Previously, I posted on the multiple regression method – in particular, the method employed in Foster and Rahmstorf (2011) – and how, when attempting to decompose the temperature evolution of my simple energy balance model into the various components (signal, ENSO, solar, and volcanic), this method encountered two large issues:
1) It did not adequately identify the longer term effect of the volcanic recovery on temperature trends, and
2) It largely overestimated the solar influence.
If you recall, I tested two scenarios in that original post. The first scenario was a linearly increasing underlying signal. The second scenario was a combination of a linearly increasing signal and an underlying low-frequency oscillation, resulting in a flattening of recent temperatures (one that was not caused by the combination of ENSO, volcanic, and solar influences). The goal was to see whether this multiple regression method could identify the flattening if it existed.
Thanks to Kevin C, who suggested and implemented a few improvements to this F&R method, noting them in the comments of that post: “…tie the volcanoes and solar together as forcings and fit a single exponential response term instead of a delay." This would allow a tail for the recovery from volcanic eruptions well beyond the removal of that actual stratospheric aerosols, and would not allow an over-fitting of the solar influence. After implementing this newer method, I would say that it is a large improvement (at least in diagnosing my simple EBM components) in the first scenario of a linearly increasing trend:
Unfortunately, due to the underlying assumption implicit in this method of a linear trend, it still has trouble identifying the recent pause present in scenario 2:
To see where exactly it is going wrong in scenario 2 vs. scenario 1, we can again look at the individual components:
As should be clear, the improvements suggested by Kevin C generally improve performance across the board. Unfortunately, in the 2nd scenario with the flattening, the multiple regression method still tries to compensate for the flattening by decreasing the diagnosed influence of volcanic recovery, therefore leading to a misdiagnosis.
Dikran Marsupial noted in the comments of that last post that “there are no free lunches.” Perhaps this helps drive the point home that assuming an underlying linear trend will lead to this misdiagnosis if the increase is not linear. I hope to investigate further the actual influence of solar + volcanic activity on recent temperatures using some GCM runs.