Applying Monte Carlo simulation to Sloan’s and Wolfendale’s use of Forbush decrease data

* Introduction
* Sloan’s and Wolfendale’s approach
* Considering standard deviation 
* Applying understanding of standard deviation to the lower cloud cover data
* Monte Carlo simulation
* Conclusions drawn from Monte Carlo Simultion

Introduction

It’s been five months since the widely remarked upon paper, Testing the proposed causal link between cosmic rays and cloud cover, by Sloan and Wolfendale, was published in the journal Environmental Research Letters.  In that short time this paper has been enshrined in the hall of fame of anthropogenic global warming dogma.  The thing that makes this paper a favorite of the climate change alarmists is its purported proof that there is no possible significant relationship between galactic cosmic ray flux, as modulated by the sun’s magnetic field and climate change.  This is important to the alarmists, because anything that might even partially exonerate CO2 must be stifled.

I spent many long hours deciphering this paper after several acquaintances alerted me of its conclusion.  I was dismayed to find out, after my careful study, that none of these acquaintances had ever actually read the paper!  I don’t know why this surprised me, they were simply victims of the global warming echo chamber.  Some of them read articles in the popular press or in web blogs that cited this paper and uncritically repeated its conclusions.

But are the conclusions that Sloan and Wolfendale based on the study of Forbush decreases (see below) correct?  Simply put, No.

The Sloan and Wolfendale paper concerns the possible link between cosmic ray flux and climate. The primary proponent of the the cosmic ray flux/climate link is Henrik Svensmark.  in his 2007 Astronomy and Geophysics paper, Cosmoclimatology, Svensmark says:

 Data on cloud cover from satellites, compared with counts of galactic cosmic rays from a ground station, suggested that an increase in cosmic rays makes the world cloudier. This empirical finding introduced a novel connection between astronomical and terrestrial events, making weather on Earth subject to the cosmic-ray accelerators of supernova remnants in the Milky Way.

This much is certain, the cosmic ray flux is modulated by the changing magnetic conditions of the sun.  Svensmark asserts that this makes the sun a very big player in changing climate conditions on the Earth.  This concept is anathema to the folks who envision gigantic climate catastrophes as a result of human activity.  Hence, the rapid dissemination of Sloan and Wolfendale’s conclusion by people who are probably very foggy on the paper’s logic.

Sloan’s and Wolfendale’s approach

A bare bones description of Sloan’s and Wolfendale’s argument is: Yes, there does appear to be some correlation between cosmic ray flux and lower cloud cover (LCC), but No, there is no causal link.  They show the correlation between cosmic ray flux (as modulated by solar magnetic activity)  and lower cloud cover with their figures 1 & 2, shown below. 

Figure 2. Sloan and Wolfendale's original caption says "The seasonally corrected LCC amount as a function of the Climax neutron monitor count rate (both monthly averaged) during cycle 22 (1983–1996). " Red annotation was added by Moriarty and will be referred to later in the text of this post.

After presenting figures 1 and 2, where, according to the authors, “The good correlation is evident,” Sloan and Wolfendale work to show that this correlation is not due to the type of causation hypothesized by Svensmark and others.  They focused on Forbush decreases, which are short term (usually several days to two weeks) reductions (a few percent to about 15%) in cosmic ray flux.   Their reasoning is that if the correlation shown in figure 2 is actually due to causation, then there should be a measureable decrease in lower cloud cover as a consequence of the the reduced cosmic ray flux during a Forbush decrease.

So, they plotted the lower cloud cover anomaly percentage change vs. the percent change in cosmic ray flux decrease during twenty-three Forbush decreases.  Their results are reproduced below in figure 3.  They claim that if the lower cloud cover is controlled by the cosmic ray flux, then the points in figure 3 should be along the line labeled “LCC CR.”  The points don’t seem to make a good fit to the line, so they conclude that there is no causation.

 One quick point before going any further.  Forbush decreases are short duration events, from a few days to a couple of weeks.  It seems odd that for 19 out of 23 the events that Sloan and Wolfendale studied they used monthly averaged lower cloud cover data.  If there is a correlation between cosmic ray decreases during Forbush events and changes in lower cloud cover, it doesn’t seem likely that monthly averaged cloud cover data is going to reveal it.

Figure 3. Sloan and Wolfendales original caption says "The measured change in the LCC plotted against the change in the Oulu neutron monitor count rate during the measurement time of 1 month for the D2 data (solid circles) and 1 week for the D1 data (open squares). The solid line shows the values expected from the smooth curve shown in figure 2. The Oulu count was observed to change by 17% due to the solar modulation during solar cycle 22." The red annotation was added by Moriarty and will be referred to later in the text of this post.

 

Considering Standard Deviation

To evaluate Sloan and Wolfendale’s conclusions, it it necessary to have an understanding of standard deviations.  If you are mathematically savvy, I don’t want to talk down to you.  So if you are clear on standard deviation, just skip this section.

The red annotation in figures 2 and 3 highlight the standard deviation of the lower cloud cover anomaly data.  Standard deviation is a statistic that tells how tightly measurements of the same thing are grouped together.  Suppose, for example, you measured the voltage between two points 50 times and the average was 3 volts, and about 2/3 of the measurements were between 2.9 and 3.1 volts, and about 95% or the measurements were between 2.8 volts and 3.2 Volts.  then the standard deviation of the measurements was about 0.1 Volts.  In general, 68% of measurements are within one standard deviation of the average and 95% of the measurements are within two standard deviations.  The voltage example is illustrated in figure 4, below.

Figure 4. Example of voltage measurements. The average is 3 V and the standard deviation is 0.1 V.

Applying understanding of standard deviation to the lower cloud cover data.

The line labeled “LCC CR” in figure 3 represents, in simple terms, the line upon which the lower cloud cover anomaly change would reside for a given cosmic ray count rate decrease if the correlation were due entirely to causation and the standard deviation for the lower cloud cover measurements were zero.  It has a slope of -0.051.  So for a 10% decrease in cosmic ray flux we would expect a -0.51% change in lower cloud cover.  This idea is shown in figure 5, below.   The x value (cosmic ray count rate decrease) for each point is the same as the corresponding point in figure 3.  The y value (Lower Cloud Cover Anomaly Change) is adjusted to reside on the line.

Figure 5. Same as figure 3, with all the same cosmic ray count rate decreases, but the lower cloud cover changes moved to the line that would be expected if changes in the lower cloud cover were 100% controlled by the cosmic ray flux and the standard deviation of the lower cloud cover measurement were zero.

If we retain the assumption that that the lower cloud cover changes are governed by changes in the cosmic ray flux, but we increase the standard deviation of the measurements of the lower cloud cover, then we can model the effect by having the points in figure 5 “wander” away form the line in the y direction.  The amount that each point is allowed to wander will be determined by the choice of standard deviation.  For example, if we choose a standard deviation of 0.1% for the measurement of the lower cloud cover change, then we will allow the points to move in such a way that 68% of them end up less than 0.1% (one standard deviation) away form the line, and 95% of the points end up less than 0.2% (two standard deviations) away from the line.  There are an infinite number of ways to do this – four of them are shown in figure 6.  In each of the four cases the newly scattered data is fit to a line which is very similar to the “LCC CR” line shown in figure 3 and figure 5.

figure 6. Each one of these four examples show data scattered away from the prefect correlation line with a standard deviation of 0.1% in the measurement of the lower cloud cover anomaly change.

For the record, the data in figure 6  and the following figures were generated using a “Gaussian white noise generator.” A Gaussian distribution is the name for the distribution shown in figure 4, above.  You may have heard of it referred to as the “normal’ distribution. 

Now we can incrementally increase the standard deviation of the lower cloud cover data.  Figure 7 shows four examples as the standard deviation goes through 0.2%, 0.4%, 0.6%, and 0.8%.  As the standard deviation increases the best line fit for each set of scattered data tends to look less and less like the lines shown in figures 3 and 5.

figure 7. Four examples of data scattered away from the prefect correlation line with standard deviations of 0.2%, 0.4%, 0.6%. and 0.8% in the measurement of the lower cloud cover anomaly change.

We can repeat the process illustrated in figures 6 and 7, but now with a standard deviation equal to the actual standard deviation of the lower cloud cover data, about 1%, as shown in the red annotation of figures 2 and 3.  Then we can compare the results to the actual measured data in figure 3.  In figure 8, below, four cases where a standard deviation of about 1% has been applied to the lower cloud cover data with the same Gaussian white noise generator.

Figure 8. Each one of these four examples shows data scattered away from the prefect correlation line with a standard deviation of 1.0% in the measurement of the lower cloud cover anomaly change.

Notice that when the standard deviation is about 1%, the best fit lines can look much different than the original line (see figures 3 and 5) from which they were scattered.  Just compare the lines in figure 8 with the lines in figures 3 and 5.  Every one of the cases in figure 8 are completely consistent with a the lower cloud cover being entirely controlled by the cosmic ray flux, but with a 1% standard deviation in the measurement of the cloud cover.  What can we say about the comparison of these four cases in figure 8 and the actual data shown in figure 3?

Monte Carlo Simulation

The four modeled plots in figure 8 are quite different from each other and different from the actual measured data in figure 3.  Yet we know the four plots in figure 8 are consistent with the lower cloud cover being controlled by the cosmic ray flux but having a standard deviation of 1%, because we created them to be that way.  The important question here is: Is the measured data, shown in figure 3, also consistent with a lower cloud cover actually being controlled by the cosmic ray flux, but having a standard deviation of 1 percent?

A Monte Carlo simulation is a perfect tool for answering this question.  A Monte Carlo simulation is the repeated calculation of some model or algorithm with random inputs from the domain of all possible inputs.  Then conclusions are drawn from the accumulated results of the repeated calculation.  A classic example is to use the Monte Carlo method to calculate Pi. 

I have used the Monte Carlo method by repeating the calculations to generate the type of data shown in figure 8.  I didn’t repeat the calculation just 4 times – I repeated it 10,000 times.  Of course, it is totally impractical and not very enlightening to show all 10,000 plots.  Instead, for each of the 10,000 sets of data I have calculated the slope and the intercept of the best fit lines.  Then these 10,000 slopes and intercepts may be analyzed and compared to the slope and intercept of the actual measured data shown in figure 3.  The best fit line for the point shown in figure 3 has a slope of 0.001 and and intercept of 0.11.

Figure 9 shows a histogram of the slopes of the best fit lines for each of the 10,000 generated sets of data.  The average slope of all of these lines is -0.051, exactly the same as the slope of the lines in figures 3 and 5, as would be expected.  The spread of the slopes yields a standard deviation of 0.055. 

Similarly, figure 10 shows a histogram of the intercepts of the best fit lines for each of the 10,000 generated sets of data.  The average intercept was, of course, 0.0, as expected.  The standard deviation of the intercepts was 0.36.

Figure 9. Histogram of the slopes of the best fit lines for each of the 10,000 cases generated with the Monte Carlo simulation. The mean slope is -0.051, the same as Sloan and Wolfendale's "LCC CR" line shown if figure 3. 68% of all cases have slopes within 0.055 of the mean (between -0.106 and +0.004). So the standard deviation of the slopes is 0.055

Figure 10. Histogram of the intercepts of the best fit lines for each of the 10,000 cases generated with the Monte Carlo simulation. The mean intercept is 0.0, the same as Sloan and Wolfendale's "LCC CR" line shown if figure 3. 68% of all cases have intercepts within 0.36 of the mean. So the standard deviation of the intercepts is 0.055

 

Conclusions drawn form Monte Carlo simulation

The pertinent question is this: Is the measured data, shown in figure 3, consistent with a lower cloud cover being controlled by the cosmic ray flux, but having a standard deviation of 1 percent?

Figure 11 shows the intercept vs. the slope for all 10,000 iterations of the Monte Carlo simulation.  It shows a wide range of possible outcomes: some with large slopes and some with small slopes, some with large intercepts and some with small intercepts.  They are arrayed about the mean slope and intercept of -0.051 and 0.0 respectively, the values that represent the “perfect fit” line labeled “LCC CR” in figures 3 and 5.  The red “X” marks the location of the slope and intercept of the 23 data points used by Sloan and Wolfendale in figure 3.  The points within the yellow region represent Monte Carlo cases where the combination (root sum square) of the slope and intercept make them closer to the “perfect fit” than the data used by Sloan and Wolfendale. Only 26% of all cases fall within the yellow region.

 So, when we start with the assumption lower cloud cover exactly follows the cosmic ray count, as indicated by the lines in figures 3 and 5, but add a standard deviaton of about 1% to the lower cloud data, then 74% of the time the rusult will be worse than the real data used by Sloan and Wolfendale.   This means, quite simply, that the measured data, shown in figure 3, is consistent with a lower cloud cover being controlled by the cosmic ray flux, but having a standard deviation of 1%. 

figure 11. Figure 11. Intercept vs. slope for each of the 10,000 interations of the Monte Carlo simulation. The vertical and horizontal lines mark the means of the slope and intercept, which are the same as the slope and intercept of Sloan and Wolfendale's "LCC CR" line in figure 3. The yellow region, which contains 26% of all iterations, highlights the cases where the root sum square of the deviations from the slope and intercept means are less than the deviations for the 23 Forbush decreases shown in figure 3 (shown by the red "X."

If fact, the badly scattered data is in figure 3 is still a better fit to the “LCC CR” line than would be expected, given the standard deviation of the lower cloud cover measurements. 

Because the standard deviation of the lower cloud cover data is so high (nearly 1%), Sloan’s and Wolfendale’s attempt to to use Forbush decreases to verify or reject the possibility that cosmic ray flux changes induce lower cloud cover changes was poorly conceived.  The data is far to noisy to tell one way or the other.  Claims in the press and the blogosphere that Sloan and Wolfendale have used Forbush decreases to show there is “no link between cosmic rays and global warming” are just plain wrong.