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A survey of the cosmic ray diurnal variation during 1973–1979—I. Persistence of solar diurnal variation
Pkm!f. SpaM sci. voi 35, No. 9, pp. 111 I-1 115, LYX7 Printed in Greut Britain.
003213633/87 $3.00 +O 00 Pergamon Journals Ltd.
A SURVEY OF THE COSMIC RAY DIURNAL VARIATION DURING 1973-1979-L PERSISTENCE OF SOLAR DIURNAL VARIATION
Department
J. F. RIKER and H. S. AHLlJWALIA* of Physics and Astronomy, The University Albuquerque, NM 87131, U.S.A. (Received in final form
of New Mexico,
18 March 1987)
Abstraer-We find that the solar diurnal variation in cosmic ray intensity is a persistent phenomenon over the years 197331979. That is, even during solar minimum, conditions in the heliosphere lead to a net flux of cosmic rays from a particular (time-varying) direction in space. If we regard the daily Ructuations in the amplitude and phase of the diurnal variation as random perturbations about the mean vector, we are able to quantify the relative magnitude of the random component.
m2(h)= m2(l)/h
INTRODUCTION
The solar diurnal variation has been the subject of considerable study in cosmic ray research. In particular, it is found that the variation undergoes dramatic changes in amplitude and phase during the solar activity cycle (Ahluwalia, 1977). In the years of solar minimum, the amplitude can be extremely small, to the point that one wonders whether the variation persists year round. It is the purpose of this paper to generalize the persistence analysis of Bartels (1935) to the case of the diurnal variation, in order to determine the components and persistence of the signal. The method utilizes the insight of Bartels that geophysical data at adjacent data points are not sequentially independent, since high values tend to follow high values and similarly for low values. Forbush et ul. (1982) adapted the method for use in analyzing the 27-day variation using simulated data, but we shall apply it to yearly averages of cosmic ray diurnal variation, during 1973-1979. We use data obtained with the vertical underground muon telescope located at Embudo, NM (vertical cut-off rigidity N 19 Gv).
(1)
where m’(l) is the variance of the original series. If instead of random ordinates, we use a series formed by repeating each ordinate some fixed number, w, of times, the standard deviation for averages of h = wh’ ordinates is m’(h) = m(h’) = m(l)/fi
= ?Fz(l)\m
(2)
where w = ~~‘z(~)/~*(l) is the number of repetitions. original series, we can compute m( 1) for single ordinates and ation m(h) for averages of (h = 2,3,. . .). By analogy with
(3)
Reverting back to the the standard deviation also the standard devih successive ordinates equation (3), we define
E(h) = /zm2(/z)/m’(l)
(4)
as the equivalent number of repetitions in the series. For truly random data, equations (1) and (4) imply e(h) = 1, which is the no conse~ation case. If actual data have conservation, equation (4) yields a measure of how much. This can be generalized to yield a statistical test for periodicity, or persistence.
CONSERVATION
The fundamental difference between a time series of cosmic ray data and a series of random ordinates is the presence of conservation in the former, as pointed out by Chapman and Bartels (1940). They quantitied the concept by considering a series of random ordinates : if one averages together h consecutive ordinates and computes the variance of the new (averaged) series, one obtains the variance -._* Now on leave at: NASA Headquarters, Code EZH, Washington,
DC 20546, U.S.A.
PERSISTENCE
If sequences of data rather than individual data points are considered, it should be possible to ascertain whether some form of behavior persists from one sequence to the next. In particular, we are interested in persistent periodicities, whereby a sinusoidal function would behave similarly (in terms of its frequency, amplitude and phase) from one sequence to the next. For example, if a year of cosmic ray data were broken into individual days, we desire to test for persistent
1111
1112
J. F.
RIKER
and
waves of given frequency, viz. the diurnal frequency (1 cpd) and its harmonics. One might suspect that if there were a truly persistent diurnal variation (i.e. one in which the frequency, amplitude and phase did not change from one day to the next), it could be detected even if the persistent wave had large random fluctuations of amplitude and phase superimposed on it. The original Bartels (1935) method employed harmonic analysis amplitudes as the data values, with a direct analogy to e(h) of equation (4) as the measure of persistence. Forbush et al. (1982) also used that method, exactly as given by Bartels. However, Bartels noted that his method could be generalized to avoid any explicit reference to harmonic analysis by dividing the original data into rows of length equal to the period under consideration and using the behavior of standard deviations. Chapman and Bartels outlined this method more completely ; this is the method that we have developed (Riker, 1986). Consider sequences of data values xi,, where the double index implies rows and columns of data as follows :
D,:x,,.x,+, D, : x(&f-
,... x2,_, ,)r,
.
(51
. .x,5_ ,a
Here, r is the period considered, N is the total number of observations, and ii4 is the number of rows. For the diurnal variation, r = 241x,N = total number of hourly observations, and M = number of days. Note that the technique does not require every day to be included in a diurnal analysis. Since the persistent wave would be present every day, with random day to day changes superimposed, skipping some days due to solar flares, equipment failure, calibration, or other causes should not change the result. If only a quasipersistent wave were present, then such missing data could be significant. The persistence test consists of determining the behavior of the sequences as they are averaged together ; first, the non-averaged sequences shown in equation (5) are used. Averages for each row are computed, yielding Lt’,,D;, . . . , Oh,. The variances
are obtained for each row, where i and ,j are the row and coIumn indices (i = I,. . . , n/r; j = 0,. . . ,r- I). These are just the variances of the individua1 ordinates in row i from 0:. We define
so that 5, is the average standard deviation for the
H. S.
AHLUWALIA
individual sets. Next we average the columns of h rows (h = 2,. . . , h,,,)together to give K = N/h average sets. These Knew rows are handled in the same manner as before, i.e. their row averages 0: and row variances 5:’ are obtained. Again, the row variances are averaged to obtain the new average standard deviation t,(h). By analogy with equation (4) we define
The quantity g,(h) is then the equivalent number of repeated sets in h successive sets, and h/cr,(h) is the effective number of independent sets. A repeated set is just one in which variations from one ordinate to another are the same in each set. If a truly persistent function is present, one expects exact behavior from one set to the next, so that the average sets behave the same as the individual sets. In that case, t:(h) = <,” and a,(h) = h. For random data, equation (1) implies t,(h) = t,/$, so that a,(h) = 1. Real geophysical data would then have o,(h) between these limits. If a,(h)-+S, as h -+ co, one has quasi-persistence, which is a tendency for similarities in successive rows to vanish gradually, so that there is no relation between rows sufficiently far apart. This would occur for data which are persistent over a few rows, but in which the amplitude and phase undergo irregular changes. There are several considerations to be noted for this persistence test. Firstly, the question of overlapping average sets should be addressed. That is, when one forms average sets of h rows, either K = N/h sets could be used (first row composed of rows 1,. ..,h,second row composed of rows 2, ...,h + 1,etc.). Overlapping sets allow averaging of more values for &(h), so that one makes a better estimate of the average t,(h). However, the same data are used repeatedly in this method. The non-overlapping method should yield a more stringent test of truly persistent functions, despite inaccuracies in the averages t,(h) for large h. Since this method is also easier to program, it was selected for our studies. Secondly, the question of how large h may be is important, since averaging only a few values of t,(k) would lead to unacceptably large errors in [,(/I). Bartels suggested that, for M harmonic analysis amplitudes, the number of independent averages K should not be too small if the expectancy of the average amplitude is to come close to the correct value. We gave h,,,= M/SO for the expectancy to be within 10% of the correct value. He also suggested that if N is the total number of available observations, then h max should be chosen so that kr = N/20. This last criterion is the one selected in the following analyses. For Embudo data, N = 4800 h typically, so
Persistence of diurnal variation In Cl = Quasi - persistent P = Persistent 2 _
R-
Random
data
average
y;(h) = $/h,
data data
,
1113 sets,
0,’
remains
unchanged,
but
so
5,2(h) = cC+y:/h e,(h) = hC(h)/5,2
= h(&/5,2)+$/5r2
= h(W,2/5,2) + 1,ct:-w:,
I
a,(h) = 1+c$(h-
Number
of averaged
sets (h)
FIG. I.
POSSIBLE FORMS OF PERSISTENCE CURVES WHEN DATA ARE COMPLETELY RANDOM (R), QUASI-PERSISTENT (Q), PERSISTENT (P), OR A MIXTURE OF THESE POSSIBILITIES.
I)/(;.
(14)
This is the equation of a straight line. Since 0,’ < t: by equation (13), the slope is less than one but is not negative. This equation allows calculation of the variance of the periodic and random parts of the data from the values of a,(h).
RESULTS
h,_r
= N/20
(9)
implies h,,, = 10 for the diurnal variation h max= 20 for the semi-diurnal variation.
and
DISCUSSION
We generated a computer program to perform the persistence analysis. Initially, numerous tests on simulated data were performed in order to verify the program. Our preliminary results were reported elsewhere (Riker and Ahluwalia, 1983). Before presenting more results, we describe the expected behavior for some simple cases. For random data, as already noted, t,‘(h) = <,?/h, so that equation (8) yields o,(h) = hl,‘(h)/t,2 Truly persistent
data have t:(h) a,(h) =
= 1.
(10)
= <,?, so that
h[5,‘1/5,’= h.
(11)
Quasi-persistent data have cr,(h) + S,, by definition. Equation (8) implies S, = /zt,‘(/r)/l,’
(large h)
t,!(h) = l,‘S,/h
(large h).
so that (12)
The various possibilities are shown in Fig. 1. For data composed only of a persistent plus a random component we have x=w+g where w is the persistent function and g is the random part. If the standard deviations of x, w and g are l,, w, and y,, then if w and g are uncorrelated, one has 5: = w,?+y,2.
(13)
We tested the persistence analysis using known persistent sinusoids of several frequencies, variable amplitude and phase functions, random data, and combinations of random plus persistent data. The last case is particularly useful, because the magnitude of the random component relative to the periodic component significantly affects the slope of the persistence curve. For such data, the curve of o(h) is always a straight line whose slope depends upon the magnitude of random fluctuations relative to the persistent function amplitude. Using the slopes obtained for data with known relative magnitudes of random to persistent variations, it is possible to specify the relative magnitude of the random component when real data are used (Riker, 1986). This approach works only so long as the real data produce straight lines for o(h). Results for actual cosmic ray muon data are summarized in Figs 2 and 3. In Fig. 2, we show the results for the data obtained at Embudo for 197331979, on one plot. All the curves are straight lines with slopes less than 1.0, from which we may conclude that the data consist of at least one periodic component of frequency 1 cpd plus random juctuations. The slopes of the curves vary in time, reaching a minimum in 1977. This implies that the slope of the curve, and hence, the relative magnitude of the observed random component, tends to mirror the behavior of the first harmonic. This correspondence is not exact, however. Figure 3 shows results obtained for 1973 when more days are averaged. The straight line behavior persists, no matter how large h,,, is taken, although statistical effects enter when h,,, is larger than about 30. By comparing the slopes of the persistence curves for each year to those for known data, we estimated the relative magnitudes of the random and periodic components. The technique is simple. First, one
J. F. RIKER and H. S. AHLUWAL,IA
1114
3
I 22 I
2 3
19f3 1974 1975 1976
b
I
3
I 2
I
I
I 3
I 5
4 Number
of days
I
I 7
6
I 8
I 3
1 IO
OverOQed
FIG.~.~RSISTENCEC~RVESA~S~OWNFOREAC~ YEAR FORT~~NDERGROUNDM~ON DATA OBTAINEV EMBUW, NM, DURING 1973..1979. All curves can be approximated by st~ight lines of slopeslessthan 1.0.
AT
2 OW 4 z
0.13
8 c %
0.11
.: 0.09 e
0
I
I
3
6
I 9
i 12
I
I
I
I
15
I8
21
24
8 27
Number of days averaged FIG. 3. PERSISTENCE CURVE FOR EMBUDO MUON DATA FOR 1973 lSSHOWN,WHEN h,,, ISLARGE. Note that the straight-line behavior persists.
obtains the usual plots of u(h) vs h for the data consisting of a sinusoid of a known amplitude “a” and random contributions of variance 5'. Next, one plots the persistence curve for numerous values of the “random fraction” k = m/a, where c is a known (arbitrary) constant of proportionality. Note that k is a ~~~le~s~o~l~~~number. It specifies the magnitude ofthe random ~uctuations relative to the known persistent amplitude. Thus if k = 1, we have a random component with variance identical to the square of the persistent amplitude. On the other hand, if k = 0, we obtain a horizontal line. Instead of plotting a(h) vs h for several values of k, it is more convenient to plot CT(~)vs k for each value of h. One then obtains nonlinear curves with c(h) = h on the k = 0 axis and m(h) = 1 as k increases. These “standard curves” are characteristic of the random fraction and should be obtained independent of the amplitude (a) and the phase of the sinusoid. We tested this technique for typical cosmic ray sinusoids and found it to be approximately true. Next, one uses the real data to obtain the usual persistence curves (see Fig. 2). For
2 0.07 3 6 I I I I I I I o.o5ISZI 1374 1975 1976 1977 1978 1979 Year FIG. 4. A COMPARISON IS MADE BETWEEN THE RELATIVE MAGNITUDES OF TNE AMPLITUDE OF THE DIURNAL VARIATION AND THE MAGNITUDE OF THE RANDOM COMPONENT PRESENT INTHEDATA. As expected, they appear to bear an inverse correlation.
each year and for each value of h, one follows the appropriate “standard curve” until one reaches the value of cr(h) found from the data. One then drops a vertical line to the abscissa to get an estimate of the magnitude of the random component for that value of h. Averaging all these estimates produces a good estimate of the relative magnitude of the random component for the year. We show our results for both the diurnal variation amplitude (before applying orbital effect correction) and the estimated random fraction (relative magnitude) in Fig. 4. We wish to point out that the observed standard deviation in the amplitude, for each of the years during 1973-1979, is -0.01%. however, the errors “attributed” to the random component appear to be much larger than the statistical errors in the data. CONCLUSIONS
The results presented in this paper give a rough estimate of the magnitude of the random component
Persistence
of diurnal
relative to that of the periodic component. While the magnitudes of the random fraction do tend to inversely follow the behavior of the first harmonic, an exact correspondence does not exist. If the random fluctuations were constant in magnitude as a function of time, we would expect that as the magnitude of the periodic part decreases, the value of the estimated random fraction should increase. This is the case for the trend over some years, but the fine structure does not necessarily follow this rule. For example, the amplitude of the diurnal variation decreases from 1973 to 1974, so we expect the estimated random fraction to increase, but it actually decreases. This effect should be analyzed further. However, we can still assert that the Embudo data do exhibit a persistent sinusoidal variation for all years 197331979, including the years near solar minimum. Also, the data apparently consist of only persistent plus random components, with the random part larger in magnitude. These results indicate for the first time that the diurnal variation is present year round, in the data
variation
1115
obtained at the underground site of Embudo, NM. We believe the technique has more general applications in terms of uncovering hidden periodicities in data which contain large random errors.
REFERENCES
Ahluwalia, H. S. (1977) Long term changes in the parameters of cosmic ray daily harmonics. 15th Intern. Cosmic Ray Conf. Papers (ICRCP) 11,298. Bartels, J. (1935) Random fluctuations, persistence, and quasi-persistence in geophysical and cosmical periodicities. J. terr. Magn. atnz. Eke. (now J. geophys. Res.) 40, 1. Chapman, S. and Bartels, J. (1940) Geomagnetism. The University Press, Oxford. Forbush, S. E., DUggal, S. P., Pomerantz, M. A. and Tsao, C. H. (1982) Random fluctuations, persistence, and quasipersistence in geophysical and cosmical periodicities : a sequel. Rev. Geophys. Space Phys. 20,971. Riker, J. F. (1986) The solar diurnal variation in cosmic ray intensitv. Ph.D. Dissertation, Universitv of New Mexico. Riker, J. F. and Ahluwalia, H. S. (1983jPersistent cosmic ray solar diurnal variations observed during 1973379. 18th ICRCP, 10, 190.
003213633/87 $3.00 +O 00 Pergamon Journals Ltd.
A SURVEY OF THE COSMIC RAY DIURNAL VARIATION DURING 1973-1979-L PERSISTENCE OF SOLAR DIURNAL VARIATION
Department
J. F. RIKER and H. S. AHLlJWALIA* of Physics and Astronomy, The University Albuquerque, NM 87131, U.S.A. (Received in final form
of New Mexico,
18 March 1987)
Abstraer-We find that the solar diurnal variation in cosmic ray intensity is a persistent phenomenon over the years 197331979. That is, even during solar minimum, conditions in the heliosphere lead to a net flux of cosmic rays from a particular (time-varying) direction in space. If we regard the daily Ructuations in the amplitude and phase of the diurnal variation as random perturbations about the mean vector, we are able to quantify the relative magnitude of the random component.
m2(h)= m2(l)/h
INTRODUCTION
The solar diurnal variation has been the subject of considerable study in cosmic ray research. In particular, it is found that the variation undergoes dramatic changes in amplitude and phase during the solar activity cycle (Ahluwalia, 1977). In the years of solar minimum, the amplitude can be extremely small, to the point that one wonders whether the variation persists year round. It is the purpose of this paper to generalize the persistence analysis of Bartels (1935) to the case of the diurnal variation, in order to determine the components and persistence of the signal. The method utilizes the insight of Bartels that geophysical data at adjacent data points are not sequentially independent, since high values tend to follow high values and similarly for low values. Forbush et ul. (1982) adapted the method for use in analyzing the 27-day variation using simulated data, but we shall apply it to yearly averages of cosmic ray diurnal variation, during 1973-1979. We use data obtained with the vertical underground muon telescope located at Embudo, NM (vertical cut-off rigidity N 19 Gv).
(1)
where m’(l) is the variance of the original series. If instead of random ordinates, we use a series formed by repeating each ordinate some fixed number, w, of times, the standard deviation for averages of h = wh’ ordinates is m’(h) = m(h’) = m(l)/fi
= ?Fz(l)\m
(2)
where w = ~~‘z(~)/~*(l) is the number of repetitions. original series, we can compute m( 1) for single ordinates and ation m(h) for averages of (h = 2,3,. . .). By analogy with
(3)
Reverting back to the the standard deviation also the standard devih successive ordinates equation (3), we define
E(h) = /zm2(/z)/m’(l)
(4)
as the equivalent number of repetitions in the series. For truly random data, equations (1) and (4) imply e(h) = 1, which is the no conse~ation case. If actual data have conservation, equation (4) yields a measure of how much. This can be generalized to yield a statistical test for periodicity, or persistence.
CONSERVATION
The fundamental difference between a time series of cosmic ray data and a series of random ordinates is the presence of conservation in the former, as pointed out by Chapman and Bartels (1940). They quantitied the concept by considering a series of random ordinates : if one averages together h consecutive ordinates and computes the variance of the new (averaged) series, one obtains the variance -._* Now on leave at: NASA Headquarters, Code EZH, Washington,
DC 20546, U.S.A.
PERSISTENCE
If sequences of data rather than individual data points are considered, it should be possible to ascertain whether some form of behavior persists from one sequence to the next. In particular, we are interested in persistent periodicities, whereby a sinusoidal function would behave similarly (in terms of its frequency, amplitude and phase) from one sequence to the next. For example, if a year of cosmic ray data were broken into individual days, we desire to test for persistent
1111
1112
J. F.
RIKER
and
waves of given frequency, viz. the diurnal frequency (1 cpd) and its harmonics. One might suspect that if there were a truly persistent diurnal variation (i.e. one in which the frequency, amplitude and phase did not change from one day to the next), it could be detected even if the persistent wave had large random fluctuations of amplitude and phase superimposed on it. The original Bartels (1935) method employed harmonic analysis amplitudes as the data values, with a direct analogy to e(h) of equation (4) as the measure of persistence. Forbush et al. (1982) also used that method, exactly as given by Bartels. However, Bartels noted that his method could be generalized to avoid any explicit reference to harmonic analysis by dividing the original data into rows of length equal to the period under consideration and using the behavior of standard deviations. Chapman and Bartels outlined this method more completely ; this is the method that we have developed (Riker, 1986). Consider sequences of data values xi,, where the double index implies rows and columns of data as follows :
D,:x,,.x,+, D, : x(&f-
,... x2,_, ,)r,
.
(51
. .x,5_ ,a
Here, r is the period considered, N is the total number of observations, and ii4 is the number of rows. For the diurnal variation, r = 241x,N = total number of hourly observations, and M = number of days. Note that the technique does not require every day to be included in a diurnal analysis. Since the persistent wave would be present every day, with random day to day changes superimposed, skipping some days due to solar flares, equipment failure, calibration, or other causes should not change the result. If only a quasipersistent wave were present, then such missing data could be significant. The persistence test consists of determining the behavior of the sequences as they are averaged together ; first, the non-averaged sequences shown in equation (5) are used. Averages for each row are computed, yielding Lt’,,D;, . . . , Oh,. The variances
are obtained for each row, where i and ,j are the row and coIumn indices (i = I,. . . , n/r; j = 0,. . . ,r- I). These are just the variances of the individua1 ordinates in row i from 0:. We define
so that 5, is the average standard deviation for the
H. S.
AHLUWALIA
individual sets. Next we average the columns of h rows (h = 2,. . . , h,,,)together to give K = N/h average sets. These Knew rows are handled in the same manner as before, i.e. their row averages 0: and row variances 5:’ are obtained. Again, the row variances are averaged to obtain the new average standard deviation t,(h). By analogy with equation (4) we define
The quantity g,(h) is then the equivalent number of repeated sets in h successive sets, and h/cr,(h) is the effective number of independent sets. A repeated set is just one in which variations from one ordinate to another are the same in each set. If a truly persistent function is present, one expects exact behavior from one set to the next, so that the average sets behave the same as the individual sets. In that case, t:(h) = <,” and a,(h) = h. For random data, equation (1) implies t,(h) = t,/$, so that a,(h) = 1. Real geophysical data would then have o,(h) between these limits. If a,(h)-+S, as h -+ co, one has quasi-persistence, which is a tendency for similarities in successive rows to vanish gradually, so that there is no relation between rows sufficiently far apart. This would occur for data which are persistent over a few rows, but in which the amplitude and phase undergo irregular changes. There are several considerations to be noted for this persistence test. Firstly, the question of overlapping average sets should be addressed. That is, when one forms average sets of h rows, either K = N/h sets could be used (first row composed of rows 1,. ..,h,second row composed of rows 2, ...,h + 1,etc.). Overlapping sets allow averaging of more values for &(h), so that one makes a better estimate of the average t,(h). However, the same data are used repeatedly in this method. The non-overlapping method should yield a more stringent test of truly persistent functions, despite inaccuracies in the averages t,(h) for large h. Since this method is also easier to program, it was selected for our studies. Secondly, the question of how large h may be is important, since averaging only a few values of t,(k) would lead to unacceptably large errors in [,(/I). Bartels suggested that, for M harmonic analysis amplitudes, the number of independent averages K should not be too small if the expectancy of the average amplitude is to come close to the correct value. We gave h,,,= M/SO for the expectancy to be within 10% of the correct value. He also suggested that if N is the total number of available observations, then h max should be chosen so that kr = N/20. This last criterion is the one selected in the following analyses. For Embudo data, N = 4800 h typically, so
Persistence of diurnal variation In Cl = Quasi - persistent P = Persistent 2 _
R-
Random
data
average
y;(h) = $/h,
data data
,
1113 sets,
0,’
remains
unchanged,
but
so
5,2(h) = cC+y:/h e,(h) = hC(h)/5,2
= h(&/5,2)+$/5r2
= h(W,2/5,2) + 1,ct:-w:,
I
a,(h) = 1+c$(h-
Number
of averaged
sets (h)
FIG. I.
POSSIBLE FORMS OF PERSISTENCE CURVES WHEN DATA ARE COMPLETELY RANDOM (R), QUASI-PERSISTENT (Q), PERSISTENT (P), OR A MIXTURE OF THESE POSSIBILITIES.
I)/(;.
(14)
This is the equation of a straight line. Since 0,’ < t: by equation (13), the slope is less than one but is not negative. This equation allows calculation of the variance of the periodic and random parts of the data from the values of a,(h).
RESULTS
h,_r
= N/20
(9)
implies h,,, = 10 for the diurnal variation h max= 20 for the semi-diurnal variation.
and
DISCUSSION
We generated a computer program to perform the persistence analysis. Initially, numerous tests on simulated data were performed in order to verify the program. Our preliminary results were reported elsewhere (Riker and Ahluwalia, 1983). Before presenting more results, we describe the expected behavior for some simple cases. For random data, as already noted, t,‘(h) = <,?/h, so that equation (8) yields o,(h) = hl,‘(h)/t,2 Truly persistent
data have t:(h) a,(h) =
= 1.
(10)
= <,?, so that
h[5,‘1/5,’= h.
(11)
Quasi-persistent data have cr,(h) + S,, by definition. Equation (8) implies S, = /zt,‘(/r)/l,’
(large h)
t,!(h) = l,‘S,/h
(large h).
so that (12)
The various possibilities are shown in Fig. 1. For data composed only of a persistent plus a random component we have x=w+g where w is the persistent function and g is the random part. If the standard deviations of x, w and g are l,, w, and y,, then if w and g are uncorrelated, one has 5: = w,?+y,2.
(13)
We tested the persistence analysis using known persistent sinusoids of several frequencies, variable amplitude and phase functions, random data, and combinations of random plus persistent data. The last case is particularly useful, because the magnitude of the random component relative to the periodic component significantly affects the slope of the persistence curve. For such data, the curve of o(h) is always a straight line whose slope depends upon the magnitude of random fluctuations relative to the persistent function amplitude. Using the slopes obtained for data with known relative magnitudes of random to persistent variations, it is possible to specify the relative magnitude of the random component when real data are used (Riker, 1986). This approach works only so long as the real data produce straight lines for o(h). Results for actual cosmic ray muon data are summarized in Figs 2 and 3. In Fig. 2, we show the results for the data obtained at Embudo for 197331979, on one plot. All the curves are straight lines with slopes less than 1.0, from which we may conclude that the data consist of at least one periodic component of frequency 1 cpd plus random juctuations. The slopes of the curves vary in time, reaching a minimum in 1977. This implies that the slope of the curve, and hence, the relative magnitude of the observed random component, tends to mirror the behavior of the first harmonic. This correspondence is not exact, however. Figure 3 shows results obtained for 1973 when more days are averaged. The straight line behavior persists, no matter how large h,,, is taken, although statistical effects enter when h,,, is larger than about 30. By comparing the slopes of the persistence curves for each year to those for known data, we estimated the relative magnitudes of the random and periodic components. The technique is simple. First, one
J. F. RIKER and H. S. AHLUWAL,IA
1114
3
I 22 I
2 3
19f3 1974 1975 1976
b
I
3
I 2
I
I
I 3
I 5
4 Number
of days
I
I 7
6
I 8
I 3
1 IO
OverOQed
FIG.~.~RSISTENCEC~RVESA~S~OWNFOREAC~ YEAR FORT~~NDERGROUNDM~ON DATA OBTAINEV EMBUW, NM, DURING 1973..1979. All curves can be approximated by st~ight lines of slopeslessthan 1.0.
AT
2 OW 4 z
0.13
8 c %
0.11
.: 0.09 e
0
I
I
3
6
I 9
i 12
I
I
I
I
15
I8
21
24
8 27
Number of days averaged FIG. 3. PERSISTENCE CURVE FOR EMBUDO MUON DATA FOR 1973 lSSHOWN,WHEN h,,, ISLARGE. Note that the straight-line behavior persists.
obtains the usual plots of u(h) vs h for the data consisting of a sinusoid of a known amplitude “a” and random contributions of variance 5'. Next, one plots the persistence curve for numerous values of the “random fraction” k = m/a, where c is a known (arbitrary) constant of proportionality. Note that k is a ~~~le~s~o~l~~~number. It specifies the magnitude ofthe random ~uctuations relative to the known persistent amplitude. Thus if k = 1, we have a random component with variance identical to the square of the persistent amplitude. On the other hand, if k = 0, we obtain a horizontal line. Instead of plotting a(h) vs h for several values of k, it is more convenient to plot CT(~)vs k for each value of h. One then obtains nonlinear curves with c(h) = h on the k = 0 axis and m(h) = 1 as k increases. These “standard curves” are characteristic of the random fraction and should be obtained independent of the amplitude (a) and the phase of the sinusoid. We tested this technique for typical cosmic ray sinusoids and found it to be approximately true. Next, one uses the real data to obtain the usual persistence curves (see Fig. 2). For
2 0.07 3 6 I I I I I I I o.o5ISZI 1374 1975 1976 1977 1978 1979 Year FIG. 4. A COMPARISON IS MADE BETWEEN THE RELATIVE MAGNITUDES OF TNE AMPLITUDE OF THE DIURNAL VARIATION AND THE MAGNITUDE OF THE RANDOM COMPONENT PRESENT INTHEDATA. As expected, they appear to bear an inverse correlation.
each year and for each value of h, one follows the appropriate “standard curve” until one reaches the value of cr(h) found from the data. One then drops a vertical line to the abscissa to get an estimate of the magnitude of the random component for that value of h. Averaging all these estimates produces a good estimate of the relative magnitude of the random component for the year. We show our results for both the diurnal variation amplitude (before applying orbital effect correction) and the estimated random fraction (relative magnitude) in Fig. 4. We wish to point out that the observed standard deviation in the amplitude, for each of the years during 1973-1979, is -0.01%. however, the errors “attributed” to the random component appear to be much larger than the statistical errors in the data. CONCLUSIONS
The results presented in this paper give a rough estimate of the magnitude of the random component
Persistence
of diurnal
relative to that of the periodic component. While the magnitudes of the random fraction do tend to inversely follow the behavior of the first harmonic, an exact correspondence does not exist. If the random fluctuations were constant in magnitude as a function of time, we would expect that as the magnitude of the periodic part decreases, the value of the estimated random fraction should increase. This is the case for the trend over some years, but the fine structure does not necessarily follow this rule. For example, the amplitude of the diurnal variation decreases from 1973 to 1974, so we expect the estimated random fraction to increase, but it actually decreases. This effect should be analyzed further. However, we can still assert that the Embudo data do exhibit a persistent sinusoidal variation for all years 197331979, including the years near solar minimum. Also, the data apparently consist of only persistent plus random components, with the random part larger in magnitude. These results indicate for the first time that the diurnal variation is present year round, in the data
variation
1115
obtained at the underground site of Embudo, NM. We believe the technique has more general applications in terms of uncovering hidden periodicities in data which contain large random errors.
REFERENCES
Ahluwalia, H. S. (1977) Long term changes in the parameters of cosmic ray daily harmonics. 15th Intern. Cosmic Ray Conf. Papers (ICRCP) 11,298. Bartels, J. (1935) Random fluctuations, persistence, and quasi-persistence in geophysical and cosmical periodicities. J. terr. Magn. atnz. Eke. (now J. geophys. Res.) 40, 1. Chapman, S. and Bartels, J. (1940) Geomagnetism. The University Press, Oxford. Forbush, S. E., DUggal, S. P., Pomerantz, M. A. and Tsao, C. H. (1982) Random fluctuations, persistence, and quasipersistence in geophysical and cosmical periodicities : a sequel. Rev. Geophys. Space Phys. 20,971. Riker, J. F. (1986) The solar diurnal variation in cosmic ray intensitv. Ph.D. Dissertation, Universitv of New Mexico. Riker, J. F. and Ahluwalia, H. S. (1983jPersistent cosmic ray solar diurnal variations observed during 1973379. 18th ICRCP, 10, 190.