sempre-caro-mi-fu

[sempre-caro-mi-fu]

ASSIGNMENT WEEK 4

Recap Weeks 1, 2 and 3

Just a recap: Weeks 1 and 2 have been devoted to the identification, within the Mars Craters database, of 5 search groups, defined by a specific range of diameters of all the craters of type “Rd” (Radial Ejecta), and within each group only a small number of craters has been included (the Main craters), each having a diameter close to the mean value of the diameters for that group in the database:

Group 1: range (0, 50) km, mean diameter 7.69 km. Selected 8 Main craters.

Group 2: range (50, 100) km, mean diameter 66.66 km. Selected 7 Main craters.

Group 3: range (100, 150) km, mean diameter 114.81 km. Selected 7 Main craters.

Group 4: range (150, 200) km, mean diameter 167.82 km. Selected 8 Main craters.

Group 5: range (200, 250) km, mean diameter 221,66 km. Selected 2 Main craters.

The idea is that each Main crater has been originated by the impact of an external body, and that impact has caused the fallout in the surrounding area of a quantity of debris, originating several secondary craters around the Main. Therefore, each Main crater within a Group is associated with a search area used to delimit the zone where to look for secondary craters. The identification of the search areas for each Main crater has been the task of Week 3, achieved using some Data Management decisions to add support variables to the original ones.

Week 4

Now, in Week 4, the final week of the course, it’s time to try to answer the two research questions of my analysis:

Is the average diameter of the secondary craters somehow dependent on the diameter of the Main crater supposedly originating them?

Is there a “privileged” direction of the secondary craters around the Main that could indicate the direction of the impact and of the consequent fallout of debris? The idea is that, at least occasionally, the impact should “push” the fallout in the direction of the impact.

To answer first research questions, the approach is as follows:

For each of the 5 Groups, consider all the Main craters within. Each Main is associated with a set of secondary around it: compute the average diameter of the secondary craters within each Group.

Compare this average diameter (relative to the secondary craters belonging to each Group) with the one of the Main craters of the same Group (see the list in the previous paragraph)

If the average diameters of the secondary craters tends to have a similar behaviour of the average diameters of the Main craters, then we can reasonably infer that the bigger the Main, the bigger the Secondary

To answer second research questions, the approach is as follows:

For each of the 32 search areas compute the direction of each secondary crater relative to the Main

Create a histogram of the directions for each Main crater using a reasonable bin amplitude, say 5°, and compute the mean value for each distribution. Plot histogram and mean value

If the histogram shows that a “preferred” direction exists in the distribution of secondary craters (and here I adopt the mean value as such an indicator), then this can suggest that the direction is the one of the impacting body.

The Program

Here is the link to the Week4 version of the program:

Results

First Question

Is the average diameter of the secondary craters somehow dependent on the diameter of the Main crater supposedly originating them?

The results are synthetized in the following graph, where the average diameters of the Main (blue) and secondary (red) craters is compared: for each of the 5 Groups

It is possible to observe that the behaviour of the average diameter of the Main and secondary craters does not appear to be correlated (the first is sharply increasing, the other is smoothly decreasing), so I would say that the answer to the research question is, reasonably NO. My expectation (and hope) was to find a positive correlation between the two sets, in the sense that even the average diameters of the secondary craters was increasing with the one of the Main, but the graph tells a different story…

Second Question

Is there a “privileged” direction of the secondary craters around the Main that could indicate the direction of the impact and of the consequent fallout of debris? The idea is that, at least occasionally, the impact should “push” the fallout in the direction of the impact.

The results are described in the following graphs: the ‘x’ marker is the angular position of the Main crater originating the Group, the blue points in “SECONDARY CRATERS MAP” are all the secondary craters identified around the Main. Where possible the ‘x’ marker is complemented by the name of the Main crater. Finally, a red arrow originating from the position of the Main crater, indicates the “prevalent” direction, assumed to be coincident with the mean.

Group 1

Group 1 contains 8 Main craters, but the secondary ones surrounding each Main are so few that the results of the analysis are not relevant at all for this Group. Considering also that Tumblr limits the number of images to be posted to 30, I have decided to privilege the other, more relevant Groups, presenting here just the bare minimum results for Group 1 to avoid to exceed this limit (3 images, for Groups (1.1), (1.4) and (1.8)). All the other images are in any case available at the same link of the program.

...(1.2) to (1.3) not shown but available at the same link of the program...

...(1.5) to (1.7) not shown but available at the same link of the program...

Group 2

Group 2 contains 7 Main craters, and the results shown here are complete. For each Main crater in this Group, the following pictures show the map of secondary craters and the corresponding angular directions:

Group 3

Group 3 contains 7 Main craters, and the results shown here are complete. For each Main crater in this Group, the following pictures show the map of secondary craters and the corresponding angular directions:

Group 4

Group 4 contains 8 Main craters, and the results shown here are complete. For each Main crater in this Group, the following pictures show the map of secondary craters and the corresponding angular directions:

Group 5

Group 5 contains 2 Main craters, and the results shown here are complete. For each Main crater in this Group, the following pictures show the map of secondary craters and the corresponding angular directions:

[sempre-caro-mi-fu]

Assignment Week 3

SOLUTION STRATEGY - CONTINUED

Data Processing

What have been achieved during Week 2 is the identification of 5 distinct groups of craters, each characterized by a specific diameter interval and mean value:

From 0 to 50 km. Mean diameter value: 7.69 km

From 50 to 100 km. Mean diameter value: 66.66 km

From 100 to 150 km. Mean diameter value: 114.81 km

From 150 to 200 km. Mean diameter value: 167.82 km

From 200 to 250 km. Mean diameter value: 221.66 km

The reason for having 5 different groups so well-spaced is because I want to verify if the average diameter of the secondary craters is related to the diameter of the Main crater. If the answer is no, I expect to have a population of secondary craters around the Main having an average diameter independent on the belonging Group, on the other side, if the answer is yes, I expect to have an average diameter value of the secondary population somehow depending on the belonging group, possibly increasing with the Group. Therefore, to answer the search question, it will be necessary to define, for each crater in the 5 Groups, the corresponding population of secondary craters, and to do this it will be necessary to identify, for each crater in each Group, the corresponding search area. This will be achieved introducing a set of new variables to each Data Frame, in accordance with what suggested in the Data Management lessons. The input dataset is numerically complete, so there is there no need for any missing variable management.

 [Note: I’m not a geologist. I’m aware that it would be necessary to be guided by some expert, for example: I don’t know exactly how to distinguish a secondary crater in the database, nor how to allocate properly secondary craters to a single Main in case of two overlapping regions but…this is a course on Data Analysis, and its focus, I hope, should be more on how one treats the data, rather than on the academic relevance of the results. Correct me if I’m wrong, please.]

The process will be as follows (in effect it should start from point 8, as it’s a continuation from Week 2):

For each one of the 5 groups, consider each crater in it, and its diameter

Consider a circular area around the crater of diameter 10 times the crater’s one, and introduce a new variable named “SEARCH_RANGE” to store it.

Introduce a new support variable containing the value of the normalized diameter of the crater, i.e. the ratio between its diameter and the diameter of the planet. Name this new variable “NORMALIZED_DIAMETER” and add it to the Data Frame.

Compute the angle to be applied to the latitude to determine the upper and lower limits of the search zone. The same angle, together with the proper correction factor, will be used to compute the upper and lower limits of the longitude of the zone. Name this angle ��DELTA_ANGLE” and add it to the Data Frame. (Note: DELTA_ANGLE = arcsin(NORMALIZED_DIAMETER))

Compute the correction factor for the longitude, as a function of crater’s latitude, name it “LONGITUDE_FACTOR” and add it to the Data Frame. (Note: LONGITUDE_FACTOR = 1 / cos(LATITUDE_CIRCLE_IMAGE)

Finally determine the extent of the first search zone (see the following picture) as :

LAT_FROM = LATITUDE_CIRCLE_IMAGE - DELTA_ANGLE

LAT_TO = LATITUDE_CIRCLE_IMAGE + DELTA_ANGLE

LON_FROM = LONGITUDE_CIRCLE_IMAGE - DELTA_ANGLE * LONGITUDE_FACTOR

LON_TO = LONGITUDE_CIRCLE_IMAGE + DELTA_ANGLE * LONGITUDE_FACTOR

Add the four corresponding new variables to each Data Frame. Having defined the 1st (angular) and 2nd (circular) search areas, the search will be conducted in two steps: in the first one the original database will be filtered to consider only the craters belonging to the angular search area computed at Step 6, in the second one, this reduced number of craters will be further filtered by distance, to consider only the ones inside the search circle of radius SEARCH_RANGE.

1st and 2nd search zones for a given crater

Program Outputs

1) Search Zones

As described before, for the 5 groups of craters identified during Week 2, in Week 3 the program identifies two types of search areas:

Angular search area, determined by the new variables LAT_FROM, LAT_TO, LON_FROM, LON_TO

Circular search area, determined by the new variable SEARCH_RANGE

All these variables have been added to the Data Frame for each crater within the 5 Groups (see Tables below). The results, for each group are represented in the following pictures and tables (Note: in the tables the new variables are indicated by green cells and the supporting ones by orange cells):

Group 1: (0, 50) km

Group 2: (50, 100) km

Group 3: (100, 150) km

Group 4: (150, 200) km

Group 5: (200, 250) km

TABLES

The summary of numerical findings for all the craters in the 5 Groups

PROGRAM

During Week 3 the program has been updated, and consists of the following files:

Week 3 Script.py: the Main program for week3 assignment

Week3Analysis.py: containing support procedures for week3 assignment processing

MarsUtilities.py: support utilities for week 2 assignment processing

MarsConstatnts.py: some useful constants

SphereDrawing.py: provides graphic support with matpllotlib

Here are the links to the files:

[sempre-caro-mi-fu]

Assignment Week 2

The Program

The Program consists of three Python Modules:

Week 2 Script.py: the main program: reads the Mars Craters Database and filters the data in order to reduce the total number of craters to be considered in the following analysis.

MarsUtilities.py: contains the function performing all the filtering described at point 1.

SphereDrawing.py: contains a set of utilities to draw the model of Mars along with craters.

1: Week 2 Script.py

2: MarsUtilities.py

3: SphereDrawing.py

SOLUTION STRATEGY

Final Codebook

After some additional verification in literature, I have decided to use an additional variable in my personal Codebook: MORPHOLOGY_EJECTA_1. This variable describes essentially the type of ejecta coming out of the crater. In my case, I’m interested in ejecta classified as “Rd”, or Radial Ejecta, interpreting this classification as properly describing the cluster of debris ejected after the impact, and radially distributed around the Main crater.

The final Codebook then becomes:

CRATER_ID:  crater  ID  for  internal  sue,  based  upon  the  region  of  the  planet

CRATER_NAME: the name of the crater (where applicable)

LATITUDE_CIRCLE_IMAGE:  crater latitude

LONGITUDE_CIRCLE_IMAGE:  crater longitude 

DIAM_CIRCLE_IMAGE: crater diameter

MORPHOLOGY_EJECTA_1  –  ejecta  morphology  classified.

Data Processing

Having defined the final Codebook, I can describe the way I intend to proceed (i.e.: the way the Program is constructed) to process the Mars Craters Data Frame in order to be able to find the relations I’m seeking for:

Read the entire Mars Craters database into a “pandas” Data Frame

Working with the Data Frame, extract just the 6 columns belonging to the final Codebook

From point 2, consider only the craters classified as “MORPHOLOGY_EJECTA_1 = Rd”

From point 3, consider only the craters with a diameter between 0 and 250 km

From point 4, divide the resulting craters in 5 groups based on diameter: (0..50) km, (50..100) km, (100..150) km, (150..200) km, (200..250) km

For each bin compute and show the distribution of crater diameters and their mean value

From each Group extract a sample of craters located around the mean value such that the resulting number of craters is not too big (say <= 10)

The data obtained at point 7 define the object of the research: for each of the identified craters in each Group, in future releases, the Program will search all the surrounding craters inside a circumference of radius 10 (TBC) times the radius of the Main crater.

Program Outputs

The following information is generated by the Program: the first 5 plots represent the histograms of the population of craters located in each of the 5 Groups defined at Step 5 before, together with the total number of “Rd” craters for that Group, and the corresponding mean value of the diameter, represented by a dotted blue line.

Position of craters on Mars surface after filtering and Tables

After the last filtering performed as described at Step 7, the number of candidate craters for the analysis has greatly reduced, especially in the first two Groups (0..50) and (50..100) km, where the original number of “Rd” craters was really high. The location of the craters is indicated on a sphere, representing Mars, by a small dot surrounded by a circle proportional to the diameter of the crater. In addition, for each Group, a Table containing the list of craters belonging to that Group, as generated by the Program, is provided.

Note: it is possible that not all the craters belonging to a group are visible in the relevant plot.

Group #1: 0 to 50 km

The craters belonging to this group are contained in the following table:

Group #2: 50 to 100 km

The craters belonging to this group are contained in the following table:

Group #3: 100 to 150 km

The craters belonging to this group are contained in the following table:

Group #4: 150 to 200 km

The craters belonging to this group are contained in the following table:

Group #5: 200 to 250 km

The craters belonging to this group are contained in the following table:

[sempre-caro-mi-fu]

First Assignment - Week 1

STEP 1: Choose a data set that you would like to work with.

Coursera propose several different data sets to work with, centred on different topics, most of which are relevant to the socio-economic conditions in America. I am a European and I am not particularly interested nor particularly competent  in working with such data, on the other side I have followed with extreme interest the ongoing exploration of Mars, therefore I have decided to select the Mars Crater Database data set to play with, even if being not a geologist, I’m not particularly familiar with the topics of crater formation, crater distribution and in general with all the related stuff. The Database is the result of a huge work of classification performed by Robbins, producing a large set of data, consisting on the classification of 384343 craters, based on 10 different variables described in the Course Codebook.

STEP 2. Identify a specific topic of interest

The 10 variables describing the craters are summarized in the following:

CRATER_ID:  crater  ID  for  internal  sue,  based  upon  the  region  of  the  planet

CRATER_NAME: the name of the crater (where applicable)

LATITUDE_CIRCLE_IMAGE:  crater latitude

LONGITUDE_CIRCLE_IMAGE:  crater longitude 

DIAM_CIRCLE_IMAGE: crater diameter

DEPTH_RIMFLOOR_TOPOG: average  elevation  of  each  of  the  manually  determined  N  points  along  (or  inside)  the  crater  rim

MORPHOLOGY_EJECTA_1  –  ejecta  morphology  classified.

MORPHOLOGY_EJECTA_2  –  the  morphology  of  the  layer(s)  itself/themselves.  

MORPHOLOGY_EJECTA_3  –  overall  texture  and/or  shape  of  some  of  the layer(s)/ejecta  that  are  generally  unique.

NUMBER_LAYERS  –  the  maximum  number  of  cohesive  layers  in  any  azimuthal  direction.

HYPOTHESIS

I assume that the large craters on Mars are caused by external bodies (asteroids or comets) impacting Mars surface. I imagine that, if the speed of the falling body was large enough, then the impact has caused a sort of “waterfall” of debris mainly (but not only) in the direction of the impact. These debris impacts should have generated a number of secondary craters around the large one (the Main). The main question associated to this hypothesis is: is it possible to identify these clusters of secondary craters around the large ones?

STEP 3. Prepare a codebook of your own

Based on the selected question, I have decided to use the following variables:

CRATER_ID  

CRATER_NAME

LATITUDE_CIRCLE_IMAGE

LONGITUDE_CIRCLE_IMAGE

DIAM_CIRCLE_IMAGE

My principal question is: is it possible to identify these clusters of secondary craters around large ones?

Answering this question requires to specify the meaning of the word “large”: just to establish a starting point, I have decided to limit the analysis to those craters in the range 150 km to 250 km. Searching within the Database, it is possible to check that there are 55 such craters.

During a second review of the codebook for the dataset that you have selected, you should:

STEP 4. Identify a second topic that you would like to explore in terms of its association with your original topic.

A secondary question is: is it possible to find a correlation between the size of the Main crater and the average size of the craters within the related cluster? My (naive!) expectation is that the bigger is the Main crater, the bigger should have been the impacting body and the larger should be the average size of craters within the cluster.

STEP 5. Add questions/items/variables documenting this second topic to your personal codebook.

Another secondary question is: is it possible to find a “privileged” orientation of the cluster aligned with the Main crater? My expectation is that this privileged direction should indicate the direction of the impact.

STEP 6. Perform a literature review to see what research has been previously done on this topic. Use sites such as Google Scholar (http://scholar.google.com) to search for published academic work in the area(s) of interest. Try to find multiple sources, and take note of basic bibliographic information.

Using Google Scholar, I have performed a search in order to identify some literature about my questions, and I have found the following articles:

1

Alfred S. McEwen, Brandon S. Preblich et al.

The rayed crater Zunil and interpretations of small impact craters on Mars [https://doi.org/10.1016/j.icarus.2005.02.009]

2

Alfred S. McEwen1 and Edward B. Bierhaus

The Importance of Secondary Cratering to Age Constraints on Planetary Surfaces [Journal Article, 2006, Annual Review of Earth and Planetary Sciences, pages 535-567]

3

Joseph M. Boyce1 and Peter J. Mouginis-Mark

Martian craters viewed by the Thermal Emission Imaging System instrument: Double-layered ejecta craters [https://doi.org/10.1029/2005JE002638]

4

Brandon Preblich, Alfred S. McEwen, and Daniel M. Studer

Mapping rays and secondary craters from the Martian Crater Zunil [J. Geophys. Res.,112, E05006, doi:10.1029/2006JE002817]

5

Stuart J. Robbins, Brian M Hynek

The secondary crater population of Mars [https://doi.org/10.1016/j.epsl.2014.05.005]

STEP 7. Based on your literature review, develop a hypothesis about what you believe the association might be between these topics. Be sure to integrate the specific variables you selected into the hypothesis.

My hypothesis is that around a “large” crater it will be possible to identify a cluster of secondary craters originated by the impact, and that the average diameter of these secondary craters will be proportional to the size of the Main crater, and that it will be possible to identify a privileged direction around which the secondary are oriented, this direction indicating the direction of the impact.