In some followup discussion with Barry Bickmore (by email) and Kevin C (at Skeptical Science) it became clear we were missing something in the analysis of Roy Spencer's climate model. Specifically, Bickmore's Figure 10:
differed significantly from Spencer's 20th century fit even though he was ostensibly using the same parameters. If you look at the year 1900 in particular, Bickmore's Figure 10 has a temperature anomaly slightly above zero, while Spencer's is pegged at -0.6 C. Bickmore did use a slightly different PDO index than Spencer for his 1000-year graph (figure 10) - but more importantly, he took Spencer's -0.6 C temperature as the starting point in the year 993 AD, rather than as a constraint on temperature in the year 1900, as it actually was in Spencer's analysis. It turns out that to actually match Spencer's 20th century temperature fit the starting temperature in 993 AD needs to be extraordinarily, far beyond impossibly, low. We'll get to the details shortly.
First, returning to the results of my previous post, Spencer's model reduces to Eq. 18:
(18) T(t) = Te + A0e-t/τ + β Q(t)/c h
(see the previous post for definitions). Note that since c h/α = τ the factor multiplying Q(t) can also be written as β/τ α. The convoluted PDO index (PDOI) function, Q(t), is defined as (see eq. 16):
(19) Q(t) = ∫-∞t PDOI(s) e- (t - s)/τ ds
and note that it depends on the PDO index going back in time (to negative infinity in the integral) - though values from more than a few multiples of the time constant τ in the past contribute only a minuscule amount to the total. Spencer seems to have used a particular source for the PDO index (downloadable here) which begins in 1900, so at least for the first few decades of the 20th century Q(t) is missing important information about pre-1900 index values. Bickmore came up with an alternate source (MacDonald et al 2005 - reconstructed from tree ring proxies) which goes back to the year 993 AD. These sources aren't identical in the 20th century where they overlap:
Figure 1: The two raw PDO indexes (R code)
but at least from about 1925 on they seem to be very close. They do strongly disagree (going in opposite directions) in the 1910-1920 period and there are some other short periods of divergence, but generally the proxy index seems to match the recent PDOI quite well.
Let's see what the convolution Q(t) looks like for these:
Figure 2: Recent PDO with convolutions Q(t), for several different time constants τ (R code)
Note that PDOI is unitless, while Q(t) has units of years. If PDOI was a constant value, then Q(t) would be constant a factor of τ years larger. If PDOI was a step function going from 0 to 1 in a certain year, Q(t) would be a smoothed step approaching that constant value τ exponentially over a few multiples of τ years. So in general the convolution should have the effect of smoothing, amplifying, and shifting forward in time the original index values, as we see in this figure.
The effect over the 1000-year record is even more dramatic:
Figure 3: Proxy PDO with convolutions Q(t), for several different time constants τ (R code).
The red curve (τ = 30 years) represents the value Spencer found in fitting 20th century temperatures, and corresponds to Bickmore's Figure 10 curve (reproduced above) except over the first few decades where the transient from model temperature initialization has an impact. In particular you can see that 20th century peaks in Q(t) is nothing special, in fact (with this 30-year time constant) it has been almost stable since about 1700, with about half the variation of the high peak around 1600 or the dip around 1200. And as Bickmore noted Q(t) had completely the wrong sign, and even been strongly negative, during the putative medieval warm period (typically attributed to the period 950 to 1250 AD).
To provide some clarity and avoid the slight differences between the recent PDO and the 1000-year proxy PDO, I spliced the two series together in the following graph, which compares Q(t) values for the recent PDOI alone, and including pre-1900 values from the MacDonald data set (this time only looking at τ = 30 years):
Figure 4: Spliced PDOI with convolution Q(t) for the spliced series, compared with the recent-data-only Q(t), for τ = 30 years (R code).
The pre-1900 data has much the same effect on Q(t) as Spencer's added initial temperature T0: the differences disappear exponentially over time, so that by mid-century the two curves are quite close. The main difference is that using the real (MacDonald) data, positive pre-1900 PDO values push temperatures in 1900 more positive, not negative as Spencer chose to do in his fit.
We can now use Spencer's parameters and equation 18 above to turn these Q(t) curves directly into fitted temperature curves. Spencer's parameter values were:
α = 3.0 W/m^2K
β = 1.17 W/m^2
τ = 30 years
A0 = -0.6 C (temperature in year 1900, if we define time as t = (year - 1900))
and using these with the spliced and recent-only Q's gives:
Figure 5: Temperature from Eq. 18 with Spencer's parameters, using two different forms for Q(t). The red curve uses the spliced Q(t) with an additional initial downward shift so A0 = -0.711245 C (R code).
Note that Spencer's model using the spliced PDO index as forcing (the green curve) shifts temperatures up slightly from the result with a PDO of zero before 1900, so the "initial" temperature with that PDO is no longer -0.6 in 1900, and the model doesn't quite match what we had for the 20th century. That can be fixed by making the A0 parameter in Spencer's model slightly more negative, resulting in the red curve here, which matches the original 20th century fit beautifully.
Bickmore was originally looking at this, with his figure 10 shown at the start here, to see how well Spencer's model did in hindcasting past temperatures. Obviously, given the Medieval Warm Period issue, it doesn't do very well to start with. But now with the analytic form we can hindcast perfectly, rather than having to initialize at some point in the year 993 and integrate forward. Spencer's model (remember fitted to 20th century temperatures) does the following when projected backward to the 19th century:
Figure 6: Temperature from Eq. 18 with same parameters as Figure 5, projecting back a century(R code).
Oops. That A0 term is wreaking havoc here. While the PDO index and Q(t) on their own are meandering around in a way that keeps temperatures somewhere close to zero anomaly, the actual model Spencer used projects that global temperatures in the year 1800 were 20 to 25 degrees C colder than present! That's several times larger than the global temperature change associated with the ice ages! I think we can rule that out.
I asked Dr. Bickmore to re-do his Figure 10 calculation to see how low a temperature was needed in his starting year 993 to match Spencer's 20th century temperature graph, with a value of -0.6 C in 1900. He sent me the following:
Figure 7: The black curve is observed temperatures since 1850; red curves are temperatures from Bickmore's matlab implementation of Spencer's model with initial temperature anomalies of -1 to -6 trillion degrees C in the year 993 AD.
It turns out you need to set the starting temperature to negative six trillion degrees in 993, in order to match Spencer's model for the 20th century. 6 trillion degrees. Wow. Now that's global warming!