Fun fact: 40 years on September 5, 1977, the first Voyager probe was launched. PBS did a fun documentary in honor of it (you may have trouble watching it in Canada).
My apologies dear readers, particularly the one that counts 😉, I had meant to get this out at the end of the week last week. Unfortunately, and somewhat fortunately, I was all over the place Thursday and friday, visiting my sister, then my mom, then we both took a two hour trip (thanks to her for driving) so I could see my grandparents (the drive back was very slow due to traffic from Irma evacuees). That was great because a delayed flight made it impossible for me to visit them during fourth of July and it had been since Christmas since I saw them last. But, in all the chaos, it completely slipped my mind until late last night when I was getting on my greyhound back from Detroit that I had forgotten to update you all.
I don’t have a lot to talk about. It boils down to two things: trying to make sure my mosaic is complete, and mapping all of Titan’s craters. Neish et al. 2016 presented a map of Titan’s craters, at which point the total crater count reached ~75.
I attempted to repeat this, without referencing this map or the catalog of known and suspected craters during my initial overview. A lot are not visible at this scale, but my current estimate is at 85 craters. I’ve started to go back and compare mine with this map and the catalog to see if there were a couple obvious ones I missed. There were, but there were also those I didn’t agree with. Which brings me to the next step on how to move forward.
Which really just consists of a few questions:
How do I check that what I’ve done is reasonable?
Can I even do that?
Should I go through the entire list of previously cataloged craters to make sure I checked all the areas?
I did not see any craters near the poles; I don’t understand how anyone can.
Should we expect craters nearer poles to be warped circles due to projections? My map is mostly complete. I have images in some areas where the Neish et al 2016 does not, but I’m still missing others.
The top right of the mosaic in particular. I checked my list of BIBQI images and I have the same images as Alyssa. I did have some BIBQH images missing that she has. Although, a lot of these are duplicates of BIBQI images. So that’s where I am at with that, but I could still use some suggestions.
I am going to keep trying to find those missing images, but then I’m going to start looking at the crater morphology unless instructed on how to improve my crater population.
In my last post, I discussed how we can try to minimize equation (7) to ascertain what the radii and d_center could be. Using Sinlap, I know the profile distance (Dp) is ~77km, and the radii could range from 50% to 200% what literature assumed based on radar imagery. I used these to get a complete picture of the parameter space. By plugging in these values, I created a 3D field for f(rl,rr,dc). I show it in the video below. This gives a value for every point in 3D space, but it’s difficult to learn much from the full plot (left). I filter out values >0.05 to locate where the function approached 0 (right).
What we see is that there exists a curved plane of possible points for these values, but even in this plane, points seem to converge to a symmetric pattern of circles. The transition from higher to lower points is clearer when looking at points less than 1.
Figure 2a. This Figure 1 from the dc by radius (same on left and right).
Figure 2b. This Figure 1 from the left radius by right radius.
These views have interesting implications. There are points of low dc or r where this function can be met, but the lower (blue) points cluster at higher values. Meaning, The center is more restricted when closer to the profile.
However, I we don’t know if this will translate to a unique range of center points. That is to say, lets visualize the field of center points this range of radii and dc values could produce.
Figure 3a. Soi Crater. All the possible center points using the range of points for rl, rr, and dc over a range of 50 increments to plot more easily.
Figure 3b. Ksa Crater. All the possible center points using the range of points for rl, rr, and dc over a range of 50 increments to plot more easily.
There is no discernible pattern in the plot of all possible center points (Figure 3). All there is are lines of possible points using the defined ranges of each variable. However, we see a pattern when we only plot the center points produced using the range of values in the curved plain of f(‘x’) -> 0 for dc, rl, and rr (Figure 4)
I think the attempt to minimize the function is flawed because, as we see above, there is no one area were the function approaches 0. However in conjunction with radar imagery, we can better rule out unlikely points. And, I think this figure suggests that might be the trick. It isn’t obvious in this picture, but a tiny black dot indicates the literature assumed center.
The function is filtered to only plot points corresponding to function f <= 0.005. The red point represents the point with the lowest value for for function f(). Presumably, I should get a similar result if I ran this function using Matlab’s minimizing function over the same constraints (min and max rl, rr, and dc).
If we repeat the same process, focusing closer to the center, the lowest point is still in the top left. It appears the absolute lowest point is a matter of its position to the topographic profile. It explains why changing the variable constraints had such a large impact on the final result. We can continue to filter out points, going as low as <=.0005. Even then we have a decent number of possible points if it is allowed to run with small enough increments. It constrains the possible points, but it will not give us a center value.
However, it is not useless. Up to now, we have relied on radar imagery to find the center, but with this we can adjust the center to better fit the topographic data. I ran the numbers for the literature values for Soi to see what value for function f it would produce and got ~0.8. Thats not far off, but it could be better.
The question becomes whether or not this is too drawn out or convoluted a process. The goal is to use the topo data to update the crater parameters, and in that sense this gives the crater parameters at least more small piece information to rely on.
In the last post I wrote about my attempt to estimate crater radius and center location using Sartopo data. Lets review.
I use this hypothetical circle to idealize what what a crater would look like. In reality, it isn’t a perfect circle. We can see in topography data that the rims are not equal distances from the center. The point closest to the center is dc, which separates the left and right into two right triangles. Each side can be thought of as some fraction x or y or the total profile length Dp. We can define Dp as follows.
Script based minimizing
I approached this thinking, how can I minimize these two equations? I thought, x, r_left, r_right, and d_c are all solvable if we assume we know where the center is. So I created a script to solve for Dp at each possible point of the center. It seemed promising, where each equation seemed to produce a circle of points that approached the desired Dp/ Subtracting the two from one another, which should produce a 0, would hone in on exactly where they meet, focusing on two points, one near the center found using the radar image and one far outside of view. Unfortunately, it was too far off. I thought it was simply a mistake somewhere in the script, so I proceeded to create a write up starting by reworking through the math I used.
We already have the equation for Dp above, but we also can find the other variables,
such that Dp=Dp. The calculations will be slightly rounded, but the net result approaches Dp because I use circular logic. This was not what I was getting. I was perplexed, and after staring for a while, Kevin lent me a hand. He pointed out that I was multiplying by x instead of dividing. That concluded this failed attempt.
Minimizing with MATLAB function
Beginning again with Dp,
This still leaves us with an function that doesn’t use Dp. I made a mistake in my logic, which I only just realized now, even after writing it up.
I moved x and y to the other side of the equation, but also kept it on the left side. This lead to incorrect and inconsistent results.
Luckily, the steps I used, of which I have more to say (later in this post) are already in place, so I can easily adjust the function and run it again. Where the new function is found to be as follows.
Keep in mind, x can be solved as a function of the other 3 variables are a matter of geometry. The other three numbers have a more direct relation to the center as a function of Dp. To my surprise, the initial results are more promising.
Literature put the center at the red dot in the center, and this estimate puts it at the yellow dot with a red rim. There is a bit more I’d like to do such as play with the constraints and plot a circle to see how well it fits the rim and radar image.
Visualizing and translating the results
This is a necessary thing to discuss because it took several days to take these variables and output values for the center lat and long. I ended up working this out multiple times from different points of view to find the center.
I thought the logic was sound, but I kept getting results that did not make sense. I have x and y values for each point on the profile, but when I found the distance between points using the x and y values, they did not translate into lat and long conversions appropriately. The biggest problem is that the y points don’t change as much as the longitude. X appears to change consistently with longitude, but I found it still to be inaccurate.
I compared changes in longitude with changes in x. I converted each of these to the other and found the two did not translate. The obvious question to ask is how I am converting between the two. Distances in kilometers were converted to lat and longs with,
%where the inputed latitude for the center is used
rLat = dy / (pi()/180 * R);
I wondered if there was something about this equation that wasn’t consistent or that didn’t apply for Titan, so I tried to compare it with another method that goes from changes in lat/lon to changes in kilometers. The method comes from one Andrew Hedges. I chose this because its the recommended method from JPL.
dlon = lon2 – lon1
dlat = lat2 – lat1
a = (sin(dlat/2))^2 + cos(lat1) * cos(lat2) * (sin(dlon/2))^2
c = 2 * atan2( sqrt(a), sqrt(1-a) )
d = R * c
I wondered if I had inputed this wrong, so I changed R to that of earth and tested it with the calculator at the site I link to above and it matched. I used this to calculate the profile length and it was off ~11km from the dbidr. Using just along x, I got a number ~5 times higher than the longitude and nearly as much higher than the dbidr value, meaning its an order of magnitude larger. dbidr ~=77km, lat/long~=66km, dx ~=300km. Therefore, lat long more reliable than the x and y values.
I don’t know if I am wrong to use these lat and lon conversions, but the x and y numbers do not work for solving in the x and y domain. So I used the initial lat and long values to to define the left rim, the dc intersection, and the center in x and y. Changes in lat and long between the left rim and dc, plus the results of the minimization give new x and y for the center. I can then find the change in x and y between dc and the center, or from lr and the center. I did both and got the same resulting lat and lon.
I feel like there must be something about the nature of the sartopo x and y data that I do not understand because the alternative is that it is wrong. Obviously the error is probably with me, but I have gone through this so many times in some many ways to try and hone in on the problem and its lead me here.
Its also worth noting, I wrote the script to use the dx/dy, dlat/dlon depending on the slope of the profile and its general location relative to the center because that will influence whether we are moving to higher or lower lat/lons. Meaning, it isn’t about how I am using the changes in x/y.
I read through Catherine’s paper on Titan crater topography, and realized that roughness wasn’t the way to go about it. I’ve since realized she left a comment saying as much on my last post. I’ve changed the roughness parameter to relative depth, the measurement that Catherine uses to see how much a crater has changed from its state (as compared to Ganymede craters).
I calculate depths using the same technique that Catherine does finding the lowest point on each side of the center of the crater. I used the depths for Ganymede (Bray and Schenck papers) I found a while back when I worked to create meshes for the tekton code. I’m still faced with the same issues in regards to the very large craters, but that isn’t the main priority at the moment. I know Catherine included Menrva in her data, so I am curious how she determined what a Ganymede depth for a 400+km crater should be.
Best Fit Circle
The circle fit that I discussed before is a function developed by Izhak Bucher in 1991. It works by using a little matrix algebra that I am having a little trouble follow. I had hoped to decipher exactly how it works before I updated you all but wasn’t successful. We start with a matrix A consisting of three columns: x, y, ones(length of x). This is divided into a vector = -x.^2+-y.^2. The result is a vector of the x and y center points and can calculate. My knowledge of matrix division is a little rusty, and from what I could find a\b = inv(a)*b but inv only works with square matrices. Without really understanding this step I can’t really explain the method exactly, merely that it tries to best fit the circle to all the given points.
I just reread it again, and I think I may have misunderstood what you were getting at. It doesn’t use RADAR imagery if that is what you are worried about. It uses the position of the rim from multiple flyovers.
If we look back at one of my figures from my last post, we can see how the rim is identified in each sartopo strip. It’s these sartopo points that the circle is being fitted to. The radar image is merely there as reference.
I thought this was a rather good way to incorporate all the data when we have multiple points available. It allows us to fit the shape to all the data and back out exactly where the center should be. Its when we have less than two fly bys, like in the example I gave last post, where it gets difficult to solve. I don’t like having to make the assumptions, but I’m finding it difficult to find a better way when we have such little data. A circle requires 3 or more points.
Single Sartopo Profile Approach
The use of the Chi-squared test is an interesting approach I didn’t think of. I haven’t done it yet because I hadn’t realized I had a response on my blog. However, I have used it before in a statistical analysis class back at Tech. I still have all my notes for this class, so I think I can do this. However, from what I can ascertain from an initial overview, it compares the modeled to the observed. I assume I would input a range of possible values for each variable to get different fits. Then the chi-squared test would be done on the fit and the actual data. It seems that I am to just compare two numbers.
The variables x and y are not exactly two unless the two radii are the same. So, we have two equations with five unknowns: r_l, r_r, d_c, x, and y. You could try a minimization process to get the ‘best guess’ for these values. Start with reasonable assumptions and see which combination produces the smallest chi-squared value when compared to the known value of D_p.
D_p = x * sqrt(r_left^2 – d_c^2) = y * sqrt(r_right^2 – d_c^2)
Here I am just calculating what the profile length (two calculated values) would be and comparing it to the single value it actually is. Why would I try to do a chi analysis when I could just find the closest value? I don’t understand what I would be performing it on.
Error and Uncertainty
I still have a ways to go with uncertainty. I updated the code to pull the random and systematic errors, but I don’t entirely understand how they fit together. Does one incorporate the other? I’ve added the two together for now and can add them to the outputted result of the code.
This will compose of two parts. First, an update of where I am at since the last post. Second, I’ll try to give a rough outline of where I hope to go over the span of the next few months.
In the first figure, we see a hypothetical crater. For a perfect circle, if we know the distance from the center to the center of the profile (from rim to rim), we have a right triangle, mirrored on the other side. The radius is r=sqrt(d_c^2+(D_p/2)^2). I use this to plot where previous literature suggests that the crater rim is.
If we find the rim using the topography, the rim won’t always aline perfectly with the center, but with only 2 points we can can’t infer where the center should be. So in the event of a single profile, we have to average the radius using the distance from the center to the rim. Obviously, r_l and r_r probably won’t be equal. The easiest estimation is to average that distance. In Figure 2 I suggest another technique.
In reality, the rims won’t match up to a perfect circle, but with only two values there isn’t much that can be done to approximate a good radius. I use the same idea that r=sqrt(d_c^2+(D_p/2)^2). I don’t actually shift the profile over, but by using the d_c and this equation, it assumes the two rims are centered around the given center. I suppose I could go a step further then find where the center would be for a circle of that radius using these two points. I didn’t do it in the Soi example (Figures 3 and 4), but I think I’ll be doing that because the center has to match up some how? But I feel like I don’t have much justification for it…so should I?
So in Figure 3 we can see what I was talking about applied to a single profile. The center of the profile is found by finding the closest point to the center (as defined by literature). Then I search for the rims using the max height over a defined profile range (based on past literature and the radar images). Soi is perplexing. Despite being perhaps the freshest crater its topography is absurdly not symmetrical. This upended what I was doing with Menrva and required I take a different approach. That was why I decided to have users look at this profile and input were to search for the rim.
I translate the data in Figure 4 to a radar view to see where the data is relative to the radar imagery. The red circle uses the center as defined by my the center point in past literature and the radius I solved for. Like I said before I could shift that to a center that matches the rim.
Now, if we move onto another example, Menrva (Figures 5 and 6) we have multiple profiles to estimate the circle.
Here, I can ascertain a radius and center by creating a best fit circle to the data. Luckily, such a function already exists on the net. There are three beautiful profiles in the lower part of the crater (30123, 30124, 30134). These are very close though so the fit using these profiles produces a fairly large crater, more than 425km and outside what we see in the radar image. Now if we include the top profile, where only a few gaps exist, a much nicer fit occurs.
The rims are in the same position as the others. It fits well with the radar image. With the bottom profile, not in Figure 5, the crater is shrank to below 380km and doesn’t fit the radar image well. I choose to stick to the top four profiles to get what we see in Figure 6. We get 395(km) using the top four.
The red circles I plot in figures 4 and 6 use the rectangle function with curvature. If we want to consider rectangular shapes there are probably approximations or fits I could do to rectangular shapes. I would need some feedback as to when and how to guide that.
In regards to crater shapes, I had hoped to read Catherines 2013 paper on crater topography by the end of this week but got lost in matlab. Moving forward I plan to read this next because I feel I need to understand how to identify where to look for rims. Because the basic shape of a rim and crater floor is not what we see here. I like what I have made above and hope to work from it moving forward, barring much pushback from Catherine or Mike.
I also need to estimate roughness. I have a few ideas on how best to do this. I could do basic statistics on the center of the crater, and I could do roughness parameters similar to what Kevin is doing. I’ll probably do some reading to figure out what past studies have done to characterize crater erosion/roughness and work with that. Depending on how time consuming that is, I’d like to finish up with working with Matlab in mid to lat July.
Finish work with Matlab and crater characterization (July 31st)
June 23rd-June 30th: Read about crater topography and crater erosion while continuing work with plot on rim and diameter estimations
July 1st-July 14th: Move on to reading about crater roughness as it relates to erosion literature review
Also 4th of July and begin prepare/review my Astronomy night presentation.
July 15th-July 21th: Apply what I learned about crater roughness to Matlab code.
July 22nd-July 28th: Finalize Matlab code and output figures.
Map Craters on Titan (August 31st)
July 29th-August 4th: Finish Titan radar map
I’m a little worried this may be more difficult than I expect. Will Mike be around? Does he know ISIS? I think I should be okay but still.
August 5th-August 31st: Map the craters on Titan
It’s difficult to asses how long this will take. I’ll probably start mapping while periodically updating you all getting feedback
I’ve reached out to Zibi again to try and make some progress with her. I’d like to meet with her a little in July, maybe early August (virtually). My goal with that is to have the code accepting the default by early August. (Consider reaching out to Mike or others?)
Finalize crater work and dig deeper into tekton again (October 31st)
11th- September 22nd: Use the crater map to estimate crater parameters for all the craters (with data).
Begin with the easiest craters. In the last week, look at the hardest to try and estimate that data as well.
A large table will review all the results. As to what craters to highlight with more detailed figures will be decided later with feedback. These are the figures I would include in a paper/thesis.
Meet with Zibi again to ensure I understand how to work with input files in Tekton.
Prepare for Titan meeting and DPS.
Also begin having to work on the Impact Craters course (23rd to 30th)
I’ve another project to work on in addition to my original project studying the relaxation of Titan craters. If you’ve followed along, you’ll know there have been a number of issues trying to get the Fortran code to run and take the input files. Catherine doesn’t have a lot of background with it, and my collaborator is very busy with a number of other projects. Unfortunately, this has made for a very slow project. We aren’t casting it aside, but it’s being moved behind my new project of identifying and characterizing all of Titan’s craters. This has been done for most of Titan’s craters. However, we’ve now completed the final Titan flyby and have all the data we are going to have, so my task is to develop a systematic approach to search for all the craters that have been discovered and for any other possible craters. Then, I will use the topography data we have using Sartopo data to characterize crater size, depth and its other characteristics. Sartopo data is topography data obtained by calibrating overlapping radar flyby profiles that give the topography in the region where it overlaps.
I’ve begun by developing an easy system of functions to call the Sartopo data for a given crater. With the function ‘crater_elevation.m’ and the data file ‘sartopo_data.mat’ you can easily call the data profiles in the region by inputing the center latitude and longitude with the crater diameter.
For example, if we use the center lat and long of Menrva (425km) we’d put into matlab the following:
It returns a map of the region with the radar images overlain with a colorize height profile where ever there is sartopo data (Figure 1).
A similar map is shown with each sartopo profile colored by profile number (Figure 2). This is derived from the sartopo data file names. SARTOPO_T00AS01_B12_V02_170315.CSV is 0000112, or 112 when inputed into matlab. Where the number is the fly-by (in this case fly-by A is 000), then the number after S and the number after B. The numbers after V are unchanging. The purpose of this plot is to just give perspective to where each profile is on the map.
All of the profiles are shown along the distance of their own profile track (Figure 3). You can use this to identify which profiles are of interest for further studying.
The next step is identifying the crater depth and rim height and possibly width and height of central peaks/pits. I’m currently developing a code that will take the center lat and long and the crater diameter. It assumes you know the diameter, but I’m only using it to tell the code where to look for data. Once I’ve done that, I’ll go about finding where the rim is highest on each side, and the crater diameter is just the rim to rim distance. The function is still in development, but right now it also takes a vector of sartopo profiles that you want it to calculate the crater information for. I’ve added a step that allows the function to run without that defined. It will determine which profiles are inside 1 radii from the center lat and long and use these profiles.
If I make progress before the end of the day, I’ll add more info. The current plan is to find the center of each profile (from rim to rim of ~1.5 of the defined diameter) then look for the max height on either side. Then look for a point of flat lowlands between these two points. I suspect we may get profiles that run along the edge, so you may see a profile that forms like a V. I’ll try to develop a way to omit these profiles because it doesn’t map the inside of the crater. Another issue I’ll have to deal with is that the profiles don’t run through the middle. Even in Figure 2, profile 30112 doesn’t go exactly through the middle. The current idea is use the crater center thats given to the function and a bit of geometry to calculate the data for directly through the center. It’ll be interesting to see this done for a crater like Menrva where we have 4 solid profiles to do the calculation on.
Over the course of 12 days, we explored parts of Arizona, Utah and Nevada for the planetary field school. I took a range of photos for each stop and have/am posting each set on facebook. Here, I’ll pick one and offer a short description, but given the large number of sites, I’ll keep the descriptions short.
We began by taking a late bus to Detroit. The weather was dreary and not the best way to start the trip. Luckily, that didn’t last. We met up in the Las Vegas Airport around lunch before driving nearly 4 hours to Mather Campground. I won’t talk much about camping, but we changed location everyday, and I posted pics from all our stops on Facebook if you’re interested.
Grand Canyon Hike
We spent the first half of the day hiking down the Grand Canyon, maybe halfway down. The grand canyon doesn’t need much background. Its creation was related to a combination of fluvial erosion, uplift, and the types of rocks in the region.
My stop was the lava river cave. The cave is the result of cooling of the outer shell of a lava river encasing the lava and insulating it. This allows lava to flow further, posing increased risk on locals. These are particularly interesting because they’re capable of maintaining constant temperatures, humidity, and other environmental factors. This makes them a prime spot to search for life on worlds like Mars. They also make for a great base on Mars or the Moon to protect from cosmic rays.
Elden mountain is a silica volcanic dome in the SF peaks, close and perhaps a part of, the stratovolcano in the region. This stop came after LRC which was such an amazing experience. I remember thinking after the GC hike, why are we doing another stop, how can we compete with the GC? RM was more impressive than I expected. I had the same experience here. LRC was amazing, and in my opinion lava domes aren’t the most exciting features. That said, I think this was the stop I started to realize even the most mundane stops had a lot of awesome things to offer.
San Francisco Volcanic Field/ Peaks/ Glaciation
We saw a variety of formations such as cinder cones in the SFVF and even a stratovolcano (top right) thats responsible for the SF Peaks (bottom), and we ended day 3 but looking at evidence of glaciation in the region (top left).
Before the trip we had an unknown location that we had to map with a satellite image. This was a our chance to see how well we predicted what happened. Grand falls is a river that was a later altered by a lava flow that went into the river channel changing the region.
Rattle Snake Crater
RSC was a surprise stop, where we tried to repeat what we did at GF without the preliminary analysis. It isn’t really a crater. Its actually a heavily altered Maar Crater. A maar crater or volcano. A MC/MV is formed when a pocket of magma interacts with water, turning the water to vapor, creating a fast and large increase in pressure, exploding to create a crater like structure. There were two sides of this. I felt very proud to climb to the top of one side (and was the first :)). Of course, it was hardly the biggest hike we did, but I loved it nonetheless.
The side I hiked
Viewing the side I hiked from the other side
The other side.
Meteor Crater is iconic, forming about 50ka. Just outside of Flagstaff, MC is an impact crater about 1.2 km in diameter and 170m deep. We saw a lot of craters that were not craters, so it was nice to see a confirmed crater. Gavin got to lead this talk (see pic).
SC is a cinder cone. At ~1ka old, it was one of the youngest structures that we saw. It’s a part of the San Francisco volcanic field, and you can see it in one of other photos.
We camped at SC, but we also had a chance to hike up it. SC is another cinder cone, not a crater. At over 300m, its a decent hike. I tried to hike up it, but I couldn’t keep up. I ended up hiking part way then going back, but a few others made it all the way to the top.
This is a volcanic plug or intrusion that feed some type of volcanic structure. Oz says we suspect it feed a Maar Volcano, but he was unable to explain why when I asked how we know.
Goosenecks State Park
This is one of the best examples we have of a meandering river.
Is it a complex impact crater or a salt diaper? At ~5km in diameter, its thought to be an impact crater, but it is indistinguishable from a salt diaper (where less dense salt makes its way up to the surface). Past authors have claimed to have found evidence of shock, but the evidence is scarce.
But aside from the science, we hiked in and it was exhausting. Just look at the stats (it was a little longer then it says because I accidentally paused it for a bit).
A confirmed salt diaper that we used to compare to upheaval dome.
Overlook at sapping valleys and paleochannels
A man made geyser due to drilling and creating excess pressure from CO2. It is a “cold” geyser but it looks just like a traditional geyser we associate with early life on earth and possibly on other worlds.
San Rafael Swell
Marscvale Volcanic Field
Coral Pink Sand Dunes
This was probably the best stop of the trip. It was like playing in a big sand box. Because we spent the night here we were able to take moonlight photos too!
Inverted Topography (St. George)
Water rich in minerals is pressurized in the ground and overtime forms these veins of gypsum.
Petrified Sand Dunes
Similar to the coral pink sand dunes, these are older, petrified into rock. This was a great contrast to the CPSD and was beautiful site to see. Although, I was careless, given that it was the last stop, and let myself get sunburnt :-/.
Day 12 and 13
Flight was canceled, so we got to rebook and planned an amazing trip in vegas! Went out and it was awesome. I gambled for the first time, and I stuck to my budget, spending half as much as I had allowed myself too. It was fantastic. I was so happy it turned out like this.