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.