Waveform cross correlation for seismic monitoring of underground nuclear explosions. Part II: Synthetic master events

A new working paper is available on arxiv.org This is a link to pdf file.

Abstract

Waveform cross correlation is an efficient tool for detection and characterization of seismic signals. The efficiency critically depends on the availability of master events. For the purposes of the Comprehensive Nuclear-Test-Ban Treaty, cross correlation can globally reduce the threshold monitoring by 0.3 to 0.4 magnitude units. In seismically active regions, the optimal choice of master events is straightforward. There are two approaches to populate the global grid in aseismic areas: the replication of real masters and synthetic seismograms calculated for seismic arrays of the International Monitoring System. Synthetic templates depend on the accuracy of shape and amplitude predictions controlled by focal depth and mechanism, source function, velocity structure and attenuation along the master/station path. As in Part I, we test three focal mechanisms (explosion, thrust fault, and actual Harvard CMT solution for one of the April 11, 2012 Sumatera aftershocks) and two velocity structures (ak135 and CRUST 2.0). Sixteen synthetic master events were distributed over a 1ox1o grid. We built five cross correlation standard event lists (XSEL) and compared detections and events with those built using the real and grand master events as well as with the Reviewed Event Bulletin of the International Data Centre. The XSELs built using the source of explosion and ak135 and the reverse fault with isotropic radiation pattern demonstrate the performance similar to that of the real and grand masters. Therefore, it is possible to cover all aseismic areas with synthetic masters without significant loss in seismic monitoring capabilities based on cross correlation.

Why Gavyn Davies is wrong about employment targeting

Gavyn Davies criticises  the Fed for wrong money policy targeting on unemployment rate, UER,  instead of the rate of employment, E/P. He  is right  that the rate of unemployment does not express well the full  capacity of the economy. However, the rate of employment  is quite different from the UER. Two figures below do show that the UER has been varying around 5% to 7% since 1948. The Fed can claim that monetary policy is effective because UER is always back to 6%. How can they say that they control E/P if it really has a strong secular component. Gavyn puts the Fed in danger to be responsible for that they can not be responsible for. It will never happen if the Fed has a bit of sense. The rate of employment is out of control not only for the Federal Reserve but also for any authority.

 

A fourty-year period of energy price oscillation

Figure 1. The difference between the headline and core CPI (both seasonally adjusted) since 1957.  The V-shape suggests the similarity of the fall and rise paths. One can estimate the next peak extrapolating the current rise by the previous fall (between 1980 and 2000).

Figure 2. The difference between the headline and core CPI. The red curve is an inverse version of the blue curve reduced by -9 and shifted by 19 years ahead.  One can estimate the distance between the peaks in 1981 and 2019 as the period  of long term oscillations, which is approximately 40 years.  

Figure 3. The difference between the energy and core CPI. The red curve is an inverse version of the blue curve reduced by -65. One can estimate the distance between the peaks as the period  of long term oscillations.  The energy price is likely on a downward trend already.

Employment in Canada revisited

Two years ago we presented a parsimonious model describing the evolution of employment/population ratio in Canada.  In essence, this model is a modified Okun’s law since there exists a trade-off between the change in unemployment and employment. Yesterday, we revisited the rate of unemployment in Canada based on a complimentary model and found an extraordinary high fit between predicted and observed curves. (A comprehensive description of both models also extended by examples in other developed countries is presented in our paper “Modeling Unemployment And Employment In Advanced Economies: Okun’S Law With A Structural Break”. )

 

 

Figure 1 compares the change in the rate of employment (the employment/population ratio), de, and the rate of unemployment, du, in Canada. As expected, the change in the rate of unemployment is slightly more volatile (also because of lower accuracy of measurements). We have retrieved all data on unemployment and employment from the U.S. Bureau of Labor Statistics.  

Figure 1. The (negative) change in the rate of employment compared to the change in the rate of unemployment in Canada.  

Two years ago we estimated several models of employment/population ratio, e. For Canada, the best-fit model has been obtained by the least-squares (applied to the cumulative sums):  

de_t = 0.40dlnG_t – 0.67, t<1984
de_t = 0.44dlnG_t – 0.56, t>1983    (1)  

where dlnG_t is the change rate of real GDP per capita at time t. Figure 2 shows the cumulative curves for the time series in (1). We did not fix the initial value in 1971 and obtained it from the regression. There is a structural break near 1984 which is expressed by a slight shift in the slope of the regression line and a 0.11 change in the intercept term. Since the latter term is cumulated over years one can consider this change as a significant one. It makes a 1.1% employment change over 10 years.  The break is needed because of the change to definitions and measurement procedures rather than actual break in the long-run link between e and G. In any case, functional dependence between these two variables stays untouched.  

The employment/population ratio varies between from ~54.5% in 1971 and ~64.1% (!) in 2008. The agreement between the actual and predicted curves in Figure 2 is excellent. We added two readings for 2011 and 2012 which practically coincide. This observation validates our original model and we will continue reporting on the evolution of employment/population ratio in Canada. 

Figure 3 present results of a linear regression with R²=0.88 for the period between 1971 and 2012. The standard error of the model is 0.82% which is chiefly related to the first ten years where measurements were rather crude.

Figure 2. The cumulative curves for the observed and predicted change in the employment/population ratio, de.

Figure 3. Linear regression of the measured and predicted curves in Figure 2.

A fairy tale about the future rate of unemployment in the US

The rate of unemployment, u, was very high (10%) in 2009. It has been often discussed that the fall in this rate is too slow historically. Figure 1 shows the evolution of u since 1980. There were two major peaks in 1982 (10.8%) and 1992 (7.8%). (All rates are seasonally adjusted.) When the troughs preceding the peaks are synchronized all three descending curves look very similar. This observation says that the current fall in u is not different from the previous. It also tells us a fairy story about the near future.  This rate will fall into the second half of the 2010s. Meanwhile, it may fall to 6.0% in 2013 or in the first half of 2014.

It is worth noting that the start troughs before two major peaks in Figure 1 were higher than the end troughs.  If this trend is retained the next trough should be ~4%.

 

Figure 1. The rate of unemployment in the US

 

Figure 2.  Two previous major peaks synchronized with the most recent. The length of fall is approximately 7 years.

The beauty of science II. Predicting the rate of unemployment in Australia

A few hours ago, I reported that Canada gives the best example of accurate quantitative prediction of unemployment in developed countries and therefore extreme satisfaction for a researcher. I changed my mind when revisited the case of Australia with new data - real GDP (GK per capita) data from the Total Economic Database and the rate of unemployment from the OECD. Two years ago, I presented a model based on data from the same sources and found relatively big discrepancy after 2000. In the new revision of the same data, this discrepancy have evaporated.

Historically, we published a paper in the Journal of Theoretical and Practical Research in Economic Fields in 2012. We presented the first version of the modified Okun’s law for developed countries including Australia. The model was estimated till 2010 and used the data available in 2011. Briefly, the model is estimated by the LSQ technique applied to the integral version of Okun’s law:

u(t) = u(t₀) + bln[G/G₀] + a(t-t₀)   (1)

where u(t) is the predicted rate of unemployment at time t, G is the level of real GDP per capita, a and b are empirical coefficients. For Australia, we estimated the model with a structural break allowed by data somewhere between 1980 and 2000. The best-fit (dynamic) model minimizing the RMS error of the cumulative model (1) with the new data revision is as follows:

du = -0.69dlnG + 1.50, t before 1992

du = -0.45dlnG + 0.75, t after 1991 (2)

Originally, the model included the structural break neat 1995. With the new data the overall fit is better and the year of break moved to 1991, which is related to major revision of unemployment definition. The new model suggests a drop in slope and a big change in the intercept around 1991. Figure 1 depicts the observed and predicted curves of the unemployment rate. The agreement is very good. Figure 2 shows that when the observed time series is regressed against the predicted one, R²=0.86 (0.84 in 2011).  The integral form of the dynamic Okun’s law (1) is characterized by a standard error of 0.72% for the period between 1971 and 2012. The average rate of unemployment for the same period is 6.9% with a standard deviation of the annual increment of 1.9%.  This is an extremely accurate prediction considering the accuracy of GDP (~1% per year) and unemployment (0.3% to 0.4%) estimates. The whole discrepancy is related to the measurement errors and thus the residual error shown in Figure 3 is an I(0) random process.  

The rate of unemployment depends on the cumulative change in real GDP per capita, as relationship (1) implies. To reduce the rate of unemployment in Australia, the rate of GDP (real per capita) growth must be above 1.7% per year.  

 

Figure 1. The observed and predicted rate of unemployment in Australia between 1971 and 2012. 

 

Figure 2. The measured time series is regressed against the predicted one. R²=0.86 with both time series likely to be stationary.

Figure 3. The residual error of the unemployment model.

So, I have to repeat. The beauty of science is the accuracy of prediction. It is difficult to express the feelings of a researcher than new observations fit his predictions based on a simple concept.  It is especially sweet when this concept is different from the mainstream one.

The beauty of science. Unemployment in Canada

The beauty of science is the accuracy of prediction. It is difficult to express the feelings of a researcher than new observations fit his predictions based on a simple concept.  It is especially sweet when this concept is different from the mainstream one. I am sure that economists never feel like that with all models flawed. Here, I present one of many cases of accurate predictions based on the link between GDP and unemployment, which is a modified Okun’s law in an integral form.

 

Canada provides an excellent set of macroeconomic data which can be described by a few deterministic links with a high level of reliability and confidence. We have retrieved real GDP (GK per capita) data from the Total Economic Database and the rate of unemployment from the OECD. In 2012, we published a paper in the Journal of Theoretical and Practical Research in Economic Fields, where presented the first version of the modified Okun’s law for developed countries including Canada. The model was estimated till 2010 and used the data available in 2011. The original model for Canada was also presented in this blog in 2011. It’s time to revisit the model and its predictions.

Overall, the model is estimated using the LSQ technique to the integral version of Okun’s law:

u(t) = u(t₀) + bln[G/G₀] + a(t-t₀) (1)

where u(t) is the predicted rate of unemployment at time t, G is the level of real GDP per capita, a and b are empirical coefficients. For Canada, we estimated the model with a structural break allowed by data somewhere between 1980 and 1990. The best-fit (dynamic) model minimizing the RMS error of the cumulative model (1) is as follows:

du = -0.28dlnG + 1.16, t<1983
du = -0.28dlnG + 0.30, t>1982 (2)

 

 
This model suggests no shift in the slope and a bigger change in the intercept around 1983. Figure 1 depicts the observed and predicted curves of the unemployment rate. Considering the accuracy of measurements for both involved variable the fit is excellent. The integral form of the dynamic Okun’s law (1) is characterized by a standard error of 0.67% for the period between 1971 and 2012. The average rate of unemployment for the same period is 8.12% with a standard deviation of the annual increment of 0.92%.  Figure 2 shows that when the observed time series is regressed against the predicted one, R²=0.87.  Here we do not test both time series for stationarity but presume that the rate of unemployment has to be a stationary time series in the long run.
One can suggest that the rate of unemployment has been driven by real economic growth and there is no much room for other macroeconomic variable to intervene. Personally, I admire the performance of this simple model and will keep reporting on it for Canada, but also for Spain, France, etc.

Figure 1. The observed and predicted rate of unemployment in the Canada between 1970 and 2010. 

Figure 2. The measured time series is regressed against the predicted one. R²=0.87 with both time series likely to be stationary.