In recent work on solar forcing, several different authors get very different results for the size of the effect. This is an attempt to try to understand those differences.

### The Problem

Solar forcing is the response of the average global temperature to fluctuations in the solar output, δT/δI, where I is the insolation at the top of the atmosphere and T is the globally-averaged temperature. A simple estimate given at the end of this post leads to the value δT/δI = 0.053 K-m^{2}/W. The problem is that the temperature data contain responses to several forcings - solar, volcanism, sea-surface temperatures, for example - and one needs a reliable way to extract the solar component. The problem is made more difficult by the fact that the fluctuations in solar input, due mostly to the 11-year solar cycle, are not large, typically less than 1 W/m^{2}. Methods that have been tried include extracting the solar part of the temperature by examining latitudinal variations, then using linear regression to fing the best value of δT/δI, and multiple regression using time series for several inputs - solar, volcanic, etc, to extract the solar component.

### Regression

The calculations described below use regression techniques. For readers who are unfamiliar with the procedure, the idea is to find the best fit to some function of time - it could be a series of data points rather than a continuous function - as a sum of one or more other functions. The basic procedure is to minimize the squared error between the function and the sum. This is perhaps most familiar in the form of finding the best-fit straight line to a time series. If the function to be fitted is δT, one forms Σ (δT - at - b)^{2}, where the symbol Σ denotes the sum over all the (time) data points. Then one chooses a and b to minimize the sum (for which there are standard formulas). More generally, one could use some other function (say, δI) to fit δT by forming Σ(δT - k δI)^{2} and finding k by minimizing. Still more generally, one could use several functions ("multiple regression") by forming Σ(δT - k _{1} δI - k _{2} δV - k _{3} δS)^{2}, where δI, δV, and δS are known functions of time, and minimizing to find the values of the k's. The set of "basis" functions that are used is usually physically motivated - the functions are those that are expected to "control" or "force" the behavior of δT, and hopefully all the important ones have been included.

### The Calculations

Camp and Tung (2008) removed the linear (global-warming) part of the surface temperature, then extracted a solar component by a spatial (latitudinal) filtering method. They did a linear regression of the result against total solar insolation, and got δT/δI = 0.18 K-m^{2}/W. A graph of their temperature and TSI data appears below:

Douglass et al (2004) did a multiple regression of satellite lower-atmosphere temperature data against sea-surface temperatures, volcanism, solar input, and a linear term. They found δT/δI = 0.1 K-m^{2}/W. Their temperature residual (after regression) with everything but the solar component removed is compared to the solar irradiance data in this graph:

The correlation is (to my eye, anyway) a lot cleaner in their results.

Benestad and Schmidt (2009) also performed a multiple regression of the temperature data against the GISS forcings, obtaining δT/δI = 0.09 K-m^{2}/W. They also did several numerical experiments on regression techniques, concluding that linear regression can overestimate the size of the effect. They also indicate that regression techniques in general can produce spurious results if the signals are noisy enough.

### Discussion

One way to think about multiple regression is as a decomposition of a vector in a multidimensional vector space, where the basis vectors aren't necessarily orthogonal. If the basis is incomplete, or if two of the basis vectors are almost parallel, the decomposition can become problematic. These vectors could be functions of space and time, so the decomposition becomes a functional expansion. For the solar forcing problem, if the basis functions are incomplete, as for linear regression (where there is only one basis function), indirect solar forcing (say, for example, in sea-surface temperatures), can show up in the direct component, making it larger than it should be. Similarly, if the signals are dominated by noise, regression techniques might start projecting noise onto noise instead of signal onto signal. It follows that the basis used for multiple regression really needs to be chosen very carefully to get acceptable results. But it is encouraging that the two calculations using multiple regression get nearly the same aswer.

Regarding the factor-of-2 difference between the simple estimate of δT/δI = 0.05 K-m^{2}/W given below and the actual results, there are several possible causes. First, the estimate didn't include water-vapor feedback (just the existing greenhouse effect, not the enhancement due to increased evaporation at the higher temperature). A factor 2 is not out of the question, but seems a bit large compared, for example, to the value of 1.6 inferred from studies of atmospheric cooling following the Mount Pinatubo eruption ( Soden et al, 2002). The IPCC climate models suggest a value between 1.5 and 2.5.

Another potential source of the difference is indirect solar effects. For example, the variations in solar output over a solar cycle are an order of magnitude larger in the UV than in the visible (1% compared to 0.07%; see, eg, Gray et al (2010) and references therein). This implies increased stratospheric heating, and there could be turbulent mixing and heat transfer between the stratosphere and the upper troposphere, eventually leading to increased heating of the lower atmosphere. This is likely to be very hard to model.

Still another channel is through cosmic-ray modulation of cloud cover. Most authors consider this effect to be too small to be observable. It is in phase with the direct effect of the solar cycle, and thus very hard to separate from the direct effect. But it *is* in the empirical temperature and not included in the simple model below.

### Conclusions

Despite the difficulties inherent in trying to extract the size of the effect of direct solar forcing on the global temperature, calculations by several authors seem to be converging on the value δT/δI ≈ 0.1 K-m^{2}/W for the total response of surface temperature to changes in top-of-atmosphere insolation. This number is within the range to be expected from theory including water-vapor feedback, and shows it is very unlikely that the observed 0.5 ^{o}C warming since about 1960 is due to solar effects: the average output of the sun has not changed by 5 W/m^{2} in that time.

### Quantifying the Effect

One can get a quick estimate of how large the effect is from global energy balance, which requires that σT^{4} = (1-α)I/4(1-f), where σ is the Stephan-Boltzmann constant, σ = 5.67x10^{-8} W/m^{2}-K^{4}, T is the absolute temperature, α ≈ 0.3 is the albedo, I is the insolation at the top of the atmosphere, I ≈ 1365 W/m^{2}, the factor 4 accounts for angular and day-night averaging, and 1/(1-f) represents the greenhouse effect of the atmosphere. If there were no atmosphere (f = 0), T ≈ 255K; with an atmosphere, T ≈ 290K. so 1/(1-f) = (290/255)^{4} = 1.69, so f = 0.41.

Now consider small changes: 4σT^{3}δT = (1-α)δI/4(1-f). At 290K, 4σT^{3} =5.53 W/m^{2}-K, so δT/δI =0.053 K-m^{2}/W.