### The association between the measure of diameter and depth among craters with or without layers.

The association between the measure of diameter and depth among craters with or without layers

### Introduction

The study about properties of craters on Mars, created by Stuart Robbins (2011), can allow us to understand crustal properties, surface ages and modification events.

This project is based on a subset of the Robbins Crater Database with 384343 craters. In our project, we studied about 2 main problems:

- The association between the size of craters (diameter) with depths and number of layers. The craters’ diameters can be larger when the craters have a large number of layers.
- The depth of craters with layers can have more effect on the diameters than the craters without layers.

We think that a large crater should have high number of layers, therefore they can have a higher depth and also make their diameter bigger. These association effects can be stronger when the number of layers is higher.

### Research Question

With that thought in mind, we propose the following 2 questions that could help us to have a better understanding about craters’ properties.

- Is number of layers associated with the measure of diameters among craters on Mars?
- Is the association between the depth and diameters simillar for all craters with or without layers?

### Methods

**Sample**

There are 384343 observations in which 364612 craters (94.8%) have no layers and the others are 19731 (5.2%), were from the **Mars Study** database.

The **Mars Study**, is researched by Stuart Robbins, presents a sample of Mars’ craters with their physics properties (e.g. location on Mars, size and depth, 3 kinds of ejecta morpholophy, number of layers).

**Measure**

Each of craters has **Crater_ID **which is identified internally based upon the region of the planet. Craters size are shown by **DIAM_CIRCLE_IMAGE **variables (units are km) which is the measurement of diameter of a non-linear least-squares circle fit to selected vertices on craters rim.

**DEPTH_RIMFLOOR_TOPOG **is calculated by taking average elevation of the determined N points along (or inside) the crater rim (units are km).

**NUMBER_OF_LAYERS** is determined by the maximum number of cohesivelayer in any azimuthal direction. There are 6 levels of number of layers (from 0 to 5). The new **Layer** category variable is symboled as 0 if the craters have 0 number of layers (called craters without layers) and 1 if the craters have more than 1 number of layers (called craters with layers).

In this study, to make it easier in analyzing data, the **DIAM_CIRCLE_IMAGE **variable is divided into 4 categories and each category has the same number of craters. The new variable **DIAM **is set values:

**DIAM**=1 if **DIAM_CIRCLE_IMAGE **less than 1.18

**D****IAM**=2 if **DIAM_CIRCLE_IMAGE** from 1.18 to 1.55

**D****IAM**=3 if **DIAM_CIRCLE_IMAGE** from 1.55 to 2.55

**D****IAM**=2 if **DIAM_CIRCLE_IMAGE** greater than 2.55

The project is analyzed by SAS program version 4.3. The source code is presented later in SAS program post.** **

### Results

**Univariate: **

There are 75% craters’ diameters less than 2.5 km. The mean of diameter (3.5 km) is greater than the median (1.5 km) and the standard deviation is 8.6 km. Therefore, most of craters have small diameters while there are some outliers have significant higher values up to 1164 km.

Univariate result for diameter

In general, the depth of craters is almost small with 99% less than 1 km. The maximum value of depth is 4.95 and the minimum is -0.42.

Univariate Result for Depth

The **diam** variable has 4 categories, each of them has nearly the same number of craters.

Nearly 95% craters have 0 number of layers. There are only few craters have 4 or five layers. The number of creaters decreases when the number of layers increases.

Frequency for number of layers

**Bivariate:**

1. Because the large amount of craters with no layers (94.8%), I can only draw a boxplot figure for craters with layers from 1 to 5. The figure below shows us the box plot of each layer with the diameter of crater. Noticed that there are small spreads within the layers and these box plots are very little overlap.

Boxplot for different number of layers, 1-5

The bar chart is then used with mean diameters of layers is added. Most of craters with no layer have small diameter.We can see that the mean diameters increase significantly with respect to the increasing level of layers from 0 to 5.

Barchart for different number of layers (0-5)

As expected, the Anova analysis showed the positive and significant association between the number of layer (category explanatory) with the measure of diameter (quantitative respone), small p-value (<0.0001) and high F value (1494.68). That is, the number of layers increases leading to the increasing diameters.

ANOVA test for the relationship between number of layers and diameter mean

2. Considering the associations between 2 variables diamter of craters (**DIAM_CIRCLE_IMAGE)** and depth of craters (**DEPTH_RIMFLOOR_TOPOG)** in 2 case: subset data on craters without layers (case 1) and on craters with layers (case 2). Based on the below scatter plot, we can deduce that there is association between the diameter and depth of craters.

Scatter plot between diameter and depth

Indeed, the two Pearson Correlation calculations give us efficient p-value (<0.0001) and the positive correlation numbers.

Then we can conclude that there is significantly positive correlation between diameter and depth of all craters. Futhermore, the correlation number in case 2 (0.73) greater than in case 1 (0.61). However we must notice that there are lot of data in case 1 and the values of craters’depth is distinct too much from the maximum and the rest. Then we can only conclude that there seems to have same deep associations in 2 cases. Futhermore, the craters with layers tends to have higher measurement of depth with higher levels of diameter.

Line Plot for the relationship between diameter and depth, with and without layer

### Discussion

**What might the results mean?**

The craters with have higer number of layers will have larger depths.

In high level of diameter, the craters having layer seems to have higher depth than the ones without.

**Strength**

Results are based on the subset sample of Robbins Crater Database.

**Limitation**

The number of craters without layer is too large compared to craters with layers. This might cause difficulty when we do test on craters with layers and without layers. Futhermore, it is quite tough to divide range of dimaters since the measure of craters is nearly 75% smaller than 2.55 km and the rest is larger.

**Recommend future research **

We can consider the role of craters’ locations impact on their porperties. That is needed to more research about the association between craters’ locations and their depths or diameters.

### Reference