I Introduction
The field of atmospheric science has numerous observation platforms that provide high space and time resolution data, but has yet to find methods which can quantify the intuitive patterns explicitly. Meanwhile, the young field of Topological Data Analysis (TDA) encompasses methods for quantifying exactly these sorts of structural intuitions seen by atmospheric scientists. This paper merges these two fields by using persistent homology, a now wellestablished tool in TDA, to quantify a diurnal cycle observed in a hurricane using Geostationary Operational Environmental Satellite (GOES) infrared (IR) satellite data.
Persistent homology, and more generally TDA methods, has found significant success in rather disparate applications by finding structure in data and using this insight to answer questions from the domain of interest. For instance, Giusti et al. used the homology of random simplicial complexes to investigate the geometric organization of neurons in rat brains
[1]. Nicolau et al. used mapper, another tool in TDA, to discover a new subtype of breast cancer [2]. More closely related to this work is the use of TDA for time series analysis and image processing. This includes using persistent homology to understand periodicity in time series arising from biological [3, 4] and engineering applications [5]. There has also been a great deal of interest in using persistence for image analysis, e.g. [6, 7, 8, 9].This paper presents an application of the use of both time series and image analysis using TDA. The diurnal cycle of tropical cyclones (TCs) has been described in previous studies [10, 11, 12, 13, 14, 15, 16, 17, 18] that provide evidence of the regularity of this cycle as well as its potential impacts. This diurnal pattern can be seen in GOES IR imagery as cyclical pulses in the cloud field that propagate radially outward from TCs at speeds of 510 m s [10, 11, 15]. These diurnal pulses, a distinguishing characteristic of the TC diurnal cycle, begin forming in the TC’s core near the time of sunset each day and appear as a region of cooling cloudtop temperatures. The area of cooling then takes on a ringlike appearance as marked cloudtop warming occurs on its inside edge and it moves away from the storm overnight, reaching several hundred kilometers from the TC center by the following afternoon. Observations and numerical model simulations indicate that TC diurnal pulses propagate through a deep layer of the TC environment, suggesting that they may have implications for TC structure and intensity [10, 11, 15, 17].
The current state of the art TC diurnal cycle measurement has a limited ability to analyze the behavior beyond qualitative observations. This paper presents a more advanced mathematical method for quantifying the TC diurnal cycle using tools from TDA, namely onedimensional persistent homology to analyze the holes in a space. This research aims to detect the presence of the diurnal cycle in GOES IR satellite imagery and to track the changes through a time series.
The first attempt, using the naive combination of persistent homology with the GOES IR imagery, did not show the recurring pattern. Due to the drastically variable values in the IR brightness temperature data, persistent homology was not able to detect any significant structure. Looking at the data, however, there is a clear circular feature visible, so we developed more sophisticated methods to extract this structure.
In this paper, we present a method applying the distance transform and onedimensional persistent homology, allowing us to quantify the cycle using maximum persistence. We show that using tools from TDA, we can detect cyclic behavior in the hurricane that repeats approximately every 24 hours.
Ii Tropical Cyclone Background
Previous research has documented a clear diurnal cycle of cloudiness and rainfall in TCs: enhanced convection (i.e., thunderstorms) occurs overnight, precipitation peaks near sunrise, and upperlevel cloudiness (i.e., the cirrus canopy) expands radially outward throughout the day, reaching its maximum areal coverage in the early evening hours [10, 11, 12, 13, 14, 15, 16, 17, 18]. To quantity the expansion and contraction of the cirrus canopy, Dunion et al. used GOES satellite IR imagery to examine the sixhour cloudtop temperature differences of major hurricanes in the Atlantic basin from 2001 to 2010 [10]. They found that an area of colder cloud tops propagated outward around 510 m s over the course of the day, with warming temperatures on its inner edge. More recently, in [15], Ditchek et al. expanded Dunion et al.’s work to include all tropical cyclones in the Atlantic basin from 1982 to 2017 and found that the diurnal pulse is nearly ubiquitous, with 88% of TC days featuring an outwardly propagating pulse.
Despite the consistent signature and documentation of this diurnal cloud signature, open questions remain as to how the diurnal cycle is linked to innercore convective processes and whether it is a columndeep phenomenon or mainly tied to upperlevel TC cloud dynamics related to incoming solar radiation [12, 13, 14, 15]. Investigating these questions is relevant to TC forecasting as the diurnal cycle of clouds and rainfall has implications for forecasting storm structure and intensity, as evidenced by the diurnal cycle in objective measures of TC intensity and the extent of the 50kt wind radius documented by Dunion et al. Additionally, and especially relevant to the current work, most of the papers above have identified the pulse using subjective measures of cloudtop temperature change and timing [10, 15]. The current work seeks to quantify the pulse to determine its true periodicity using persistent homology, a topological tool that is particularly effective at capturing the type of patterns visible in the pulse.
Iii Math Background
Persistent homology is a tool from the field of TDA which measures structure in data. This data can start in many forms, including as point clouds or, as in the case of this work, as a function on a domain. In this section, we will briefly review the necessary background to understand cubical homology and persistent homology, and refer the interested reader to [19, 20, 21, 22, 23] for a more complete introduction. Additionally, we will introduce tools used in our method, including the distance transform, mathematical morphology, and the Fourier transform.
Iiia Cubical complexes
In this section, we largely follow Chapter 2 of [19] with the caveat for the informed reader that because we use homology with coefficients, we can be lazy about orientations of cubes. In addition, our data consists of 2D images, so we need only define cubes up to dimension 2.
An elementary interval is a closed interval of the form or for , which are called nondegenerate and degenerate respectively. An elementary cube is a product of elementary intervals . The dimension of , , is the number of nondegenerate components of . Note that 0dimensional cubes are just vertices at the points on the lattice in , 1dimensional cubes are edges connecting these vertices, and 2dimensional cubes are squares. Let denote the set of all elementary cubes in and the set of dimensional cubes. A set is cubical if it can be written as a finite union of elementary cubes. Then we denote the associated cubical complex as , with the dimensional subset denoted . If , then we say is a face of , denoted . If , then is a proper face of , denoted , and is additionally a primary face of if .
A greyscale image, or more generally an matrix, can be viewed as a function where
We will model this as a function defined on a particularly simple cubical set ; see the left of Fig. 1 for an example. For simplicity, we denote by the square .
So, given a matrix , we equivalently think of this data as a function where we set equal to the matrix entry and set for all lower dimensional cubes . Note that we will abuse notation and use to denote both the original matrix and the view of this matrix as a function with domain .
IiiB Distance transform
The distance transform is a tool used in image processing and is computed on binary images [24, 25]. It is used in various applications in many disciplines, such as guiding robots to navigate obstacles [26], computing geometric representations such as Voronoi diagrams [27] as well as being a useful method in many other image processing tools [28].
This operation assigns each pixel in the foreground (pixels with value 1) a value based on its distance to a pixel in the background (pixels with value 0). This can be computed using various distance metrics, most frequently the or distance.
Using the notation from the previous section, represents the pixel, in the image represented as a matrix of pixels, . Given any
where is a 0valued pixel and is any distance metric. Given two pixels, we calculate the distance, also called the chessboard distance, between them as
This defines a distance on the pixels, which are the 2cells in the cubical complex. The distance can be extended to the lower dimensional cells in the same manor as described in Sec. IIIA.
IiiC Homology
is a standard tool in algebraic topology which provides a vector space
^{1}^{1}1Normally a group, however, we are working with field coefficients. for each dimension for a given topological space . The different dimensions measure different properties of the space. In particular, for this work we are interested in dimensional homology; i.e. when . The 1dimensional homology group measures the number of loops in the space; equivalently, we can think of this as the number of holes in the space. In particular, if we look at the black region in each of the examples in Fig. 2, the rank of the first homology for each is (1,1,1,2,2,3).The exact definition of homology is as follows. For any cubical set (which for the purposes of this discussion will always be a subset of ), we have sets giving the cubes of different dimensions: for . An chain is a formal linear combination of simplices in ,
with coefficients . We can of course add these objects by setting and multiply by a constant. Thus, the collection of all chains forms a vector space .
We define a linear transformation
called the boundary map, by setting where the sum^{2}^{2}2Again, notice that because we are working with coefficients, the bookkeeping normally needed for orientation is unnecessary. is over the primary faces . The kernel of , , (that is, the set of elements of which map to 0) is generated by closed loops in . The image of , , is generated by boundaries of 2cells. Then the dimensional homology group is defined to be . An element of this group , represents an equivalence class of loops which can differ by collections of 2cells; see the right side of Fig. 1 for an example of two loops which are equivalent in .
IiiD Persistent Homology
For a static space , measures information about the number of loops. Persistent homology takes as input a changing topological space, and summarizes the information about how the homology changes.
Let an valued matrix be given. Fix a function value and let . That is, is the subset of squares in which have value at most in the matrix, along with all edges and vertices which are faces of any included square. is often called a sublevel set of . See the right side of Fig. 1 for an example of the structure of and see Fig. 2 for regions corresponding to the matrix on the left of Fig. 2. This shows drawn in black for and .
These spaces have the property that for , thus we can consider the sequence
(1) 
for any set of numbers . This sequence of spaces is called a filtration. For each of the spaces, we can compute the homology group . The inclusion maps of Eqn. (1) give rise to linear maps
It is these maps that we study to understand how the space changes. In particular, when we are focused on 1dimensional homology () as in this study, a loop is represented by an element . We say that this loop is born at if it is not in the image from the previous space; that is, . This same loop dies at if it merges with this image in ; that is, where we abuse notation by using to both refer to the class in and the image of this class under the sequence of maps in . We refer to as the lifetime of the class.
A persistence diagram,as seen in the right of Fig. 2, is a collection of points where for each class which is born at and dies at is represented by a point at . The intuition is that a class which has a long lifetime is far from the diagonal while a class with a short lifetime is close. In many cases, a long lifetime loop implies that there is some sort of inherent topological feature being found, and thus that this point far from the diagonal is important, while short lifetime loops are likely caused by topological noise due to sampling or other errors in the system. In the example of Fig. 2, there is a prominent offdiagonal point which shows that the function defined by the matrix has a circular feature. Thus, a common measure for looking at the persistence diagram when investigating a single, circular structure is the maximum persistence, defined as
for a given persistence diagram, .
IiiE Mathematical morphology
Mathematical morphology is a broad field based on analyzing shapes of objects using mathematical tools from areas including set theory and geometry. This field is particularly useful for analyzing geometric structure in images and image processing. Specific applications include classification of digital images of cancerous tissue [30], restoration of old films [31], and ridge detection in finger prints [32].
Two major tools in mathematical morphology are erosion and dilation. Both tools involve a kernel moving through a binary image. In erosion, a pixel in the original image will remain a 1 only if all pixels under the kernel are 1’s, otherwise it becomes a 0. This process removes small clusters of pixels, often considered noise, and pixels near the boundary.
Dilation is the opposite of erosion. A kernel moves through the binary image and a pixel is assigned a 1 if at least one pixel under the kernel is a 1, otherwise it is assigned a 0. Therefore, erosion followed by dilation will remove noise and rebuild the area around the boundary. This process of erosion followed by dilation is called opening. We use opening in Sec. VC to test the influence of noise on our method. The choice of kernel for these methods can vary in size and shape depending on the application. For a more intensive explanation of these methods as well as the mathematical properties, we direct the reader to [33].
Opening is included in the python module cv2. Opening is specifically implemented using the function cv2.morphologyEx using cv2.MORPH_OPEN as the second input.
IiiF Fourier transform
The Fourier transform is a common method for investigating the periodicity of time series. It does so by decomposing a wave into a sum of sinusoids with different frequencies. We will provide a short description of the Fourier transform; for a more detailed explanation, see [34]. Given a real valued function , the Fourier transform is
This converts a function from the time domain to the frequency domain. In particular, when working with discrete data, we use the discrete Fourier transform. Let be the time between discrete samples, then let where . Then the discrete Fourier transform is
The inverse discrete Fourier transform can then be calculated as
The discrete Fourier transform reveals periodic components of the input data as well as the strength of each periodic component.
The power spectrum can be estimated using the discrete Fourier transform by calculating the square of the absolute value of the Fourier transform,
. Plotting this gives a visualization of the strength of each frequency of the periodic components in the input data. See Fig. 5 for an example. If the strongest frequency is , then the period for this component is .Iv Method
The data was given in the form of two sets of stormcentered GOES IR (10.7 m) satellite imagery. These two data sets have the same native spatial resolution, but differ in temporal resolution. The first set (hereafter the GOES12 dataset), utilizes brightness temperatures derived directly from GOES12 4km IR satellite imagery and consists of data in hourly increments, spanning 2 to 4 September 2007 with the exception of 0415 UTC and 0515 UTC each day (due to the GOES12 satellite eclipse period). Imagery was remapped such that each pixel has a spatial resolution of 2 km and each image covers a total area of approximately 1500 km 1500 km. This is represented as a 752 752 matrix. The second data set is the GridSatGOES and consists of data in 3hour increments, spanning 31 August to 6 September 2007 with the exception of 0600 UTC each day [35]. Each pixel has a resolution of approximately 8 km and each image covers a total area of approximately 2400 km 2400 km. This is represented by a 301 301 matrix. This data is cropped to a 191 191 matrix to approximately match the area covered by the first set of data. The cropped version covers a total area of approximately 1530 km 1530 km.
The GridSatGOES data set requires some additional processing. A different normalization is used with this data; thus, in order to convert it, the following equation is applied to the GridSatGOES brightness temperatures
Some images in the GridSatGOES data set also contain missing values where the brightness temperature for certain pixels was not recorded and is instead assigned a fill value. In order to prevent these values from impacting our results, we interpolate values for these pixels. For a given pixel with a missing value, we compute the average value of a
grid centered at the pixel, not including the pixels in this range that also have missing values.The TC diurnal pulse is propagating outward through the day; thus, in order to see the movement and changes in the GOES satellite brightness temperature, we consider the difference in matrices six hours apart [10, 15]. For all times , given the original brightness temperature image , we compute the sixhour differences, ; see Fig. 3 for an example. While circular features are visually prominent in the data, simply using persistence on the difference data, as discussed in Sec. III, did not show any relevant features due to the extreme differences in the function values between the circular sections. The right side of Fig. 3 shows the corresponding persistence diagram where no significant circular structure is detected.
Thus, we define a new function on the difference matrix using the following method. Fix a threshold and let be the subset of which has function value less than . This method results in a binary matrix defined entrywise as,
We will address this choice of threshold in Sec. VB; however, we will focus on the case degrees for most of our analysis. Note that because is a difference of two images, the threshold is not isolating all pixels above a certain temperature, but rather those pixels that increase in value by at least 70 degrees over the six hours.
We apply the distance transform to this binary matrix, as described in Section IIIB, giving a new matrix .
To calculate the distance transform, we use the python submodule scipy.ndimage, specifically the function distance_transform_cdt with the chessboard metric. Therefore, each entry in corresponds to the minimal distance to an entry where . The distance transform is then scaled by the resolution for each dataset in order to convert the units to kilometers instead of pixels. For the GridSatGOES data, we scale by a factor of 8 km/pixel and for the GOES12 data, we scale by a factor of 2 km/pixel We then compute sublevel set persistence on the function using the cubtop method in Perseus [21, 36]. Fig. 4 shows an example of each step in the method.
For each sixhour difference in each data set, we apply the steps described above, then calculate maximum persistence as defined in Sec. IIID. By plotting maximum persistence over time, we can see how the most prominent circular feature changes through the progression of the day and life of the TC. The red and blue solid lines in Fig. 6 represent this plot for both data sets. The plot shows an oscillatory pattern for both data sets that appears to repeat approximately daily.
V Results
Va Maximum persistence
In order to quantify the oscillatory pattern seen in Fig. 6, we use the discrete Fourier transform (DFT). The DFT is used to calculate the most prominent frequency in the data, in order to determine how often the cyclic behavior repeats. Note, to use the discrete Fourier transform the time steps must be equal; however, because of the missing times in our data, this is not the case. Therefore, we approximate the maximum persistence at these values by adding a point along the line between the times immediately before and after the missing time. Additionally, we truncate the maximum persistence to only include the days where we have the data for the entire day. This means truncating the GridSatGOES data set to include only 14 September, and the GOES12 data set to include only 12 September. The discrete Fourier transform was calculated using the python submodule numpy.fft.
We first calculate the Fourier transform using the function fft, then calculate the frequency bins using fftfreq. Using this information we plot the approximate power spectrum (the square of the absolute value of the amplitude) for each data set, as can be seen in Fig. 5. Note, our data is all real, so the power spectrum will be symmetric for positive and negative frequencies; therefore, we only need to look at the positive frequencies. Picking the frequencies corresponding to the highest peaks in each of the power spectrums gives a frequency of 0.976 cycles/day for the GridSatGOES data set, and 0.979 cycles/day for the GOES12 data set. This means that the cycle is repeating every 24.6 hours for the GridSatGOES data set, and every 24.5 hours for the GOES12 data set.
Using the most prominent frequency for each data set, we calculate the inverse Fourier transform using ifft and plot these smoothed versions over the original data in Fig. 6. These smoothed curves closely resemble the patterns exhibited by the original maximum persistence versus time plots; therefore, these approximately 24 hour patterns seen in the plots are actually the most prominent signals in the data. This verifies the claim that our method is detecting a daily cycle.
VB Choice of threshold
We claim that the choice of threshold is not difficult and that the approximately daily cycle will be detected for a variety of thresholds. In Fig. 7 are surfaces that represent maximum persistence versus time for thresholds, . There is a clear periodic pattern for both data sets regardless of the threshold chosen in this range. In fact, for all thresholds tested between 35 and 75, the period is consistent at 24.6 hours for the GridSatGOES data set and 24.5 hours for the GOES12 data set. For , the Fourier transform is unable to pick up the daily pattern in the GridSatGOES data set, and instead just sees one period for the entire five days. Because a wide range of threshold values consistently detect a daily pattern, we conclude the choice of threshold does not need to be precise.
VC Removal of Noise
While the above method detects a daily cycle, there are some instances where the sixhour differencing introduces noise because of varying behavior in the center of the hurricane. The top images in Fig. 8 shows an example of this. There is a small area of pixels in the center of the image that are above the threshold, causing the distance transform to fill in the center of the circular region. This dot in the center will change the value of the maximum persistence; therefore, we used the method of opening described in Sec. IIIE using a pixel kernel for the GOES12 data set and a pixel kernel for the GridSatGOES data set to remove noise such as these center pixels. Note, the difference in size of the kernel is due to the differences in spacial resolution between the two data sets. We use the python module cv2 for these computations. Opening is specifically implemented using the function cv2.morphologyEx using cv2.MORPH_OPEN as the second input. The bottom images in Fig. 8 show the results when opening is used on the thresholded matrix and then the distance transform is applied. The center pixels are removed so the distance transform is no longer filled in.
Using this extra step in the method, we recalculate maximum persistence for all times and the estimated period using Fourier transforms in the same manner described in Sec. VA. Fig. 9 shows maximum persistence plotted vs time for the both data sets using our original method, and the method including the additional opening step. While the new maximum persistence values vary a little from the originals, the general oscillatory behavior remains unchanged. Using the Fourier transform, we get the same results as in Sec. VA; the cycle still repeats every 24.6 hours for the GridSatGOES data set, and every 24.5 hours for the GOES12 data set. Thus, the noise is not affecting the periodicity we are detecting and the opening step is therefore unnecessary.
Vi Discussion
This paper presents a novel method for detecting and analyzing the diurnal cycle of tropical cyclones using methods from TDA. While current state of the art TC diurnal cycle measurement is mostly qualitative, our method provides a mathematically advanced method for automatic detection and measurement. While our method involves a choice of a parameter for the threshold, we present evidence that the choice is not important as there are many possible choices of thresholds that would yield the same results. Additionally, our method is robust to noise in the GOES satellite imagery as blurring the image does not yield different periodicity than the original method.
While this method was only tested on Hurricane Felix (2007), it was able to detect almost identical patterns in two sets of GOES satellite data for that hurricane that varied in spatial and temporal resolution. In the future, we hope to apply this analysis to more TCs and apply it to other satellite channels to further test our method.
Acknowledgment
The authors would like to thank William Dong, who was instrumental in creating code for the first iteration of this project. The work of ST and EM was supported in part by NSF grant DMS1800446; EM was also supported in part by NSF CMMI1800466. The work of JD was supported in part by the Office of Naval Research Program Element (PE) 0601153N Grant N000141410132 and by NOAA Grant NA14OAR4830172 from the Unmanned Aircraft Systems (UAS) Program. KC was supported by NSF Grant AGS1636799.
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