Math Modeling: Day 18

Last time

Next time

Announcements
- Your exam revisions are due Friday.
  - You may discuss the problems with each other, and with me.
  - Print off a fresh copy of the exam, and submit both exams. Do not alter the original.
  - Note: I'd like them all back, even if you choose not to revise.
- Also due Friday: your data entry. I want each group to submit their data to me, electronically, so that we can start a group database.
  Submit one file.
Last time:
- We talked a little about the graph of methane. Interestingly enough, I just read this article yesterday: Oil and gas firms 'have had far worse climate impact than thought'
  What caught my eye was this line: Atmospheric methane had started to flatten off at the turn of the century, but rose again after a surge in fracking activity in the US and elsewhere. The industry, however, continues to claim that the energy source can be used as a "bridge fuel" because it has lower carbon emissions than oil or coal, but this fails to account for leaks and flares of methane and other gases during extraction.
- Per our discussion, I decided to model it with a cubic, with a sinusoidal oscillation:
  Methane emissions:
- We formed new groups for the first phase of the Fletcher project, and brain-stormed a little about what we are expecting to see in the data. Some good ideas, in answer to the questions
  1. What would we expect to see in the absence of any effect of climate change?
  2. How might we imagine the impact of climate change on the distributions of daily temperatures throughout the year, over these 140 or so years?
  - We brought in the idea of a "null model": no climate change
  - In the absence of climate change, each decade is equivalent. We would expect to see about equal numbers of records in each decade.
  - Several argued that, with global warming, we should see more "max daily maxes" in more recent decades.
  - Jon talked about the daily temperature distributions are different over the course of the year. That, with additional hours of sunlight, we might expect to see longer stretches of warm temperatures during the summer, and shorter stretches during the winter.
Today:
- Today I want to begin discussing non-linear regression, and I start in what might seem an unusual place: with Newton's method, which some of you will have seen before (and others of you may not have seen).
  The key to non-linear regression is linear regression! That may seem odd, but I will mention again that frequently in mathematics we often cast a problem from the non-linear world over to where we obtain a linear problem, then bring that solution back (and hope for the best).
  In the case of the linearizations we have done, we only did this once. In the case of non-linear regression, we will do it multiple times (iteratively), hoping for constant improvement.
  After I describe Newton's method, I'll
  - go through an actual example with you: my non-linear regression derivation; and then
  - give you a chance to play with it a bit, using this code.
    One important point about non-linear methods is that we may get more than one solution. I would like to walk you through that, as we begin.
Links:
- The history of fracking.
- Study: Fracking prompts global spike in atmospheric methane: "If we can stop pouring methane into the atmosphere, it will dissipate", he said. "It goes away pretty quickly, compared to carbon dioxide. It's the low-hanging fruit to slow global warming."
- The Fracking Industry's Methane Problem Is a Climate Problem (shows some of the methane leak impacts of fracking).
- Robert W. Howarth lecture (12/14/2019) on methane, fracking, and climate change: "Professor of Ecology & Environmental Biology at Cornell University, asking "Is Shale Gas a Major Driver of Recent Increases in Global Atmospheric Methane?" Howarth, who participated in the 2015 Paris Climate Agreement, among other things, is a world authority on the effects of human activity on the environment." He addresses the question of the shape of the methane curve at 1 hour, 5 minutes or so.
- "The IPCC stands by its equation of 450 ppm = 2 degrees C [of warming]....": For years scientists have said that if atmospheric levels of carbon dioxide reach 450 parts per million (ppm) the planet would heat up by an average of 2 degrees Celsius above pre-industrial levels. They have also said that if the world crosses that threshold, ecosystems worldwide would suffer serious damage.
  [ael: by the way -- ecosystems worldwide are already suffering serious damage.... The world won't "suddenly heat up" if we reach 450 ppm -- it's heating up right now, and we're seeing bumblebees struggle with the heat, for example. They pollinate our food -- think about it....]

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