitude) for the type of convective storms. However, the estimation of precipitation intensities for meteorological conditions in tropic latitudes, using the auto-estimator technique, needs to be re-adjusted and calibrated. It is necessary to improve this type of technique that allows decision-makers to have hydro-informatic tools capable of improving early warning systems in tropical regions (15–25◦ Mexican tropic latitude). The main objective of the work is to estimate rainfall from satellite imagery in the infrared (IR) spectrum from the Geostationary Operational Environmental Satellite (GOES), validating these estimates with a network of surface rain gauges. Using the GOES-13 IR images every 15 min and using the auto-estimator, a downscaling of six hurricanes was performed from which surface precipitation events were measured. The two main difficulties were to match the satellite images taken every 15 min with the surface data measured every 10 min and to develop a program in C+ that would allow the systematic analysis of the images. The results of this work allow us to get a new adjustment of coefficients in a new equation of the auto-estimator, valid for rain produced by hurricanes, something that has not been done until now. Although no universal relationship has been found for hurricane rainfall, it is evident that the original formula of the auto-estimator technique needs to be modified according to geographical latitude.

Keywords: rainfall intensity; hurricane Dean; hurricane Ernesto; auto-estimator; GOES; Mexico

1. Introduction The measurement of the space–time variability of rainfall is essential for the progress of hydrologic studies, such as the water balance in a watershed, or the execution of projects and actions related to urban development in the field of hydraulic networks [1,2]. There is an increasing demand to improve rainfall estimates from satellite systems on a different range of scales in time and space. Remote sensing of the earth and its atmosphere in the infrared spectrum has become a mainstay of environmental monitoring for weather and climate [3]. The trend towards increasingly new applications in the field of hydrometeorology requires precise estimates of rainfall for global or local coverage [4–6]. Also, the new meteorological radar systems with improved beam resolution have increased signal-noise sensitivity. In fact, one of the most important limits of hydrological prediction is due to the low

Forecasting 2020, 2, 5; doi:10.3390/forecast2020005 www.mdpi.com/journal/forecasting Forecasting 2020, 2, 5 86 of 101 resolution of the input of hydrological models. This input is given by the measurements of the rain-gauge, so a dense network rain-gauge would allow the progress in radar rainfall estimates [7]. In any case, it is of great importance to have the appropriate characteristics of the passive microwave radiometers and the computational capacity to analyze the information retrieved from precipitation events [8,9]. In the case of radars, the algorithms to know the rain-rate or the intensity of the rainfall (RR) must be calibrated for a specific geographical location. It is also very important to know the frequencies in which the algorithms derive RR. Usually, the advanced microwave sounding radiometers (AMSR) measure at 6.9, 7.3, and 10.65 GHz [10]. For tropical regions, it could be from 10 to 150 GHz [11]. Radar measurements are an excellent way to develop warning systems; however, not all the countries of Latin-American and the Caribbean (LAC) have access to radar data or automatic telemetric rain-gauges. In this region, hurricanes are undoubtedly the most extreme events to be studied. Significant progress has already been made in implementing the monitoring of monsoon and tropical cyclone rainfall. For example, in India, a new technique has been developed to estimate rainfall on a very fine scale (hourly rain rate), using the infrared (IR) and water vapor (WV 6.7 µm) channel from satellite images. However, eventually, these values need to be compared with values measured on a radar [12]. The useful images to report cloud coverage during the day are of 0.6–1.6 µm (visible), 3.9–7.3 µm (infra-red/water vapor), and 8.7–13.4 µm (thermal images). However, the access and use of satellite images are promising in America for the LAC region with the Geostationary Operational Environmental Satellite (GOES) and the Meteosat for Africa. Meteosat is a geostationary weather satellite launched by the European Space Agency (ESA). In Florida, USA, the stream-flow simulated by satellite rainfall data was slightly better than when driven by rain-gauge data and was similar to the case of using radar data, reflecting the potential applications of satellite rainfall in basin-scale hydrologic modeling [13]. These methodologies use the brightness temperature of satellite images and then transform that brightness temperature into cold cloud temperature. Finally, the brightness temperature is transformed into precipitation intensity [14,15]. Geostationary satellites are especially important for their unique ability to simultaneously observe the atmosphere and its cloud cover from the global scale down to the storm scale at high resolution in both time (every 15 min) and space (1–4 km) [16]. A feature of the geo-stationary satellite is that it measures the amount of energy emitted by the atmosphere. The infrared sensors recorded the thermal properties of the Earth’s soil and ocean surface [17]. The top of the cold clouds is the main focus of the IR sensors in order to get rainfall quantification. There are a lot of satellite images available for tracking and forecasting extreme storms. However, in Mexico, since there is no extensive radar coverage, it is necessary to use satellite images to forecast rainfall intensities during the trajectory of hurricanes. The GOES-13 orbits the Earth at an altitude of approximately of 35 thousand kilometers above the equator; at this altitude, the satellite speed is equal to the angular speed of the Earth. The satellite rotates 360◦ in 24 h, this characteristic allows us to have constant monitoring of the same area, and in this case, the whole Mexican territory is monitored all day long at a 15-min frequency. The temperature of the cloud top can be estimated by the brightness temperature value of the image from the satellite in the infrared (IR) channel. This is the satellite is used in Mexico to visualize convective storms and hurricanes. A total of five geostationary satellites provide operational imagery: these currently include the Meteosat Second Generation satellites (MSG) from the European Organization for the Exploitation of Meteorological Satellites (EUMETSAT), two Geostationary Operational Environmental Satellites (GOESs) and the Japanese Multifunctional Transport Satellite (MTSAT) series [18]. However, the National Weather Service (Servicio Meteorologico Nacional Mexico, SMN) and the Mexican Space Agency (AEM), through an agreement signed in 2016 with the National Weather Service, USA (NWS), have access to GOES images. This allows the use of images with all their characteristics and all their attributes. The most recognized methodology to estimate rainfall intensity from IR images was developed by Vicente et al. [19]. This methodology was applied for deep convection storms in the plains of the United States of America (32–49◦ latitude). Those locations are significantly different in latitude, longitude, and topographic conditions from those located in the tropical areas (15–25◦ Mexican tropic latitude). Forecasting 2020, 2, 5 87 of 101

In order to apply this method in the Mexican territory, an adjustment to the formula proposed by Vicente et al. [19] is required. It is also relevant to develop this type of regionalization especially for deep convection storms, “convective systems which produce the greatest rainfall intensity”; these storms occur when the environment, specifically a parcel of air mass above the ground and under the cloud top, begins to get colder and starts to expand [20]. Kalinga and Gan [13] introduced rain and no-rain discrimination in their study, improving the estimates from IR imagery; in this case, the value of the discrimination brightness temperature was 109. At this brightness temperature value, we can observe rainfall intensity measured by rain gauges. It is important to mention that, to date, the algorithm proposed by Vicente et al. [19] only applies to convective storms. The use of this equation should be reviewed if it is to be applied to hurricane storms. The aim of this work is based on the development of a new equation for each case study from IR satellite images. Using GOES 13 images and information from automatic rain-gauge stations (MGS). As already mentioned, the brightness temperature value of the image is converted to temperature and then to precipitation intensity. This paper is organized according to the following lineaments. Section2 presents the imagery processing based on the brightness temperature value from the geo-localization of the zone of study (in the IR channel) and to get the brightness temperature values from a convective storm, also to recollect rain gauge data (rainfall inferred from IR satellite data) and validate the estimations with gauges measurements (time-lapse desegregation of satellite series). Then, six case examples are briefly described in Section3. The first one, Hurricane Dean, on 21 August 2007, and the second, Hurricane Ernesto, with satellite images from 9 August 2012. These hurricanes were selected because they were the most intense in recent years. These case studies allow us to prove the limitations of the auto-estimator technique, with respect to the limit of rainfall rate (RR) to be considered. Section4 presents a spatial-temporal description of the hurricanes analyzed. Based on the results of estimating precipitation intensity during a hurricane, the validity of the formula proposed by Vicente et al. [19] is discussed. The last section summarizes the main results of the research.

2. Materials and Methods

2.1. The Auto-Estimator in Tropical Regions The auto-estimator is the most commonly used technique in the forecasting of rainfall intensity estimation from satellite images. However, it was developed for specific locations at 32–49◦ latitude, and only for convective storms. Nevertheless, the estimation of rainfall intensities for meteorological conditions in tropic latitudes, using the auto-estimator technique, needs to be re-adjusted and calibrated. First, hurricane events are selected with information available from GOES images and MGS surface data. Then, the satellite images are processed to get the brightness temperature values of each pixel. Then, using the original auto-estimator equation and coefficients, the brightness temperature is transformed into cloud top temperature, and the RR–satellite image is calculated. The next step is to match these obtained values with the surface data from RR–MGS. To carry out this comparison, it is necessary to disaggregate the rainfall data at the same time intervals. The comparative of the RR values allows us to know if the original formulation of the auto-estimator can properly forecast the RR. By applying the logarithmic conversion, it is possible to convert the auto-estimator equation into a linear expression, and thus to calculate the two coefficient (parameters) by multiple regression. In this way, two new parameters of the auto-estimator equation are calculated from the hurricanes analyzed. It can also be solved through B-spline estimation or least squares [21] getting the same set of results. Next, a regionalization of all the parameters that are obtained in this way is carried out and these are plotted in a Mexico map. Finally, a validation of the new auto-estimator coefficients is carried out with data from other hurricanes. Forecasting 2020, 2, 5 88 of 101

2.2. Imagery Treatment Processing In Mexico, the SMN has records of MGS rain-gauges and has a receiving station of the IR images Forecastingfrom GOES-13 2019, 1 FOR with PEER a frequency REVIEW of approximately 15 min. All this data, MGS and image records4 include a period of 13 years. In order to assign each pixel, its digital level value between 0 and records include a period of 13 years. In order to assign each pixel, its digital level value between 0 255, according to its resolution of 8 bits. The first step is to obtain the brightness temperature value and 255, according to its resolution of 8 bits. The first step is to obtain the brightness temperature from a convective storm (in the IR channel), geo-localize the zone of study, as well as to recollect value from a convective storm (in the IR channel), geo-localize the zone of study, as well as to rain-gauge data and, with the help of the auto-estimator technique, validate the RR with surface recollect rain-gauge data and, with the help of the auto-estimator technique, validate the RR with gauge measurements. The first step involves reading the brightness temperature values from cloud surface gauge measurements. The first step involves reading the brightness temperature values from tops for each pixel, and then convert them into temperature and subsequently into RR. To carry out cloud tops for each pixel, and then convert them into temperature and subsequently into RR. To this, a hydro-informatics tool was developed (Sat-Viewer®). Other software is available to do this carry out this, a hydro-informatics tool was developed (Sat-Viewer®). Other software is available to analysis. However, the development of our own software allows us to get basic information since do this analysis. However, the development of our own software allows us to get basic information it is required to systematize the analysis and verify the results obtained step by step. Commercial since it is required to systematize the analysis and verify the results obtained step by step. software would work as a “black-box”, where the possibility of following the processes is limited [22]. Commercial software would work as a “black-box”, where the possibility of following the processes This hydro-informatic tool obtains the value of the brightness temperature in the pixels from de satellite is limited [22]. This hydro-informatic tool obtains the value of the brightness temperature in the images. This hydro-informatics tool consists of a screen of conformed visualization of two-parts, on the pixels from de satellite images. This hydro-informatics tool consists of a screen of conformed left is placed the satellite image previously converted from pcx to bmp format, later with the aid of the visualization of two-parts, on the left is placed the satellite image previously converted from pcx to hydro-informatic tool of selection, on the right part the values of the digital level of the brightness bmp format, later with the aid of the hydro-informatic tool of selection, on the right part the values temperature are extracted in tabular format for all the pixels of the selection and are placed with a of the digital level of the brightness temperature are extracted in tabular format for all the pixels of nominative Bi,j for their identification, the sub-index i corresponds to the column, and j corresponds to the selection and are placed with a nominative B for their identification, the sub-index i the row of the location of the pixel (Figure1). , corresponds to the column, and j corresponds to the row of the location of the pixel (Figure 1).

® Figure 1. Extraction of the brightness temperature value of the pixels, using the Sat-Viewer ®.. The data extracted in the table are exported in txt format facilitating the estimation of the The data extracted in the table are exported in txt format facilitating the estimation of the precipitation by means of the auto-estimator technique of the area of study through worksheets to precipitation by means of the auto-estimator technique of the area of study through worksheets to develop the pertinent statistical and multivariate analysis. develop the pertinent statistical and multivariate analysis. 2.3. Rainfall Inferred from IR Satellite Data 2.3. Rainfall Inferred from IR Satellite Data There are several methods to estimate rainfall from satellite images, i.e., convective stratiform There are several methods to estimate rainfall from satellite images, i.e., convective stratiform technique (CST), modiﬁed convective stratiform technique (mCST), auto-estimator (AE), and quantile technique (CST), modified convective stratiform technique (mCST), auto-estimator (AE), and analysis formed by regression function of sorted rainfall intensity [23]. The formulas that relate the quantile analysis formed by regression function of sorted rainfall intensity [23]. The formulas that brightness temperature of the cloud-top with the associate temperature are provided by the National relate the brightness temperature of the cloud-top with the associate temperature are provided by Oceanic and Atmospheric Administration (NOAA). The value of rainfall intensity can be determined the National Oceanic and Atmospheric Administration (NOAA). The value of rainfall intensity can be determined by each pixel from the IR images. If B represents the brightness temperature value in the IR image, then the temperature in Kelvin degrees (K) is given by T = 418 − B for B > 176 (1)

T = 330 − (B⁄ 2) for B ≤ 176 (2) Forecasting 2020, 2, 5 89 of 101 by each pixel from the IR images. If B represents the brightness temperature value in the IR image, then the temperature in Kelvin degrees (K) is given by

T = 418 B for B > 176 (1) − T = 330 (B/2) for B 176 (2) − ≤ There are some formulas that can find rain rate or rainfall intensity RR (mm/h) depending on the temperature (K). One of the well-known techniques was developed to estimate it with images in the band 10.7 µm (channel 4), coming from the GOES-8 and GOES-9 satellites, but they are also valid and can be used with GOES-13 data. Particularly, events related to summertime and deep convection [19] found the curve fitting between these two variables with α = 11.183 1010 and β = 0.036382; × − the equation that describes the curve fitting with a power-law relationship is RR = α exp β T1.2 when 195 < T < 260 K (3) · · 2.4. Time Lapse Desegregation of Satellite Series It is important to take into account that the time lapses between the automatic rain gauge station (MGS) and the satellite data differ a few minutes. In this case, the MGS has data every 10 min, while satellite images are received every 15 min. For this reason, a previous adjustment of the data is required to be able to compare the values in real-time. An alternative is to use only the data obtained at 30 and 60 min, which are the only values that can be compared by satellite images and MGS. However, in this study, all the data measured on the surface are to be used; a temporal disaggregation of the precipitation is made. An alternative is to use Huff curves; unfortunately, these are not available for Mexico. We propose using a wavelet transform-based method to fit the daily rainfall data [24]; see AppendixA for details. This allows us to estimate rainfall intensities from satellite images (brightness temperature values) at each delay and to compare the hyetograph measure in the ground by the MGS. Disaggregation over time is of great importance since the auto-estimator is used in real-time (every 15 min), unlike other studies where downscaling is done daily or monthly [25,26].

2.5. Targeted Hurricanes Mexico is a country with a great diversity of natural protected areas; however, this biodiversity is exposed to a large number of meteorological phenomena such as hurricanes that every year disturbs the balance of environmental processes. Hurricanes such as Emily 2005, Wilma 2005, Dean 2007, Rick 2009, and Patricia 2015 have caused serious damage to ecosystems and forests. The criteria for the identiﬁcation of an erosive event are: (i) the cumulative rainfall of an event should be greater than 12.7 mm, or (ii) the event has at least one peak that is greater than 6.35 mm in 15 min (25.4 mm/h) [27]. Just to mention a few examples. Emily 2005; Campeche-MGS: 17 July from 23:20 to 23:20 recorded 8.38 mm (50.3 mm/h). Cancun-MGS: 18 July from 03:40 to 03:50 recorded 32.77 mm (196.6 mm/h). Wilma 2005; Campeche-MGS: 20 October from 23:40 to 23:50 recorded 8.89 mm (53.3 mm/h). Dean 2007; Cancun-MGS: 21 August from 10:00 to 10:10 recoded 15.49 mm (92.9 mm/h). Rick 2009; El Fuerte-MGS: 12 October from 17:00 to 17:10 recorded 7.4 mm (44.4 mm/h). Tomatlan-MGS: 21 October from 10:00 to 10:10 recorded 10.41 mm (62.46 mm/h). Odile 2014; Cabo San Lucas-MGS: 15 September from 02:10 to 02:20 recorded 18.54 mm (111.2 mm/h). Patricia 2015; Atoyac-MGS: 23 October from 23:30 to 23:40 recorded 7.4 mm (44.4 mm/h). This is why it is essential to know the intensity of the precipitation from hurricanes (RR > 25 mm/h). In Mexico, there are also meteorological radars; however, their coverage is limited and, for this reason, satellite images are used [28]. In Mexico, there are 175 MGS; however, with two million square kilometers of territory, coverage is still limited. Even so, some MGS can monitor the hurricanes that aﬀect both Mexican coasts. Many of these MGS are even destroyed during the passage of these extreme events. Therefore, the few data obtained from hurricane storms are very valuable. Table1 shows the Forecasting 2020, 2, 5 90 of 101 hurricanes analyzed in this study. Some of the MGS that record the precipitation height generated by the hurricanes in the studies are listed. In fact, it would be better if there were more automatic stations. However, in Mexico and the main Latin American and Caribbean countries, the network of stations is not the same as in the developed countries. However, this does not prevent an analysis of proposed formulas in non-hurricane countries with the available data.

Table 1. Relation of automatic rain-gauge stations (MGS) and the hurricanes studied.

H. Patricia H. Dean H. Odile H. Manuel H. Paul H. Ernesto MGS 20–24 Oct 13–27 Aug 9–18 Sep 12–20 Sep 12–18 Sep 1–10 Aug Name 2015 2007 2014 2013 2012 2012 Chinipas D15 1 - D15 D15 D15 - Guachochi D15 - - - D15 - Urique D15 - D15 D15 D15 - Maguarichi D15 - - D15 D15 - Cabo San L - - D15 D15 D15 - Cancun - D15 - - - D15 Alvarado - D15 - - - D15 Atoyac D15 D15 - D15 D15 D15 Huichapan D15 - D15 D15 D15 - Chinatu D15 - D15 D15 D15 - Las Vegas D15 - D15 D15 D15 - Obispo D15 - D15 D15 D15 - San Juan D15 - D15 - - - El Fuerte - - - D15 D15 - Alamos D15 - D15 D15 D15 - 1 D15, data available every 15 min. - MGS did not record any rain at that site.

3. Results As stated above, there are several methods to estimate rainfall from satellite images [23]. Once the method proposed here is implemented, it is possible to carry out a comparison of the estimated RR values with satellite images and those measured in MGS. It is possible to desegregate the time series in two examples cases. The ﬁrst one, Hurricane Dean, 21 August 2007, and data of the MGS Cancun in Yucatán. The second case, Hurricane Ernesto, with satellite images from 9 August 2012, and data of the MGS Alvarado in Veracruz. For Hurricane Ernesto, disaggregated time series was for 10 min as in MGS, while for Hurricane Dean, the desegregation time was for 5 min because the images used a breakdown of 5 min less. All times are referred to as GMT-6.

3.1. Hurricane Dean MGS Cancun, Yucatán (21 August 2007) Dean was named as the second major hurricane, cataloged category V in the Saﬃr–Simpson scale, of the top 20 major hurricanes in Mexico, according to records in the years from 1970 to 2007. Dean caused damage of more than 450 million dollars. The strong winds of up to 230 km/h and the rain caused by Hurricane Dean caused 32 deaths. This hurricane hit the Mexican coast twice. The ﬁrst impact inland was as a hurricane category V on 21 August at 3:00. The eye of the hurricane Dean impacted land with maximum sustained winds of 260 km/h and gusts of 315 km/h. The Hurricane Center was located at 65 km east of Chetumal, Quintana Roo. After causing devastation in Jamaica and other Caribbean islands, Dean reached the Mexican coast near the Caribbean Coast, not far from the Belize border; the hurricane was moving westward at a speed of 32 km/h. Figure2 compares the values between the rain intensity recorded by MGS Cancun on 21 August 2007, (from 9:40 to 11:10) with the RR value of the satellite image, and with the new auto-estimator setting proposed by Equation (4). It is important to highlight that the original formula proposed by Vicente et al. [19] that limits the rainfall intensity (RR) to 72 mm/h; however, for this case, the maximum intensity recorded was Forecasting 2020, 2, 5 91 of 101

92.94 mm/h recorded at 10:00 (192 K temperature value). In all cases, the original form of the equation proposed by Vicente et al. [19] is preserved. RR = 0.00737358743 1010 exp 0.025 T1.2 (4) Forecasting 2019, 1 FOR PEER REVIEW × · − · 7

Figure 2. The relationship (R2 = 0.55) between rainfall, the intensity from MGS Cancun 21 August 2007, Figure 2. The relationship (R2 = 0.55) between rainfall, the intensity from MGS Cancun 21 August and temperature of cloud top from the satellite image on the same date of Hurricane Dean. 2007, and temperature of cloud top from the satellite image on the same date of Hurricane Dean. 3.2. Hurricane Ernesto MGS Alvarado, Veracruz (9 August 2012) 3.2. HurricaneHurricane Ernesto Ernesto MGS Alvarado, was formed Veracruz from a tropical (9 August wave 2012) developed on 27 July on the African coast; the tropical wave was intensified to a depression storm on 1 August, next to the evolution of this Hurricane Ernesto was formed from a tropical wave developed on 27 July on the African coast; system in Tropical Storm on 2 August, and by 7 August, Ernesto was cataloged as a hurricane category the tropicalII on the wave Saffi r–Simpsonwas intensified scale. The to principala depression affectation storm was on the 1 heavy August, rainfall next in theto statesthe evolution of Yucatán, of this systemQuintana in Tropical Roo, Campeche,Storm on Tabasco, 2 August, and Veracruz.and by Ernesto7 August, made Ernesto landfall inwas the southcataloged of Quintana as a Roo,hurricane categorymoving II on west the andSaffir–Simpson impacting the scale. Campeche The coast.principal Afterwards, affectation it made was a secondthe heavy landfall rainfall in Veracruz; in the states of Yucatán,this region Quintana received Roo, more Campeche, than 300 mm Tabasco, of rain, and mainly Veracruz. on the central Ernesto coast made of Veracruz. landfall Thein the state south of Quintanaof Veracruz Roo, moving has seven west MGS, and but justimpacting three of themthe Campeche registered the coast. event, Afterwards, and just one registeredit made thea second landfallmaximum in Veracruz; rainfall this intensity. region Alvarado received Station more received than 300 80.76 mm mm of/h onrain, 9 August mainly 2012 on atthe 14:00 central (175 K). coast of The beginning of the measurements was taken from 12:32 an hour before the maximum was registered. Veracruz. The state of Veracruz has seven MGS, but just three of them registered the event, and just For the Hurricane Ernesto, the series of time was greater because rainfall records had a duration of one registered3 h 10 min. the Figure maximum3 compares rainfall the valuesintensity. between Alvarado the rain Station intensity received recorded 80.76 by MGS mm/h Alvarado on 9 August 2012 aton 14:00 9 August (175 2012K). The (from beginning 06:30 to 16:40) of the with measur the RRements value of was the taken satellite from image, 12:32 and an with hour the newbefore the maximumauto-estimator was registered. setting proposedFor the Hurricane by Equation Ernesto, (5). the series of time was greater because rainfall records had a duration of 3 h 10 min. Figure 3 compares the values between the rain intensity RR = 0.0010929761 1010 exp 0.0246 T1.2 (5) recorded by MGS Alvarado on 9 August 2012 (from× 06· :30 −to 16:40)· with the RR value of the satellite image, andFigure with 4the compares new auto-estimator the values between setting the rainproposed intensity by recorded Equation by (5). MGS Cabo San Lucas on 15 September 2014 with the RR value of the satellite image of hurricane Odile, and with the new . auto-estimator setting proposedRR = 0.0010929761x10 by Equation (6). ∙exp(−0.0246 ∙ T ) (5)

Figure 4 compares the values between the rain intensity recorded by MGS Cabo San Lucas on 15 September 2014 with the RR value of the satellite image of hurricane Odile, and with the new auto-estimator setting proposed by Equation (6). Forecasting 2019, 1 FOR PEER REVIEW 8

ForecastingForecasting 2019, 1 FOR2020 ,PEER2, 5 REVIEW 92 of 101 8

Figure 3. The relationship (R2 = 0.67) between rainfall intensity from MGS Alvarado on 9 August 2012 and the temperatureFigure 3. The relationshipof cloud tops (R2 = from0.67) betweenthe satellit rainfalle image intensity on from the MGSsame Alvarado date of onHurricane 9 August 2012 Ernesto. 2 Figure 3.and The the relationship temperature of(R cloud= 0.67) tops between from the satellite rainfall image intensity on the fr sameom dateMGS of Alvarado Hurricane Ernesto.on 9 August 2012 and the temperature of cloud tops from the satellite image on the same date of Hurricane Ernesto. 160 MS Cabo San Lucas, 15th Sep 2014 140 160 Vicente Fitting, 1998 MS Cabo San Lucas, 15th Sep 2014 120 140 Cabo San Lucas Fitting, 15th Sep 2014 Vicente Fitting, 1998 100 120 Cabo San Lucas Fitting, 15th Sep 2014 80 100 (mm/h) 60 80 (mm/h) 40 60

20 40 Rainfall rate registered in Meteorological Station Cabo SL SL Cabo Station Meteorological in registered rate Rainfall 0 20170 180 190 200 210 220 230 240 250 Top Cloud Temperature observed from GOES (K) Rainfall rate registered in Meteorological Station Cabo SL SL Cabo Station Meteorological in registered rate Rainfall 0 Figure 4. The170 relationship 180 (R2 = 1900.57) between 200 rainfall 210 intensity 220 from 230 MGS Cabo 240 San Lucas 250 on Top Cloud Temperature observed from GOES (K) Figure 4.15 The September relationship 2014 and (R the2 = temperature0.57) between of cloud rainfall tops fromintensity the satellite from image MGS at Cabo the same San date Lucas of on 15 SeptemberHurricane 2014 and Odile. the temperature of cloud tops from the satellite image at the same date of 2 HurricaneFigure3.3. 4. Validation TheOdile. relationship (R = 0.57) between rainfall intensity from MGS Cabo San Lucas on 15 September 2014 and the temperature of cloud tops from the satellite image at the same date of The validation process consists of accepting the estimated RR values from the satellite images Hurricane Odile. 3.3. Validation(RR–satellite) and comparing them with the RR values obtained from the surface MGS stations (RR–MGS). The validation is made every 10 min when the two RR time series are already matched. The validation process consists of accepting the estimated RR values from the satellite images 3.3. ValidationA difference is calculated between the measured value in MGS and the RR value of the satellite image. (RR–satellite)Using anand optimization comparing process them with with the generalizedthe RR values reduced obtained gradient (GRG)from nonlinearthe surface algorithm MGS of stations The validation process consists of accepting the estimated RR values from the satellite images (RR–MGS).the ExcelThe validation Solver tool, theis made difference every between 10 min the twowhen RR valuesthe two is minimized. RR time series are already matched. A(RR–satellite) differenceOther is and calculated procedures comparing between can them be used, the with formeasured example,the RR valuevalues a procedure in obtained MGS based and onfrom thethe maximum theRR surfacevalue likelihood of MGS the satellitestations image.(RR–MGS). Usingmethod The an [29validation optimization]. Optimization is made process algorithms every with 10 can min the be when effgeneralizedective the for two optimizing reducedRR time the seriesgradient training are of(GRG)already artificial nonlinear matched. algorithmA difference of the is calculatedExcel Solver between tool, the the differe measurednce between value thein MGS two RRand values the RR is minimized.value of the satellite image.Other Using procedures an optimization can be used, process for example,with the ageneralized procedure reducedbased on gradient the maximum (GRG) likelihoodnonlinear methodalgorithm [29]. of theOptimization Excel Solver algorithms tool, the differe can ncebe betweeneffective thefor twooptimizing RR values the is minimized.training of artificial intelligenceOther proceduresmodels [30,31], can bemulti-layer used, for perceptronexample, a procedureneural networks based on[32], the and maximum machine likelihood learning modelsmethod [33]. [29]. The Optimization use of a meta-heuristic algorithms techniquecan be effective as a harmonic for optimizing search was the proposed training [34].of artificial In this way,intelligence new coefficients models [30,31], of Formula multi-layer (3) are perceptronobtained for neural each hurricanenetworks analyzed.[32], and Itmachine is important learning to mentionmodels [33].that Theonly use the of alpha a meta-heuristic and beta coefficients technique are as changed.a harmonic The search exponent was proposed(−1.2) of the[34]. original In this way, new coefficients of Formula (3) are obtained for each hurricane analyzed. It is important to mention that only the alpha and beta coefficients are changed. The exponent (−1.2) of the original Forecasting 2020, 2, 5 93 of 101

Forecastingintelligence 2019, 1 FOR models PEER REVIEW [30,31], multi-layer perceptron neural networks [32], and machine learning 9 models [33]. The use of a meta-heuristic technique as a harmonic search was proposed [34]. In this way, formulanew is coepreservedfficients ofso Formula that the (3) new are obtainedcoefficients for each have hurricane a similarity analyzed. with It the is important original formula to mention and can that only the alpha and beta coefficients are changed. The exponent ( 1.2) of the original formula is be compared. − preserved so that the new coefficients have a similarity with the original formula and can be compared. To confirmTo confirm the new the new proposed proposed equations, equations, satellite images images of of the the analyzed analyzed hurricanes hurricanes wereused were used ® at 15-minat 15-min intervals. intervals. With With the the support support of of the the Sat-Viewer ® tool, tool, the the intensity intensity of precipitation of precipitation for each for each hurricanehurricane was wascalculated. calculated. On On the the other hand, hand, the the precipitation precipitation intensities intensities were obtained were fromobtained the MGS from the MGS stationsstations that that recorded recorded the precipitationthe precipitation sheets during sheets the during tracks of the the tracks hurricanes. of the Figures hurricanes.5 and6 show Figures 5 and 6 theshow results the ofresults this procedure of this procedure as a validation as ofa thevalid proposedation of equations. the proposed Figure 5equations. shows the validationFigure 5 shows the validationwith Hurricane with Hurricane Odile for days Odile 14 (23:50) for days to 15 14 (02:10) (23:50) of Septemberto 15 (02:10) 2014, of compared September with 2014, the datacompared measured at the Cabo San Lucas station, BCS. According to Equation (6): with the data measured at the Cabo San Lucas station, BCS. According to Equation (6): RR = 0.0085641613 1010 exp 0.1553 T1.2 (6) RR = 0.0085641613x10× · ∙exp−(−0.1553· ∙ T.) (6)

70 Estimated (mm/h)

Real (mm/h) 20 20 30 40 50 60 70

Figure 5. Validation of the methodology using data from Hurricane Odile, 14 (23:50) to 15 (02:10) 2 FigureSeptember 5. Validation 2014 at of MGS the Cabomethodology San Lucas, BCS.using (R data= 0.9387). from Hurricane Odile, 14 (23:50) to 15 (02:10) 2 SeptemberUsing 2014 the sameat MGS procedure, Cabo San Figure Lucas,6 shows BCS. (R the= validation 0.9387). using the data from Hurricane Patricia in October 2015 on the 23rd (22:00) to the 24th (06:30) compared with the data from the Atoyac station, UsingGuerrero. the Accordingsame procedure, to Equation Figure (7): 6 shows the validation using the data from Hurricane Patricia in October 2015 on the 23rd (22:00) to the 24th (06:30) compared with the data from the RR = 0.0074832483 1010 exp 0.1522 T1.2 (7) Atoyac station, Guerrero. According to Equation× (7): · − ·

RR = 0.0074832483x10 ∙exp(−0.1522 ∙ T.) (7)

Forecasting 2020, 2, 5 94 of 101 Forecasting 2019, 1 FOR PEER REVIEW 10

40 Estimated (mm/h)

Real (mm/h) 20 20 25 30 35 40

Figure 6. Validation of the methodology using data from Hurricane Patricia, 23 (22:00) to 24 (06:30) OctoberFigure 6. 2015, Validation in the MGSof the Atoyac, methodology Guerrero. using (R2 data= 0.9605). from Hurricane Patricia, 23 (22:00) to 24 (06:30) October 2015, in the MGS Atoyac, Guerrero. (R2 = 0.9605). 4. Discussion 4. Discussion The auto-estimator technique is typically used to estimate precipitation in deep convective systems collectedThe inauto-estimator 24 h by infrared technique information is typically [35]. Oneused ofto theestimate main restrictionsprecipitation of in the deep auto-estimator convective techniquesystems collected is related in to the24 precipitationh by infrared from information clouds with [35]. more One cold-tops of the low,main like restrictions the nimbostratus, of the whichauto-estimator is ignored technique if there is is related no convection to the precipitation present in from the environment clouds with more close. cold-tops In events low, of intenselike the precipitation,nimbostratus, the which contribution is ignored of if these there clouds is no toconvec the totaltionrecord present of in precipitation the environment is small close. in proportion, In events butof intense in other precipitation, cases, it is not the negligible. contribution In addition,of these clouds there areto the cases total of precipitationrecord of precipitation from clouds is small stratus in inproportion, the absence but of in convection, other cases, which it is not occur negligible. mainly atIn extra-tropical addition, there latitudes, are cases in of which precipitation the technique from doesclouds not stratus either in assign the absence precipitation of convection, [36]. This which finding occur has mainly important at extra-tropical implications latitudes, for developing in which thethe usetechnique of the does auto-estimator not either techniqueassign precipitation in the LAC [36]. region, This wherefinding mostly has important convective implications rains can for be detecteddeveloping by satellitethe use of images. the auto-estimator However, the technique cold cloud-top in the cannot LAC region, detect rainywhere system mostly information, convective andrains the can radar be detected would have by satellite to be used images. (as commented, However, the countries cold cloud-top in the LAC cannot region detect have rainy poor system access toinformation, radar data). and In the the radar United would States, have the to detection be used (as of cloudycommented, areas countries is carried in out the with LAC a region network have of meteorologicalpoor access to radar radars data). that cover In the the United national States, territory the detection [37], and of cirrus cloudy and ar clusterseas is carried in the dissipationout with a stage,network which of meteorological do not produce radars precipitation that cover or the are rathernational low, territory are filtered [37], inand this cirrus way. and Without clusters the in help the ofdissipation radars, the stage, auto-estimator which do techniquenot produce confuses precipitation these cold or cloudare rather tops inlow, areas are of filtered cirrus andin this remains way. ofWithout clusters, the with help precipitating of radars, the systems, auto-estimator which wastechnique reported confuses by Rozumalski these cold [ 38cloud]. These tops resultsin areas are of consistentcirrus and withremains those of of clusters, other studies with precipitating and suggest thatsystems, all the which remote-sensing was reported products by Rozumalski underestimate [38]. theThese rainfall results as are compared consistent to the with rain those gauge of measurementsother studies and when suggest evaluated that rainfallall the estimatesremote-sensing from groundproducts radar underestimate network and the satellite rainfall algorithmsas compared for to typhoons the rain atgauge various measurements spatio-temporal when scales evaluated from 0.04rainfall to 0.25 estimates◦ and hourly from toground event totalradar accumulation network an [d39 satellite]. algorithms for typhoons at various spatio-temporalIn the case ofscales hurricanes, from 0.04° the to auto-estimator 0.25° and hourly overestimates to event total the accumulation precipitations [39]. that come from CumulonimbusIn the case cloudsof hurricanes, that generate the auto-estimator large systems ov sucherestimates as hurricanes, the precipitations as shown in Figuresthat come2 and from3. TheCumulonimbus results of this clouds study that show generate that withlarge nosystems radar such data as available, hurricanes, the as alternative shown in isFigures to change 2 and the 3. coeTheffi resultscients inof Equationthis study (3) show to adjust that the with RR no values radar to data surface available, precipitation the alternative records. is to change the coefficientsOn the otherin Equation hand, in (3) Equation to adjust (3) the proposed RR values by Vicenteto surface et al. precipitation [19], the maximum records. intensity of rainfall intensityOn the is reached other hand, between in Equation 195◦ and (3) 200 proposed◦ K, while by in Vicente the two et casesal. [19], above, the maximum present lower intensity rainfall of intensityrainfall intensity at those temperatures.is reached between In other 195° words, and higher 200° K, intensity while in occurs the two at colder cases temperatures above, present (less lower than 190rainfall◦ K). Thisintensity difference at those in temperature temperatures. is associated In other withwords, latitudes higher near intensity the equator. occurs In at Figure colder3, ittemperatures can be observed (less thatthan the190° potential K). This fittingdifference for Ernesto in temperature values is is closeassociated to zero with or minimumlatitudes near rainfall the intensityequator. In decreases Figure 3, the it can potential be observed slope. Precipitationthat the potential is a well-knownfitting for Ernesto phenomenon, values is and close it canto zero make or someminimum simplifications, rainfall intensity establishing decreases a range the potential of temperatures slope. Precipitation where rainfall is a well-known intensity is equalphenomenon, to zero, atand the it samecan make time some getting simplifications, a fitting in the establishing potential curve a range with of the temperatures recorders in where the ground. rainfall Inintensity Figure2 is, Hurricaneequal to zero, Dean’s at fitthe shows same atime tendency getting that a correspondsfitting in the to potential a power-law curve curve; withaccording the recorders to [19 in], itthe is ground. In Figure 2, Hurricane Dean’s fit shows a tendency that corresponds to a power-law curve; Forecasting 2019, 1 FOR PEER REVIEW 11

ForecastingForecasting2020 2019,,2 1, 5FOR PEER REVIEW 95 of 10111 according to [19], it is notable that at temperatures colder than 210° K, rainfall intensity increases accordingexponentially. to [19], A relevant it is notable aspect that to considerat temperatures is the Saffir–Simpson colder than 210° category K, rainfall for hurricanes. intensity increases notableexponentially.For that example, at temperaturesA relevant Hurricane aspect colder Ernesto, to thanconsider categorized 210◦ isK, the rainfall Saffir–Simpson as hurricane intensity increases categoryexponentially. forII, hurricanes.shows a tendency A relevant of aspectcoldestFor to temperatures considerexample, isHurricane the related Saffir–Simpson with Ernesto, minimal categorycategorized rainfall for intensity; hurricanes.as hurricane in contrast, category the II, hurricanes shows a tendencycataloged ofas coldestmajorFor hurricanes, temperatures example, higher Hurricane related than with Ernesto,category minimal categorizedIII, presentrainfall theintensity; as highest hurricane in rainfall contrast, category intens the II, hurricanesity shows in warmer a tendencycataloged cloud top ofas coldestmajortemperatures, hurricanes, temperatures very higher similar related than to with thecategory minimalrelationship III, present rainfall demons intensity;the highesttrated inby rainfall contrast, Scofield intens the[40]. hurricanesity In in this warmer paper, cataloged cloud only topthe as majortemperatures,results hurricanes, for a few very hurricanes higher similar than to are categorythe presented. relationship III, presentHowever, demons the 175 highesttrated storms by rainfall Scofield caused intensity [40].by 14 In major in this warmer paper,hurricanes cloud only topthatthe temperatures,resultshave affected for a few veryMexico hurricanes similar were to analyzed.are the presented. relationship The resultsHowever, demonstrated show 175 a storms pattern by Scofield caused in the [byalpha40 ].14 In majorand this beta paper,hurricanes coefficients. only that the resultshaveFigures affected for 7 and a few Mexico8 show hurricanes werethat there areanalyzed. presented. is a spatial The However,results pattern sh inow 175 the a storms distributionpattern caused in the of byalphathe 14 parameter majorand beta hurricanes values coefficients. of that the haveFiguresequation aff ected7 proposedand Mexico8 show by werethat Vicente there analyzed. etis al.a spatial [19]. The This resultspattern means show in the that a distribution pattern the limit ins the ofof alphaVincent’sthe parameter and equation beta values coe ffimustcients. of the be Figuresequationadjusted7 and forproposed 8tropical show by that latitudes. Vicente there isFurtheret a al. spatial [19]. studies, patternThis meanswh inich the takethat distribution thethese limit variabless of of the Vincent’s parameterinto account, equation values will must ofneed the beto equationadjustedbe undertaken proposedfor tropical the by extremelatitudes. Vicente rainfall etFurther al. [ 19from studies,]. This hurrica meanswhichnes thattake and thethese the limits coefficientsvariables of Vincent’s into of account, auto-estimator. equation will must need The be to adjustedbefindings undertaken forof tropicalthis the study latitudes.extreme have rainfall Further a number studies,from hurricaof whichimportantnes take andthese implications the variables coefficients intofor account,theof auto-estimator.future will practice need to The beof undertakenfindingsauto-estimator of the this extremein studythe LAC rainfall have region. froma number hurricanesFurther of investigation important and the coe implicationsffiandcients experimentation of auto-estimator. for the intofuture Thethe practice findingsAdvanced ofof thisauto-estimatorBaseline study Imager have ain number(ABI) the LACis ofstrongly importantregion. recommended. Further implications investigation The for theABI future isand being experimentation practice developed of auto-estimator as theinto future the inAdvanced image the LAC on region.Baselinethe Geostationary Further Imager investigation (ABI) Operational is strongly and experimentationEnvironmental recommended. intoSatelliteThe theABI Advanced isGOES-R being developed Baseline[41]. The Imager asGOES-R the (ABI)future rainfall is image strongly rate on recommended.thealgorithm Geostationary is an The infrared-based Operational ABI is being algorithmEnvironmental developed calibra as theSatelliteted future in real-timeGOES-R image on against[41]. the GeostationaryThe passive GOES-R microwave Operationalrainfall raterain Environmentalalgorithmrates [42]. is an Satellite infrared-based GOES-R [ 41algorithm]. The GOES-R calibra rainfallted in ratereal-time algorithm against is an infrared-basedpassive microwave algorithm rain calibratedrates [42]. in real-time against passive microwave rain rates [42].

Figure 7. Spatial distribution of alfa parameter of Equation (3). Figure 7. Spatial distribution of alfa parameter of Equation (3). Figure 7. Spatial distribution of alfa parameter of Equation (3).

FigureFigure 8.8. SpatialSpatial distributiondistribution ofof betabeta parameterparameter ofof EquationEquation (3).(3). Figure 8. Spatial distribution of beta parameter of Equation (3). Forecasting 2020, 2, 5 96 of 101

5. Conclusions For the equation that relates the rainfall intensity with the cloud top temperature, it is important to recall that this equation was developed in the United States of America. The method can be applied in the tropics, but it is important to consider that there are different atmospheric conditions. For example, the highest rainfall intensity that was measured in the hurricanes does not take in count in the original formulation of auto-estimator (limited to 72 mm/h). Since space technology can be accessible for countries that have not developed their own meteorological satellite technology, the sharing information is a relevant hydro-informatics tool for those who see and prognosticate the weather in a wide time–space resolution. The image from GOES provides important information about meso and local scale of hydrometeorological events, which impact the local weather generating severe rainfall, flash floods, and floods. Since we can read this GOES-13 image at almost real-time, it can be read as rainfall intensity in all Mexican territory, providing a beneficial tool, which improves the decision-making in different parts of Mexico. The next step is to develop a connection between the image bright-temperature values to the MGS having the auto-adjusted relationship in real-time. While there are several alternatives to use satellite images, in Mexico, the access is with GOES images. With images from other satellites, the result of applying the technique of auto-estimator will be the same; the only variation would be the resolution of the images. There are few studies of the coupling of satellite images during the occurrence of hurricanes. Not only is it enough to check their track, but also to match and verify it with surface measurements. Certainly, this work, although it has the limitation of only having used information for Mexico, expects to be the beginning of the verification in the downscaling of the instantaneous precipitation data measured in surface with the satellite data [43]. There is an important opportunity in developing new and innovative techniques and technology based on satellite data adjusted to the tropical latitudes. Further investigations consist of the classification of climatic regions, more specifically, rainfall similarity zones, which include patrons of storms and rainfall intensity in all Mexican territories and the greatest rainfall intensity for each region will be set; as a result, the auto-estimator can be used to monitor rainfall in all states in Mexico where there is no meteorological information available. The hydro-informatics tools developed by www.redciaq.uaq.mx help decision-makers to have a better understanding of hydrometeorological events; they can visualize the possible storms getting close to a specific area in order to act and send help to critical areas. These kinds of hydro-informatics tools give new and reliable information that can support decision-making. At the same time, people can check the weather from home, work, school, and mobile phones. It is important to conduct more investigations and research in this field. Satellite images offer wide time-space coverage, and it is possible to get enough information that can help scientists and researchers to get different values that give understanding and prediction to weather phenomena as a result of climate change.

Author Contributions: Conceptualization, M.M.-R. and A.G.-L.; methodology, M.M.-R.; validation, M.M.-R. and A.G.-L.; formal analysis, A.G.-L.; investigation, M.M.-R.; data curation, M.M.-R.; writing—original draft preparation, M.M.-R.; writing—review and editing, A.G.-L. All authors have read and agree to the published version of the manuscript. Funding: This work was financially supported by Consejo Nacional de Ciencia y Tecnología (CONACYT). This research received funding from Secretaria de Educacion Publica (SEP-Mexico). Programa para el Desarrollo Profesional Docente (PRODEP) and was supported by the Fund for the Reinforcement of Research UAQ-2018 Research and Postgraduate Division (FOFIUAQ/FIN201911). Acknowledgments: The authors are grateful to the Risk Management Unit of the UNESCO Regional Office of Science for Latin America and the Caribbean. Special thanks to SMN (Servicio Meterologico Nacional) for the information provided, imagery, and meteorological stations database, to NOAA-NASA GOES Project for sharing data and information, and to Jose Eduardo Hernandez Paulin, Manon Breda, Juan Pablo Molina, Anna Gancarczyk, Claire Fourniret, and Marion Palleschi for their help in data processing. We express our sincere gratitude to Ivonne Cruz and Giselle Galvan Lewis for their detailed style review. Conflicts of Interest: The authors declare that they have no conflict of interests regarding the publication of this paper. Forecasting 2020, 2, 5 97 of 101

Appendix A. Wavelet Transformation Engineers involved in the design of waterworks and the exact natural sciences usually see the time series of hydro-meteorological variables referred to in the time and frequency domain. The origin of the wavelet transformation is due to the need to get the time-frequency disaggregation of a time series. The Fourier transform provides information about which frequencies are present in the time series but does not respond to their exact location in time. The wavelet solution requires a time window of a certain width for all the evaluated frequencies. With the help of the τ -scale parameter, it is possible to change the width of the window. With the s parameter, called the position-parameter, the location of the series on the time axis is changed [44].

1 t s ψτ (t) = ψ − τ, s R, τ , 0 (A1) ,s √τ τ ∈

Z + ∞ 1 W f (τ, s) = f (t) ψτ (t) dt (A2) √τ ,s −∞ In this way, the wavelet function transforms an input function f (t) of t variable into its two-dimensional representation W(τ, s) of τ and s variables. The variable τ refers to the scale and the variable s represents the time displacement of ψ(t). With the hypothesis that a time series can be expressed (or disaggregated) according to its two basic components: h(t) the trend represents the main behavior of the RR–MGS, and s(t) is the periodic movement of the series in RR–satellite image. The wavelet’s continuous transformation is deﬁned as

Z + ∞ ˆ W(τ, s) = f (t)ψτ,s(t) dt (A3) −∞ ˆ ψτ,s(t) denotes the complex conjugate of the function ψτ,s(t). Equation (A3) can be easily represented as a set of linear, displaced, and invariant filters over time. These filters are described as a ˆ convolution ψτ,s(t) of the filtered signal s(t) with its response to the pulse h(t):

Z + ∞ h(t) s(t) = h(t)s(t τ)dt;W(τ, s) = f (s) ψˆ τ ( s) (A4) · − · ,s − −∞ The steps to follow to ﬁnd the wavelet transform of a signal are i. It starts with a scale value, e.g., τ = 1 for the wavelet signal; this is placed at the beginning of the data record in t = 0 (see the output in Figure A1). ii. The two time series are multiplied together, and the result is integrated over the time-frame. The result of this integration is multiplied by the inverse of the square root of τ (see the output in Figure A2). iii. The wavelet function in the same scale s = 1 is displaced in time to the right in τ. The procedure of Step (i) is also repeated until the end of the analyzed series is ﬁnished (see the output in Figure A3). Forecasting 2019, 1 FOR PEER REVIEW 14

Forecasting 2019, 1 FOR PEER REVIEW 14 2.00 standardized RR value ForecastingForecasting 2019, 12020 FOR, 2 ,PEER 5 REVIEW 98 of 101 14 1.50 2.00 standardized RR value 1.41 0.88 1.00 1.502.00 standardized RR value 1.41 0.70 0.71 0.88 1.000.501.50 0.37 time (min) 1.41 0.70 0.71 0.88 0.500.001.00 0 5 100.37 15 20 25 30 35 40 45 0.71 time (min) 0.70 -0.42 -0.500.000.50 -0.53 0-0.74 5 100.37 15 20 25 30 35 40 45 time-0.84 (min) -1.00 -0.42 -0.500.00 -0.53 0-0.74 5 10 15 20 25 30 35 40 45 -0.84 -1.50 -0.42 -1.00-0.50 -1.55 -0.53 -0.74 -0.84 -2.00 -1.50-1.00 -1.55

Figure A1. Temporarily-2.00-1.50 -1.55 disaggregated series of the RR–MGS Huichapan (red line) and RR–satellite image (blue line)-2.00 during the Hurricane Manuel (16 Sep 2013), Step (i). Figure A1. Temporarily disaggregated series of the RR–MGS Huichapan (red line) and RR–satellite image Figure(blue line) A1. Temporarily during the disaggregatedHurricane Manuel series of (16 the Sep RR–MGS 2013), Huichapan Step (i). (red line) and RR–satellite Figure A1. Temporarily disaggregated series of the RR–MGS Huichapan (red line) and RR–satellite image (blue2.00 line)standardized during RR the value Hurricane Manuel (16 Sep 2013), Step (i). image (blue line) during the Hurricane Manuel (16 Sep 2013), Step (i). 2.001.50 standardized RR value1.41 1.40

2.00 1.501.00 standardized RR value1.41 0.88 1.40 0.70 0.71 1.50 1.41 0.88 1.000.50 1.40 0.70 0.71 time (min) 1.00 0.88 0.500.00 0 5 10 15 200.70 25 300.71 35 40time (min) 45 0.50 -0.500.00 -0.53 0 5 10 15 20 25 30 35 40 45 -0.74 time-0.81 (min) -0.84 0.00 -0.50-1.00 -0.53 0 5 10 15 20 25 30 35 40 45 -0.74 -0.81 -1.25 -0.84 -0.50 -1.00-1.50 -0.53 -0.74 -0.81 -1.25 -0.84 -1.50-1.00 Figure A2. Temporarily-1.25 disaggregated series of the RR–MGS Huichapan (red line) and RR–satellite image Figure(blue -1.50line) A2. Temporarily during the disaggregated Hurricane Manuel series of (16 the Sep RR–MGS 2013), Huichapan Step (ii). (red line) and RR–satellite Figureimage A2. Temporarily (blue line) during disaggregated the Hurricane series Manuel of (16the Sep RR– 2013),MGS Step Huichapan (ii). (red line) and RR–satellite image (blue line) during the Hurricane Manuel (16 Sep 2013), Step (ii). Figure A2. Temporarily disaggregated series of the RR–MGS Huichapan (red line) and RR–satellite 2.00 standardized RR value image (blue line) during the Hurricane Manuel (16 Sep 2013), Step (ii). 2.001.50 standardized RR value1.56

1.41 0.99 2.00 1.501.00 standardized RR value1.56

1.41 0.990.70 1.50 1.56 0.59 1.000.50 1.41 0.990.70 time (min) 1.00 0.59 0.500.00 0 5 10 15 200.70 25 30 35 40 45 0.59 time (min) 0.50 -0.500.00 -0.53 0 5 10 15 20 25 30 35 40 45 -0.74 time (min) -0.84 0.00 -0.50-1.00 -0.53 0-1.04 5 10 15 20 25 30 35 40 45 -0.74 -1.17 -0.84 -0.50 -1.00-1.50 -1.04 -0.53 -0.74 -1.17 Figure A3. Temporarily disaggregated series of the RR–MGS Huichapan (red line) and-0.84 RR–satellite -1.50-1.00 -1.04 Figureimage A3. Temporarily (blue line) during disaggregated the Hurricane series Manuel of (16 the Sep RR– 2013),MGS ﬁnal Huichapan step. (red line)-1.17 and RR–satellite image (blue line)-1.50 during the Hurricane Manuel (16 Sep 2013), final step. Figure A3. Temporarily disaggregated series of the RR–MGS Huichapan (red line) and RR–satellite image (blue line) during the Hurricane Manuel (16 Sep 2013), final step. ReferencesFigure A3. Temporarily disaggregated series of the RR–MGS Huichapan (red line) and RR–satellite image (blue line) during the Hurricane Manuel (16 Sep 2013), final step. References1. Berne, A.; Uijlenhoet, R. Path-averaged rainfall estimation using microwave links: Uncertainty due to spatial rainfall variability. Geophys. Res. Lett. 2007, 34, doi:10.1029/2007gl029409. 1.References Berne, A.; Uijlenhoet, R. Path-averaged rainfall estimation using microwave links: Uncertainty due to 2. Karabatić, A.; Weber, R.; Haiden, T. Near real-time estimation of tropospheric water vapour content from spatial rainfall variability. Geophys. Res. Lett. 2007, 34, doi:10.1029/2007gl029409. 1. groundBerne, A.; based Uijlenhoet, GNSS data R. andPath-averaged its potential rainfall contribution estimation to weather using now-castingmicrowave links:in Austria. Uncertainty Adv. Space due Res. to 2. Karabatić, A.; Weber, R.; Haiden, T. Near real-time estimation of tropospheric water vapour content from 2011spatial, 47 rainfall, 1691–1703, variability. doi:10.1016/j.asr.2010.10.028. Geophys. Res. Lett. 2007 , 34, doi:10.1029/2007gl029409. ground based GNSS data and its potential contribution to weather now-casting in Austria. Adv. Space Res. 2. Karabatić, A.; Weber, R.; Haiden, T. Near real-time estimation of tropospheric water vapour content from 2011, 47, 1691–1703, doi:10.1016/j.asr.2010.10.028. ground based GNSS data and its potential contribution to weather now-casting in Austria. Adv. Space Res. 2011, 47, 1691–1703, doi:10.1016/j.asr.2010.10.028. Forecasting 2020, 2, 5 99 of 101

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