1. CALCULATION OF THE Dst INDEX
Presentation at LWS CDAW Workshop Fairfax, Virginia March 14-16, 2005
R.L. McPherron
Institute of Geophysics and Planetary Physics
University of California Los Angeles

2. GEOGRAPHIC COORDINATES USED IN MAGNETIC MEASUREMENTS
Dipole is tilted and inverted relative to rotation axis
Dipole field lines are nearly vertical above 60 latitude
Cartesian geographic coordinates are defined in a plane tangent to earth at observer’s location
X component is towards geographic north pole
Y component is east along a circle of latitude
Z component is radially inward or down

3. LOCAL VIEW OF VARIOUS COORDINATE SYSTEMS USED IN GEOMAGNETISM
Origin is located at observer
X points north, Y points east, Z points down in the local tangent plane
F is the total vector field
H is the horizontal projection of the vector F
D is the east declination of H from geographic north in tangent plane
I is the inclination of F below the tangent plane
X, Y, Z are the geographic Cartesian components of F

4. DISTRIBUTION OF RING CURRENT AND ITS PERTURBATION IN A MERIDIAN
Most of the current is concentrated close to the equator
Eastward current inside and westward outside
Perturbations curl around the volume of current
The perturbation over the earth is nearly uniform and axial

5. Origin of Dst
Moos, N.A.F., Colaba Magnetic Data, 1846 to 1905, 2, The Phenomena and its Discussion, Central Government Press, Bombay, 1910.
Figure below taken from the following reference to illustrate work by Moos
Chapman, S., and J. Bartels, Geomagnetism, Vol 1, Clarendon Press, Oxford, 1962.
Use a large set of storms with start time uniformly distributed in local time
For each hour after an ssc (storm time) find the average departure of H at a single station from its mean value in the corresponding months (disturbance) obtaining the disturbance versus storm time or Dst
Separate the storms by the local time at which the ssc occurred to illustrate the asymmetry of the development as seen by a single station

6. IGY Calculation of Dst Measured field at ith station ) ( t D L Sq H +t D L Sq H i + =
Disturbance Variation
Lunar Variation
Solar Quiet Day Variation
Main Field Variation
Main field and its secular variation
Secular Variation from average
IGY Average

7. Average Variation over Longitudinal Chain
For each hour average the preceding equation over 8 stations around the world at fixed latitude
Average Disturbance
Average Lunar variation ~ 0
Average Sq at 8 stations
Average secular variation at 8 stations
Round-world average variation at time t

8. Average Sq Variation
For ith station in month m take the average of the five international quiet days (25 hours) defined by Greenwich time
From this average quiet day subtract a linear trend connecting midnight at the two ends of the Greenwich day
For each month average the quiet day variations over all stations
Model the residual average Sq variation with a double Fourier series in time T and month M. Estimate up to 6th harmonic.
Use this series to estimate the average Sq variation at any hour of any day of year. Subtract the estimate from DH(t).

9. Average Secular Variation
Plot the residual variation versus time. Found that there was no trend in its baseline, i.e. the average secular variation was constant
Determine the average level of the baseline during quiet intervals not affected by magnetic storms
Subtract this constant from the previous residual obtaining
Assume that the last term in {} is approximately zero so that
Assume that only the ring current contributes to the average disturbance so that we have found the disturbance as a function of storm time

10. Modern Dst Calculation Sugiura, M. , and T
Modern Dst Calculation Sugiura, M., and T. Kamei, Equatorial Dst index , IAGA Bulletin No 40, pp. 7-14, ISGI Publications Office, Saint-Maur-des-Fosses, France, 1991.
Use four stations distributed in longitude near 25 magnetic latitude
SECULAR VARIATION
For each station calculate annual means from the 5 quiet days of each month
Use current and four preceding annual means to determine a polynomial fit to the quiet days. Let t be time relative to some reference epoch.
Use the preceding 5-year fit to predict the baseline value on first day of current year. Include this value as a data point in the current 5-year fit
Create the deviation of H from the secular trend for each hour of current year
QUIET DAY VARIATION
Use the five local days closest to the Greenwich monthly five quiet days plus 1-hour at each end of these days
At each hour calculate monthly averages of the local quiet days
Subtract a linear trend passing through average of first and last hours
Fit a double Fourier transform in hour of day and day of year to the 12 sets of 24 hourly values
Use the fit coefficients to calculate the quiet day variation at every hour of year
Subtract the estimated Sq variation from the deviation time series
Calculate Dst as the latitude weighted average disturbance variation

11. Creation of the Secular Variation
For every calendar month select 10 international quiet days
Determine the monthly median value at local midnight (red dots)
Take 2-year running average of midnight medians
Fit a cubic smoothing spline to the filtered data (black line)

12. Creation of Monthly Quiet Day Curve
Create ensemble of the 10 international quiet days for a month
Subtract value at local midnight
Subtract linear trend through left and right local midnight
Calculate median variation as function of time

13. Solar Cycle Effects on Sq Variation
Calculate monthly median quiet day for each month of four solar cycles (44 years)
For each year of an 11-year solar cycle calculate mean of four monthly medians
Compare all means (all years in lower right panel)
There appears to be little effect of solar cycle on the median quiet day
We can ignore effect of phase of solar cycle in Sq

14. Kakioka Monthly Quiet Days 1960 to 2004
For each month in 44 years find median Sq of 10 quiet days
Find median of each month for all years
Arrange as a map of variation as function of local time and month
Use coefficients of two-dimensional Fourier transform to calculate Sq

15. Fourier Synthesis of Arbitrary Quiet Day
Use data from four solar cycles
Find the median quiet day for each month
Load data into a 12 month by 24 hour 2-D array
Perform a double Fourier transform
Expand array to 366 by 24 and move the Fourier coefficients to correct location
Inverse transform to obtain quiet day for each day of year

16. Removal of Secular and Quiet Variations
Select the portion of the secular variation curve for the interval
Synthesize the quiet day variation for the appropriate days of year
Subtract both from the original data to obtain the disturbance variation for the given station component

17. LONGITUDINAL PROFILE OF Bj FROM MAGNETOSPHERIC CURRENTS
Symmetric ring should create nearly constant longitudinal profile in H component
Local time average of H at equator approximates B at center of Earth
But other magnetospheric currents create local time dependent deviations from symmetry
Assume asymmetric component has zero mean when averaged over local time
Define the disturbance storm time index Dst as local time average of observed H profile

18. Relation of Dst to Stream Interface
The figure shows the relation of several solar wind and magnetospheric variables to CIRs The main stream interface at leading edge of high speed stream is taken as epoch zero in a superposed epoch analysis
The colored patches show upper and lower quartiles of the variable as function of epoch time
The heavy red line is the median curve
Stream interfaces cause weak storms

19. COMPARISON OF SEVERAL OBSERVED AND PREDICTED QUIET DAYS AT GUAM IN 1986
-30
-20
-10 10 20 30 40 50 60 70
Observed
Quiet
Disturbance (nT)
Residual 40 41 42 43 44 45 46 47 48 49 50
Day in 1986

20. CORRECTED H AT GUAM DURING RECOVERY FROM A MAGNETIC STORM60 40 20
Quiet H
-20
-40
Disturbance (nT)
-60
-80
Corrected H
-100
-120
Observed H
-140 40 41 42 43 44 45 46 47 48 49 50
Day in 1986

21. The End!

22. SCHEMATIC ILLUSTRATION OF EFFECTS OF RING CURRENT IN H COMPONENT
Magnetic effects of a symmetric equatorial ring current
Projection of a uniform axial field onto Earth’s surface

23. In
Out

24. REMOVAL OF SECULAR TREND FROM HOURLY VALUES OF H AT GUAM DURING STORM
115
120
125
130
135
140
3.575
3.58
3.585
3.59
395.5
x 10 4
Observed H (nT)
COMPARISON OF GUAM H WITH SECULAR TREND IN 1986
-100
-50 50
Day in 1986
Transient H (nT)
DEVIATION OF GUAM H FROM SECULAR TREND IN 1986
Secular Trend

25. MINOR MAGNETIC STORM RECORDED AT SAN JUAN - 11/24/96

26. THE DESSLER-PARKER-SCKOPKE RELATION

27. CONTRIBUTIONS TO THE VARIATION IN THE H COMPONENT

28. ESTIMATION OF THE SECULAR TREND IN H COMPONENT AT SAN JUAN
1978
1983
1988
2.7
2.705
2.71
2.715
2.72
2.725
2.73
2.735
2.74
x 10 4
Year
H (nT)
Fourth Order Trend
Daily Average
80% Point

29. QUIET VALUES DURING STORM USED IN QUIET DAY (Sq) ESTIMATION80
Flagged Point
Quiet Value 70 60 50 40 30
Transient H (nT) 20 10
-10
-20
115
120
125
130
135
140
Day in 1986

30. QUIET GUAM H TRACE AT EQUINOX AND SOLSTICE 1986
Spring
Summer
Fall
Winter 5 10 15 20
-10 30 40 50 60
Local Time
Pred Quiet H (nT)

31. Sq FOR H AT SAN JUAN IN 1978 AS FUNCTION OF DAY OF YEAR AND UT-5
350 -5 10
300 20 25 15 5
250
200 20
Day of Year 15 5
150 15 10 20 25
31.1 30
100 50 20 25 -5 5 5 10 15 20
UT Hour -5 5 10 15 20 25 30
Diurnal Variation (nT)

32. REMOVAL OF STORM EFFECTS IN QUIET DAY (Sq) ESTIMATION
COMPARISON OF DETRENDED GUAM H TO MIDNIGHT SPLINE 50
Disturbance (nT)
-50
Midnight Spline
-100
H Comp
115
120
125
130
135
140
DETRENDED AND STORM CORRECTED GUAM H IN 1986 80 60 40
Residual H (nT) 20
-20
115
120
125
130
135
140
Day in 1986

34. CURRENTS CONTRIBUTING TO MIDLATITUDE MAGNETIC PERTURBATIONS
View is from behind and aabove earth looking toward Sun
Current systems illustrated
Symmetric ring current
Dayside magnetopause current
Partial ring current
Tail current
Substorm current wedge
Region 1 current
Region 2 current
Current systems not shown
Solar quiet day ionospheric current
Secular variation within earth
Main field of Earth

35. EFFECTS OF MAGNETOPAUSE ON THE Dst INDEX
Balance magnetic pressure against dynamic pressure 10 8 6
Neutral 4
Point 2
Z (Re) -2
Solar
Wind -4 -6 -8
-10 15 10 5
X (Re)

36. A SHEET CURRENT MODEL OF EFFECT OF TAIL CURRENT ON Dst
Magnetic Effects
Tail Current Model
xxx x x x -6 -4 -2 2 4 6
-35
-30
-25
-20
-15
-10 -5
-Xgsm (Re)
Bz (nT)
Normal Tail
Inner Edge
Total
Earth Ri Ro Bz
xxx x x x

37. MAGNETIC EFFECTS OF A SUBSTORM CURRENT WEDGE
Transverse currents in the magnetosphere are diverted along field lines to the ionosphere
Viewed from above north pole the projection of the current system has a wedge shape
Midlatitude stations are primarily affected by field-aligned currents and the equatorial closure (an equivalent eastward current)
The local time profile of H component is symmetric with respect to the central meridian of wedge
The D component is asymmetric with respect to center of wedge

38. STEPS IN THE CALCULATION OF Dst INDEX
Define the reference level for H component on a monthly basis
Fit a polynomial to reference H values (secular variation)
Adjust H observed on a given day by subtracting secular variation
Identify quiet days from same season and phase of solar cycle
Remove storm effects in quiet values and offset traces so that there is zero magnetic perturbation at station midnight
Flag all values recorded during disturbed times and interpolate from adjacent quiet intervals
Create some type of smoothed ensemble average of all quiet days
Subtract average quiet day from adjusted daily variation to obtain disturbance daily variation for station
Repeat for a number of stations distributed around the world at midlatitudes
Project the local H variations to obtain axial field from ring current and average over all stations

39. Magnetograms from several midlatitude stations during storm

40. MAJOR SUBSTORMS DURING MAGNETIC STORM OF APRIL 3-5, 197912 24 36 48 60 72
-2000
-1000
1000
AU and AL index (nT)
125
627
1610
1803
2118
2351
252
757
1143
1702
2200
700
900
1032
1641
2147
-200
-100
100
Time from 0000 UT on April 3
Dst Index (nT)

41. CONCLUSIONS
The Dst index is defined to be linearly proportional to the total energy of particles drifting in the radiation belts (symmetric ring current)
Dst must be estimated from surface measurements of the horizontal component of the magnetic field
Surface field measurements include effects of many electrical currents other than the symmetric ring current
These effects must be estimated or eliminated by the algorithm that calculates the Dst index
Extraneous currents include: secular variation, Sq, magnetopause, tail, Region 1&2, partial ring current, substorm current wedge, magnetic induction
There are numerous assumptions and errors involved in Dst calculations and the index contains systematic and random errors as a consequence
Be aware of these problems and take them into account in interpreting Dst!

42. EXAMPLE OF MIDLATITUDE MAGNETIC DATA DURING MAGNETIC STORM

43. INTERPLANETARY MAGNETIC FIELD, AE AND Dst INDICES DURING STORM
Coronal mass ejection produce intervals of strong southward Bz at the earth
Magnetic reconnection drives magnetospheric convection
Convection drives currents along field lines and through ionosphere
Ground magnetometers record effects of ionospheric currents in H and other components
H traces are used to construct the AE and Dst index

44. MAGNETIC EFFECT OF A RING CURRENT AT EARTH’S CENTER
Axial field from a circular ring current Z
Field at center of ring X
LRRe
Westward
Ring Current
Convenient units

45. THE SOLENOIDAL EFFECT OF THE RADIATION BELT CURRENTS
A more realistic model of the ring current
Shows the magnetic perturbations
Shows the distortion of dipole current contours
Perturbation field from ring current

46. DESSLER-PARKER-SCKOPKE DERIVATION