A More Precise Cooling Experiment

  Two weeks ago I redid the cooling experiment paying more attention to the precision measurement of the temperature through the use of a thermistor. The experimental setup is shown below with the thermistor inserted in a tall glass of hot tap water.

The thermistor was insulated with heat shrink tubing and held in place by a foam board card. The thermistor resistance was measured with a digital multimeter. The thermistor was obtained from Radio Shack a few years ago and contained calibration data on the back of the card.

A cubic fit of the characteristic curve for the temperature vs the natural log of the relative resistance allowed the resistance measurements collected to be converted to temperatures in °C. The electrical properties of a thermistor can be found in Millman & Halkias, Electronic devices and circuits, 1967, McGraw-Hill, p.132.

The resulting measurements were very consistent.

Using an electrical analogy for the cooling mechanism one can derive a formula involving exponential functions that one can fit to the data. The fit is quite good and also compares well with the rate of internal cooling which is proportional to the negative rate of change in temperature.

  The data for the last plot used a numerical formula to estimate the rate of change in temperature.

The results for the experiment were quite good but that doesn't necessarily mean they were accurate since we couldn't check thermistor calibration against a standard gas thermometer or, equivalently, the thermodynamic temperature scale. The ambient temperature of the room remained within a few tenths of 70.1 °F throughout the entire time the data was collected and there may have been a some slight warming towards the end of the data. Although I recorded the data manually the entire experiment could be done with a computer recording the data using a USB port if the appropriate drivers are installed on the computer.

One wonders how accurately the temperatures are measured using weather temperature recording programs?

An Outline of a Derivation of an Algebraic Expression for cos(1°)

  Yesterday I posted a tweet containing a closed-form algebraic expression for the cosine of 1°. One might wonder how one can arrive at this formula. The answer is that one has to use the formulas of trigonometry for combining angles, or the formulas for the product of complex numbers or alternatively formulas for successive rotations. We'll start by giving a few formulas for rotations. The following matrix rotates a point on the x,y-plane an angle counter clockwise through an angle θ.

A number of useful formulas are the following in which a product is equated to the identity matrix which corresponds to a complete rotation of 360 degrees or another matrix for some other known rotation.

Notice that one can use the cube of a rotation matrix to get a cubic formula in reduced form involving only the cosine of the unknown angle and that of the known angle. One can use the Cardano formula to solve for κ. The last formula is for the difference between two known angles. These formulas can be used to find the sines and cosines of the following angles.

The solution of the first four matrix power formulas is simplified by using the components equal to zero and eliminating unneeded factors. One has to be careful about the κ in the fourth power equation since the solution is κ=0 for θ=90°. A check shows that the formulas used in the tweet work as desired.

The Effect of the Stress Index on Ridgecrest Earthquakes is Still Inconclusive

  At first glance the number of recent earthquakes at Ridgecrest don't appear to be affected by stress index (SI).

But a closer look shows changes for both the activity (A) and the peak magnitude (M_pk) at about the time the stress index was greatest on day of year 212 but they could just be another coincidence.

The changes in behavior is made clearer by the expected values shown in dark blue for the given interval indicated along with their upper and lower bounds. Projections from the baseline values are shown in light blue. For some reason the activity was greater than expected while the peak magnitudes were less than expected.

If one fits a plane through the activity and peak magnitudes when plotted normal to the doy,SI plane one see a slight dependence on SI in both cases. The coefficients were computed using all the available data as standard values with a least squares fit. Note that the doy and SI terms pass through the standard zeroes so their effect is to rotate data in their plane with the vertical and the apparent correlation with it may be due to an imbalance in the data.

An explanation of why we didn't see another large earthquake about the time when the SI was greatest may be that all the available pent up stress had been released.

Update on Recent Ridgecrest Area Earthquake Activity

  The pattern of aftershocks of the M7.1 Ridgecrest earthquake appears to be continuing. If one uses the standard deviation of the activity A to estimate error bounds on the daily number of earthquakes one gets a good fit to the data except for the time of the major earthquake.

The daily peak magnitudes continue to steadily decrease.

If the trend continues one would expect the number of earthquakes above M2.5 to range from about 3 to 11 and the peak magnitudes to range from about M2 to M4 over the next few days. But we are approaching similar stress index (SI) values to that of the time of the M7.1 earthquake and the same is true for a supermoon index (SMI) for the coincidence of lunar perigee and the alignment the Moon with the Sun over the next few days.

Anything above M4 would be unlikely based on the statistics from the series of aftershocks.

Update on the Latest Earthquakes in the Ridgecrest Area

  There was a M4.7 earthquake at the beginning of UTC doy 207 that one might consider unusual since it was about 3 standard deviations from the most likely number of earthquake and the count for the day is not complete. It is difficult to characterize this since the some other source of deviation in the magnitudes other than random error.

The peak magnitude of this earthquake also appears to be unusual since it is greater than 2 standard deviations from the most likely magnitudes.

It is also within 2 standard deviations of the of the most likely peak magnitudes of the stress index correlation curve.

It looks like the faults in the area are having a little trouble deciding which way they want to go. It should be noted that one might be able to improve on the stress index to get better agreement with observations. There still seems to be problems with it and its interpretation.

note: I seem to have gotten the doy of year wrong. What was previously posted was the number of elapsed days since the beginning of 2019. I've now corrected to the usual doy.

Why We Still Need to Monitor the Ridgecrest Earthquakes

  A error in one of the averages cropped up as a result of adding data to existing tables. The result is that correlation for the activity and the daily tallies of earthquakes agrees slightly better with the observed values.

The formulas for the correlation of activity with time suggest some sort of relaxation phenomenon taking place.

The stress index had to be eliminated as a good indicator of daily earthquake activity but it still might play a role in indicating peak earthquake magnitudes. There is a lot of variation in the daily peak magnitudes so it is difficult to rule out the possibility. This morning's M4.13 earthquake in the Ridgecrest area points out that the peak magnitudes need to be watched.

The M7.1 earthquake appears to be somewhat exceptional for the area since its deviation from the SI peak magnitude is greater than two standard deviations.

Edit (Jul 23): Miscalculated the 2σ error bounds. Used the std dev of pmag_obs when I should have used that for the difference of the two pmags. Replaced the pmag plot with a corrected version. The M7.1 earthquake appears to be rather unusual.

Supplemental (Jul 24): An alternative hypothesis is the peak magnitudes are part of the population of aftershocks and are declining exponentially. If so, one would conclude that yesterday's M4.13 earthquake was within expectations.

Rejection of the SI Hypothesis and Provisional Acceptance of the Aftershock Hypothesis

  On UTC doy 202 there were just 8 earthquakes in the Ridgecrest area data from the USGS earthquake catalog. This appears to be a rare event for the stress index hypothesis that it is an accurate indicator of general earthquake activity.

The observed counts still appear to be following the aftershock hypothesis so it is provisionally accepted. Note that initially there were significant deviations from this hypothesis.

An Alternative Hypothesis for Recent Earthquake Activity in the Ridgecrest Area

  Let's say our first hypothesis is that the stress index SI is an accurate indicator of earthquake activity, A, and an alternative hypothesis is that the activity is determined by the elapsed time from that of the M7.1 earthquake. We find that there is a negative correlation although it is quite good because its magnitude is close to unity. The correlations for the primitive hypotheses are approximately 0.9 for the same interval of time which excludes the M7.1 earthquake.

But the correlation is related to the slope of a straight line drawn through the data from which we can estimate the activity and the number of earthquakes since the means from which the correlation was determined are known. When we compare the alternative hypothesis with the first hypothesis we find that the observed number of earthquakes is within about two standard deviations of both curves so we don't have a valid reason for rejecting either hypothesis. We are still in suspense about their validity.

In a few days the two hypotheses will become mutually exclusive so it's likely one will have to be rejected. The alternative hypothesis can be interpreted as all the earthquakes since the M7.1 are aftershocks.

Supplemental (Jul 21): Note that the correlation is the slope in standardized coordinates. The black diagonal lines are A_std=±t_std.

Supplemental (Jul 21): Today's search of the USGS earthquake catalog of the Ridgecrest area gave 14 earthquakes for UTC doy 201. This is slightly outside the 2σ bounds but the σ used is for a normal distribution and equal to √n̄. The bounds are likely to be slightly larger since for smaller counts the spread is larger than that of a normal distribution. The number of earthquakes is probably inside 3σ bounds so at the risk of being over cautious we are still reluctant to reject the first hypothesis.

A Factor Analysis of Earthquake Activity in the Ridgecrest Area

  The combined factors or stress index used in the last blog didn't correlate well with the earthquake activity defined as the base 10 log of the number of earthquakes in a day.

Modifying the angle factors by adding one and dividing the result by two worked better.

One can use the correlation coefficient to estimate the number of earthquakes per day for a given stress index, SI. When we compare this estimate with the earthquake histogram we get a fairly good agreement for the background activity.

There is a deficit of earthquakes before the M7.1 earthquake and a surplus afterwards that may be associated with aftershocks. The new stress index lacks the peaks for when Moon's declination is negative.

The definition of the modified factors is as follows.

The stress index suggests that we might start to see a noticeable increase in earthquake activity in the next few days. The correlation coefficients for the individual factors with the earthquake activity have values similar to that of the SI.

Forming Hypotheses About the Chances of an Earthquake

  One can study the USGS Ridgecrest area earthquake data and try to come up with some hypotheses about factors indicating the likelihood of the occurrence of an earthquake. The horizontal axis is the day of the year (doy).

When we compare this data with JPL's Horizons Lunar data we encounter some factors that seem to correlate well with the occurrence of the M6.4 and M7.1 and we get a number of coincidental events such as the relative force acting on the Earth's surface, the declination of the Moon and the Moon-Earth-Sun angle (∠M-E-S). The ratio of the average range of the Moon and its value at some time can be represent by the dimensionless factor φ and ϕ² is a measure of the relative force acting on a portion of the Earth's surface. The relative force was near a maximum at the time of the two earthquakes.

The sine of the Moon's declination and the cosine of M-E-S angle were also near maximums.

The product of the first and third factors might be used to indicate the lunar perigee and syzygy alignment which tells us when there will be a Supermoon. The product of the first and second is a measure of the relative torque acting on the Earth's equatorial bulge. If we combine all three factors we also get a good agreement of their maximum with the time of the two earthquakes.

If we look at this indicator over the entire year we see that it was near a global maximum at the times of the two earthquakes.

It will be interesting to see what the plots of the earthquake data look like over the next couple of months or so.

Supplemental (Jul 19): Replaced the histogram above with one that included some missing days at the end. One can modify the definitions of the indices to let them represent relative quantities better such as replacing sin(dec) with μ=sin(dec)/sin(ι) where ι=23.473 deg is the inclination of the ecliptic. This makes the combined indicator a little more meaningful.

The doy of the indicator peaks can be converted into calendar dates.

It should be noted that one cannot accurately evaluate the indices based on a small sample of data over a short period of time. The peaks of the indices are likely to diverge as the time increases since there are different periods associated with each index. We may be able to determine which of the indices best match the pattern of earthquake activity over time.

Using Macros to Assist Plotting USGS Earthquake Data

  The macros that one uses in worksheets can be quite sophisticated. One can use them to record one's actions to simplify doing a complicated repetitive task. For example, one can search the USGS earthquake catalog for the earthquakes that occurred in a specified region in given time span and save the results in a .csv which Excel can open. The macros need to be saved in an otherwise blank worksheet and to which the .csv data can be copied and pasted to. The first macro was designed for eliminating unneeded files and converting the date and time of an earthquake to decimal UTC day of year (doy). The next macro does the data needed for the histogram. The blank worksheet also contains macros to plot earthquake magnitudes vs doy and one to plot the histogram data.

Recording and Modifying Macros in Excel

  One can record macros in Excel to perform repetitive tasks. If you've never done this before you may need to look up "record macro" in Help. To record a macro activate the Developer tab and go to the code section. When one clicks on "Record Macro" one is asked to supply a name and complete a shortcut by adding a letter that will help activate it. Here is the VBA code for a recorded macro that copies a range of numbers and pastes the values into another column.

The " _" at the end of the 4th line indicates that the line is continued on the next line. The update macro used to repeat the copy and paste values operation and increment the tallies a given number of times uses VBA code to modify a similarly recorded macro that was give the name "update."

A macro that will zero the tallies is also useful.

If you've done everything correctly this is the what one can do.