16.3 MONTE CARLO for PhYSICS APPLICATIONS

PhYSICS APPLICATIONS (Search & Rescue)

Click the image below to watch a video about how Monte Carlo methods can help save lives in search-and-rescue missions!!

Using Probability in Search-and-Rescue Operations | Prediction by the Numbers | PBS LearningMediaExplore how probability can be used to help find people lost at sea, even when rescuers have very little information, in this video from NOVA: Prediction by the Numbers. To improve its search-and-rescue efforts, the U.S. Coast Guard has developed a system that uses Bayesian inference, a mathematical concept that dates back to the 18th century. The Search and Rescue Optimal Planning System (SAROPS) uses a mathematical approach to calculate probabilities of where a floating person or object might be based on changing ocean currents, wind direction, or other new information. Use this resource to stimulate thinking and questions about appropriate uses of statistical methods.

Now it's your turn to help find the paddleboarder

Monte_Carlo 1 search and rescue intro.mp4

CALCULATING DISTANCE

Let's recall:

Velocity = Distance / Time

If you know a set velocity that an object is moving as it goes from one point to another, and you know the time it takes, how do you find the distance traveled?

We can rearrange it to say:

Distance = Velocity * Time

Let's Make some Assumptions & Calculate

Assumptions help us frame the problems and set bounds and limits to what we are trying to accomplish.

Let's assume:

The velocity of the paddleboarder is 10 km/h (and it remains constant).
The direction of the wave current carrying the paddleboarder is from the South to the North only.
The time that we are interested is 1hr of drifting with the wave currents.

Where will the paddleboarder be after 1 hour?

How far away from the original location?
And will they be found north, south, east, or west of the original location?

We will return back to these questions and answers when it comes time to validate our code!

MONTE CARLO:

But What IF... We don't know the exact velocity

It would be wonderful if we always knew the velocity at every single second of a given situation, but that isn't always the case, such as in the padddleboarder search and rescue mission. This is where Monte Carlo calculations comes into play to simulate many, many possible estimations. Let's discuss how we will implement these theories.

1) Historical data can be used to set the credible maximum and minimum current velocities; and

2) Random numbers within these ranges can be used to estimate these variables that are unknown or changing (i.e. current speeds and directions in our case).

3) Large data samples can be calculated and examined to help ensure a higher accuracy of the results

KEY POINT: Although Monte Carlo methods involve random sampling (since we are using a random number generator), the results are not arbitrary. The outcomes represent expected possibilities based on random inputs within a reasonable range. We can then calculate the probabilities of each of these outcomes using our simulation! In this way, Monte Carlo methods allow us to quantify uncertainty and make predictions, even in situations with incomplete or uncertain information!

Create YOur OWN CODE

INiTIAL COORDINATeS & RANDOM NUMBER GENERATORS

Monte_Carlo 1_paddleboarder step 1.mp4

Step 1: Location

Create a value for the initial location.

latitude: 41.3

longitude: -70.8

Step 2: Randomized Current Speeds

Create a randomized number for a current_speed. ranging from0 to 20km/h (depending on weather conditions).

Step 3. Randomized Current Directions Create a randomized number for current_direction ranging from 0 to 360 degrees.

Note: The labels "Code 1" and "Code 2" will come into play in a future example and don't correspond to the steps! Steps 1 and 2 will be done within the "Code 1" block and Step 3 will be done within "Code 2" :)

Coding Assistance: Click this link if you need help solving this problem.

SET UP MONTE CARLO RUN(s)

Step 4. From your previous trinket steps, copy "Code 1" and paste at the appropriate location in this new Trinket Window.

Step 5. Do the same for "Code 2" which should be placed a little lower in the code.

(Be sure to pay attention to the correct indenting needed for Code 2).

Monte_Carlo 1_paddleboarder step 2 - copying code1and2.mp4

You should see the location & drift.

If you zoom in, you should see points for 15 min, 30 min, and 1hr elapsed time.

Note: This trinket (seen below) will be used for Steps 4-9. Therefore, you will need to come back to this window as you progress through the problem.

Coding Assistance: Click this link to get the paddleboarder's route for 15 min, 30 min, and 1hr elapsed times.

Validating Your Code

Now that we have code that is working, how do we know we can trust it? How do we know the data and information is correct?

We must run some test cases to prove it is correct. This process is called code validation and it is an important part of the engineering process. This is basically setting initial conditions and parameters where you KNOW the exact outcome and checking that the resulting data is matching those values.

code validation - lET'S try it

Validating Code for Constant Speed of 10 km/hr with a current to the north.

Monte_Carlo 1_paddleboarder step 3 - Validating Code.mp4

Current to the north.

In order to validate the code, let's set a few items to constant values and check the new location (how far it is displaced from the original location and in what direction).

Step 6a: First, let's set the direction of the wave currents to be only south to north.

- - What would the bearing angle be for the direction of these waves?
  - How do you use the min_bearing & max_bearing variables to make this a constant direction?
  - Where would you expect to find the paddleboarder? (North, South, East, or West of the original location)?

Step 6b. Second, let's set the wave current speed to a constant speed of 10 km/hr.

- - How far would the paddleboarder have drifted after an hour? Let's test it!
  - What should the current_speed_min & current_speed_max values to set to in order to keep a constant speed of 10km/hr?

Click on the map you produced to find the starting point coordinates and the final point coordinates.

Enter these values into the calculator to find the total distance traveled. Does it equal 10 km?

Validating Code for Constant Speed of 10 km/hr, current to the east.

Current to the east.

Step 6c: Third, test it for a different wave current direction (i.e. only east to west).

Once you have both cases (North only & East only currents}) working, you can increase the Monte Carlo runs to multiple iterations in the next section to get more estimated locations.

Running Multiple Monte Carlo Iterations

Step 7. Reset the min & max bearing angles and current speeds to variable values (non-constant values).

Step 8. Find the "nums_simulations" variable.

Change this to 5 iterations. Run the code and observe.

Step 9. Set the same variable "nums_simulations" to 100 iterations and re-run the code. Make some notes on your observations.

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