I don’t think this is a simple topic.

The essence of the problem is this:

Can we measure the top of atmosphere (TOA) radiative changes and the surface temperature changes and derive the “climate sensivity” from the relationship between the two parameters?

First, what do we mean by “climate sensitivity”?

In simple terms this parameter should tell us how much more radiation (“flux”) escapes to space for each 1°C increase in surface temperature.

### Climate Sensitivity Is All About Feedback

Climate sensitivity is all about trying to discover whether the climate system has positive or negative feedback.

If the average surface temperature of the earth increased by 1°C and the radiation to space consequently increased by 3.3 W/m², this would be approximately “zero feedback”.

Why is this zero feedback?

If somehow the average temperature of the surface of the planet increased by 1°C – say due to increased solar radiation – then as a result we would expect a higher flux into space. A hotter planet should radiate more. If the increase in flux = 3.3 W/m² it would indicate that there was no negative or positive feedback from this solar forcing (note 1).

Suppose the flux increased by 0. That is, the planet heated up but there was no increase in energy radiated to space. That would be positive feedback within the climate system – because there would be nothing to “rein in” the increase in temperature.

Suppose the flux increased by 5 W/m². In this case it would indicate negative feedback within the climate system.

The key value is the “benchmark” no feedback value of 3.3 W/m². If the value is above this, it’s negative feedback. If the value is below this, it’s positive feedback.

Essentially, the higher the radiation to space as a result of a temperature increase the more the planet is able to “damp out” temperature changes that are forced via solar radiation, or due to increases in inappropriately-named “greenhouse” gases.

Consider the extreme case where as the planet warms up it actually radiates less energy to space – clearly this will lead to runaway temperature increases (less energy radiated means more energy absorbed, which increased temperatures, which leads to even less energy radiated..).

As a result we measure sensitivity as W/m².K which we read as *Watts per meter squared per Kelvin*” – and 1K change is the same as 1°C change.

### Theory and Measurement

In many subjects, researchers’ algebra converges on conventional usage, but in the realm of climate sensitivity everyone has apparently adopted their own. As a note for non-mathematicians, there is nothing inherently wrong with this, but it just makes each paper confusing especially for newcomers and probably for everyone.

I mostly adopt the Spencer & Braswell 2008 terminology in this article (see reference and free link below). *I do change their α (climate sensitivity) into λ (which everyone else uses for this value) mainly because I had already produced a number of graphs with λ before starting to write the article..*

The model is a very simple 1-dimensional model of temperature deviation into the ocean mixed layer, from the first law of thermodynamics:

C.∂T/∂t = F + S ….[1]

where C = heat capacity of the ocean, T = temperature anomaly, t = time, F = total top of atmosphere (TOA) radiative flux anomaly, S = heat flux anomaly into the deeper ocean

What does this equation say?

Heat capacity times change in temperature equals the net change in energy

- this is a simple statement of energy conservation, the first law of thermodynamics.

The TOA radiative flux anomaly, F, is a value we can measure using satellites. T is average surface temperature, which is measured around the planet on a frequent basis. But S is something we can’t measure.

What is F made up of?

Let’s define:

F = N + f - λT ….[1a]

where N = random fluctuations in radiative flux, f = “forcings”, and λT is the all important climate response or feedback.

The forcing **f** is, for the purposes of this exercise, defined as something added into the system which we believe we can understand and estimate or measure. This could be solar increases/decreases, it could be the long term increase in the “greenhouse” effect due to CO2, methane and other gases. For the purposes of this exercise it is **not feedback**. Feedback includes clouds and water vapor and other climate responses like changing lapse rates (atmospheric temperature profiles), all of which combine to produce a change in radiative output at TOA.

And an important point is that for the purposes of this theoretical exercise, we can remove **f** from the measurements because we believe we know what it is at any given time.

**N** is an important element. Effectively it describes the variations in TOA radiative flux due to the random climatic variations over many different timescales.

The climate sensitivity is the value **λT**, where λ is the value we want to find.

Noting the earlier comment about our assumed knowledge of ‘f’ (note 2), we can rewrite eqn 1:

C.∂T/∂t = - λT + N + S ….[2]

remembering that - λT + N = F is the radiative value we measure at TOA

### Regression

If we plot F (measured TOA flux) vs T we can **estimate** λ from the slope of the least squares regression.

However, there is a problem with the estimate:

λ (est) = Cov[F,T] / Var[T] ….[3]

= Cov[- λT + N, T] / Var[T]

where Cov[a,b] = covariance of a with b, and Var[a]= variance of a

Forster & Gregory 2006

This oft-cited paper (reference and free link below) calculates the climate sensitivity from 1985-1996 using measured ERBE data at 2.3 ± 1.3 W/m².K.

Their result indicates positive feedback, or at least, a range of values which sit mainly in the positive feedback space.

On the method of calculation they say:

This equation includes a term that allows F to vary independently of surface temperature.. If we regress (- λT+ N) against T, we should be able to obtain a value for λ. The N terms are likely to contaminate the result for short datasets,

but provided the N terms are uncorrelated to T, the regression should give the correct value for λ, if the dataset is long enough..

[Terms changed to SB2008 for easier comparison, and emphasis added].

### Simulations

Like Spencer & Braswell, I created a simple model to demonstrate why measured results might deviate from the actual climate sensitivity.

The model is extremely simple:

- a “slab” model of the ocean of a certain depth
- daily radiative noise (normally distributed with mean=0, and standard deviation σ
_{N}) - daily ocean flux noise (normally distributed with mean=0, and standard deviation σ
_{S}) - radiative feedback calculated from the temperature and the actual climate sensitivity
- daily temperature change calculated from the daily energy imbalance
- regression of the whole time series to calculate the “apparent” climate sensitivity

In this model, the climate sensitivity, λ = 3.0 W/m².K.

In some cases the regression is done with the daily values, and in other cases the regression is done with averaged values of temperature and TOA radiation across time periods of 7, 30 & 90 days. I also put a 30-day low pass filter on the daily radiative noise in one case (before “injecting” into the model).

Some results are based on 10,000 days (about 30 years), with 100,000 days (300 years) as a separate comparison.

In each case the estimated value of λ is calculated from the mean of 100 simulation results. The 2nd graph shows the standard deviation σ_{λ}, of these simulation results which is a useful guide to the likely spread of measured results of λ (if the massive oversimplifications within the model were true). *The vertical axis (for the estimate of λ) is the same in each graph for easier comparison, while the vertical axis for the standard deviation changes according to the results due to the large changes in this value.*

First, the variation as the number of time steps changes and as the averaging period changes from 1 (no averaging) through to 90-days. Remember that the “real” value of λ = 3.0 :

*Figure 1*

Second, the estimate as the standard deviation of the radiative flux is increased, and the ocean depth ranges from 20-200m. The daily temperature and radiative flux is calculated as a monthly average before the regression calculation is carried out:

*Figure 2*

As figure 2, but for 100,000 time steps (instead of 10,000):

*Figure 3*

Third, the estimate as the standard deviation of the radiative flux is increased, and the ocean depth ranges from 20-200m. The regression calculation is carried out on the daily values:

*Figure 4*

As figure 4, but with 100,000 time steps:

*Figure 5*

Now against averaging period and also against low pass filtering of the “radiative flux noise”:

*Figure 6*

As figure 6 but with 100,000 time steps:

*Figure 7*

Now with the radiative “noise” as an AR(1) process (see Statistics and Climate – Part Three – Autocorrelation), vs the autoregressive parameter φ and vs the number of averaging periods: 1 (no averaging), 7, 30, 90 with 10,000 time steps (30 years):

*Figure 8*

And the same comparison but with 100,000 timesteps:

*Figure 9*

### Discussion of Results

If we consider first the changes in the standard deviation of the estimated value of climate sensitivity we can see that the spread in the results is much higher in each case when we consider 30 years of data vs 300 years of data. This is to be expected. However, given that in the 30-year cases σ_{λ} is similar in magnitude to λ we can see that doing one estimate and relying on the result is problematic. This of course is what is actually done with measurements from satellites where we have 30 years of history.

Second, we can see that mostly the estimates of λ tend to be lower than the actual value of 3.0 W/m².K. The reason is quite simple and is explained mathematically in the next section *which non-mathematically inclined readers can skip*.

In essence, it is related to the idea in the quote from Forster & Gregory. If the radiative flux noise is **uncorrelated** to temperature then the estimates of λ will be unbiased. By the way, remember that by “noise” we don’t mean instrument noise, although that will certainly be present. We mean the random fluctuations due to the chaotic nature of weather and climate.

If we refer back to Figure 1 we can see that when the averaging period = 1, the estimates of climate sensitivity are equal to 3.0. In this case, the noise is uncorrelated to the temperature because of the model construction. Slightly oversimplifying, today’s temperature is calculated from yesterday’s noise. Today’s noise is a random number unrelated to yesterday’s noise. Therefore, no correlation between today’s temperature and today’s noise.

As soon as we average the daily data into monthly results which we use to calculate the regression then we have introduced the fact that monthly temperature **is correlated** to monthly radiative flux noise (note 3).

This is also why Figures 8 & 9 show a low bias for λ even with no averaging of daily results. These figures are calculated with autocorrelation for radiative flux noise. This means that past values of flux are correlated to current vales – and so once again, daily temperature will be correlated with daily flux noise. *This is also the case where low pass filtering is used to create the radiative noise data (as in Figures 6 & 7)*.

### Maths

x = slope of the line from the linear regression

x = Cov[- λT + N, T] / Var[T] ….[3]

It’s not easy to read equations with complex terms numerator and denominator on the same line, so breaking it up:

Cov[- λT + N, T] = E[ (λT + N)T ] – E[- λT + N]E[T] ….[4], *where E[a] = expected value of a*

= E[-λT²] + E[NT] + λ.E[T].E[T] – E[N].E[T]

= -λ { E[T²] – (E[T])² } + E[NT] – E[N].E[T] …. [4]

And

Var[T] = E[T²] – (E[T])² …. [5]

So

x = -λ + { E[NT] – E[N].E[T] } / { E[T²] – (E[T])² } …. [6]

And we see that the regression of the line is always biased if N is correlated with T. If the expected value of N = 0 the last term in the top part of the equation drops out, but E[NT] ≠ 0 unless N is uncorrelated with T.

*Note of course that we will use the negative of the slope of the line to estimate λ, and so estimates of λ will be biased low.*

As a note for the interested student, why is it that some of the results show λ > 3.0?

### Murphy & Forster 2010

Murphy & Forster picked up the challenge from Spencer & Braswell 2008 (reference below but no free link unfortunately). The essence of their paper is that using more realistic values for radiative noise and mixed ocean depth the error in calculation of λ is very small:

Figure 10

The value b_{a} on the vertical axis is a normalized error term (rather than the estimate of λ).

Evaluating their arguments requires more work on my part, especially analyzing some CERES data, so I hope to pick that up in a later article. [*Update, Spencer has a response to this paper on his blog, thanks to Ken Gregory for highlighting it*]

### Linear Feedback Relationship?

One of the biggest problems with the idea of climate sensitivity, λ, is the idea that it exists as a constant value.

From Stephens (2005),* reference and free link below*:

The relationship between global-mean radiative forcing and global-mean climate response (temperature) is of intrinsic interest in its own right. A number of recent studies, for example, discuss some of the broad limitations of (1) and describe procedures for using it to estimate Q from GCM experiments (Hansen et al. 1997; Joshi et al. 2003; Gregory et al. 2004) and even procedures for estimating from observations (Gregory et al. 2002).

While we cannot necessarily dismiss the value of (1) and related interpretation out of hand, the global response, as will become apparent in section 9, is the accumulated result of complex regional responses that appear to be controlled by more local-scale processes that vary in space and time.

If we are to assume gross time–space averages to represent the effects of these processes,

then the assumptions inherent to (1) certainly require a much more careful level of justification than has been given. At this time it is unclear as to the specific value of a global-mean sensitivity as a measure of feedback other than providing a compact and convenient measure of model-to-model differences to a fixed climate forcing (e.g., Fig. 1).

[Emphasis added and where the reference to "(1)" is to the linear relationship between global temperature and global radiation].

If, for example, λ is actually a function of location, season & phase of ENSO.. then clearly measuring overall climate response is a more difficult challenge.

### Conclusion

Measuring the relationship between top of atmosphere radiation and temperature is clearly very important if we want to assess the all-important climate sensitivity.

Spencer & Braswell have produced a very useful paper which demonstrates some obvious problems with deriving the value of climate sensitivity from measurements. Although I haven’t attempted to reproduce their actual results, I have done many other model simulations to demonstrate the same problem.

Murphy & Forster have produced a paper which claims that the actual magnitude of the problem demonstrated by Spencer & Braswell is quite small in comparison to the real value being measured (as yet I can’t tell whether their claim is correct).

The value called climate sensitivity might be a variable (i.e., not a constant value) and it might turn out to be much harder to measure than it really seems (and already it doesn’t seem easy).

### Articles in this Series

Measuring Climate Sensitivity – Part Two – Mixed Layer Depths

Measuring Climate Sensitivity – Part Three – Eddy Diffusivity

### References

The Climate Sensitivity and Its Components Diagnosed from Earth Radiation Budget Data, Forster & Gregory, *Journal of Climate* (2006)

Potential Biases in Feedback Diagnosis from Observational Data: A Simple Model Demonstration, Spencer & Braswell, *Journal of Climate* (2008)

On the accuracy of deriving climate feedback parameters from correlations between surface temperature and outgoing radiation, Murphy & Forster, *Journal of Climate* (2010)

Cloud Feedbacks in the Climate System: A Critical Review, Stephens, *Journal of Climate* (2005)

### Notes

**Note 1** – The reason why the “no feedback climate response” = 3.3 W/m².K is a little involved but is mostly due to the fact that the overall climate is radiating around 240 W/m² at TOA.

**Note 2** - This is effectively the same as saying f=0. If that seems alarming I note in advance that the exercise we are going through is a theoretical exercise to demonstrate that *even if* f=0, the regression calculation of climate sensitivity includes some error due to random fluctuations.

**Note 3** - If the model had one random number for last month’s noise which was used to calculate this month’s temperature then the monthly results would also be free of correlation between the temperature and radiative noise.