# The Mid-Latitude Positive Bay and the MPB Index of Substorm Activity

## Abstract

Substorms are a major source of magnetic activity. At substorm expansion phase onset a westward current flows through the expanding aurora. This current is the ionospheric closure of the substorm current wedge produced by diversion of tail current along magnetic field lines. At low latitudes the field-aligned currents create a systematic pattern in the north (X) and east (Y) components of the surface magnetic field. The rise and decay in X is called a midlatitude positive bay whose start is a proxy for expansion onset. In this paper we describe a new index called the midlatitude positive bay index (MPB) which monitors the power in the substorm perturbations of X and Y. The index is obtained by removing the main field, storm time variations, and the solar quiet (Sq) variation from the measured field. These are estimated with spline fits and principal component analysis. The residuals of X and Y are high pass filtered to eliminate variations with period longer than 3 hours. The sum of squares of the X and Y power is determined at each of 35 midlatitude stations. The average power in night time stations is the MPB index. The index series is standardized and intervals above a fixed threshold are taken as possible bay signatures. Post processing constrains these to have reasonable values of rise time, strength, and duration. Minima in the index before and after the peak are taken as the start and end of the bay. The MPB and AL indices can be used to identify quiet intervals in the magnetic field.

### Keywords

Substorm Substorm current wedge Field-aligned current Quiet day variation Midlatitude positive bay index (MPB) MPB onsets## 1 Introduction

### 1.1 Magnetic Activity and the “Quiet Day”

The Earth’s magnetic field is constantly changing. On a scale of a million years the main field reverses polarity (Merrill and McFadden 1999). On shorter scales the field components are slowly varying due to changes in currents internal to the Earth. This is called secular variation (Barraclough et al. 1992). On a daily time scale an observatory on the Earth’s surface rotates underneath an ionospheric current caused by solar and lunar tides. This current system is approximately fixed about local noon so diurnal rotation carries a station beneath different parts of the system altering the surface magnetic field. The pattern and strength of this current depend on ionospheric conductivity and so the measurements change with time of day, season, and solar cycle. These changes are called the solar quiet day (Sq) variation (see Chap. 7 in Chapman and Bartels 1962). All studies of the Earth’s main field must remove the effects of Sq from the data. At a midlatitude station the variation in the X component (north) is roughly symmetric about local noon and the Y component (east) is antisymmetric. The amplitude of these variations is about 50 nT as compared to a total field of about 50,000 nT.

The solar wind also drives currents in the ionosphere through a process called magnetospheric convection (Cowley 2000). These currents are linked to the magnetosphere through magnetic field-aligned currents (FAC) (Iijima and Potemra 1976a). The current along field lines, the current in the ionosphere, and currents induced in the Earth cause field variations. These are often much stronger than the Sq current and their pattern is more complex and variable. In the auroral zone the disturbances can be as large as 3000 nT comparable to the horizontal component of the main field. Much of the history of solar-terrestrial physics has been concerned with attempts to find patterns in these variations and to understand their causes.

A fourth source of magnetic variation is electric currents in space around the Earth. The magnetopause is the boundary between the solar wind and the Earth’s magnetic field and is created by a sheet current flowing in a pattern very similar to that of the Sq current (Chapman and Ferraro 1931; McPherron 1987). It causes an increase in the surface field everywhere on the Earth (Mead 1964). The strength of this effect depends on the dynamic pressure of the solar wind so it can change quite suddenly when the solar wind changes. The magnetotail is produced by a current that has the shape of two solenoids stretched out behind the Earth. This current is generated by the tangential drag of the solar wind on the magnetosphere stretching field lines out behind the Earth. This current generally causes decreases in the surface magnetic field and it is also highly sensitive to the properties of the solar wind. The ring current is a doughnut shaped current around the Earth that produces a southward axial field parallel to the dipole axis (Akasofu et al. 1961). It is created by the westward drift of positive ions (mainly protons) and eastward drift of electrons in the Earth’s main field. During its growth the current is asymmetric with maximum depression near dusk (Cummings 1966; Clauer and McPherron 1974a, 1980). The ring current can produce southward equatorial disturbances as large as \(-600~\mbox{nT}\) that persist for many days (Akasofu and Chapman 1964). Such a disturbance is called a magnetic storm. Sudden increases in solar wind drag on the magnetosphere create smaller disturbances called magnetospheric substorms (Coroniti et al. 1968). During a substorm a fraction of the tail current is diverted through the ionosphere for a short time causing strong disturbances in the auroral zone and at midlatitudes. This short-lived current system is called the substorm current wedge (SCW) (McPherron et al. 1973a). The magnetic perturbations of the SCW at midlatitudes are the primary focus of this paper as their presence provides a good indicator that the magnetic field is not quiet.

Studies of the Earth’s main field depend on measurements free of effects of currents external to the Earth. A day in which there is no magnetic storm and an interval in which there are minimal variations caused by magnetospheric currents is called a quiet day. On the dayside of the Earth the effects of the Sq system are always present and so must be subtracted from the measurements. On the nightside there is little effect from Sq, but magnetospheric activity may cause changes in the magnetic field which we refer to as magnetic activity. To minimize effects of such activity it is essential to use different indicators of quiet conditions and to select the quietest intervals possible. Generally magnetic indices are used to accomplish this. Today there are a number of available indices, each designed to monitor a specific current system. A description of these indices and their creation is given in a companion paper in this collection (Kauristie et al. 2016). There are several distinct current systems that produce most geomagnetic activity as discussed next.

### 1.2 The DP 1 and DP 2 Current Systems

### 1.3 The Substorm Current Wedge

The real current wedge is known to be more complex than indicated by this diagram (Kepko et al. 2014a). The currents actually flow on field lines distorted by the tail current. The FAC are volume distributions of current better approximated by sheet currents. There is an additional wedge of opposite polarity inside the wedge that is not shown in Fig. 2. This current is weaker than the outer current so what is seen on the Earth at midlatitudes is the difference in effects of the two current loops. In space between the two loops the magnetic perturbations are much larger than seen on the ground. Currents induced in the Earth by these loops add to the effects measured at the surface. If a spacecraft magnetometer at low altitude travels through the interior of the wedge it will measure effects of the ionospheric closure as well as amplified effects from the two loops. Fortunately the current wedge is only present for a short time during substorms and a good index of its presence should allow one to eliminate times when it is present.

### 1.4 Region 1 and Region 2 Currents

A three dimensional rendering of the R-1 and R-2 current system is presented in Online Resource 2 (Le et al. 2010). In the drawing dawn is toward the upper left and dusk toward the lower right. The diagram makes the point that the DP-2 system flows at right angles to the Pedersen current. The electrojets are Hall currents concentrated between the two sheets of FAC. At the universal time of the drawing the eastward electrojet flows toward midnight in the lower right of the drawing. The Region-1 and Region-2 are present all the time changing in strength as the solar wind changes.

### 1.5 Magnetic Indices

The polar cap index PC is derived from a single observatory close to a magnetic pole (Troshichev et al. 1988; Stauning 2013). It is a measure of the magnetic perturbation created by the sheet current flowing from midnight to noon in the DP-2 system. The actual derivation is quite complex but it is designed to be proportional to a quantity dependent on the solar wind electric field at the magnetopause. Panel (a) of Fig. 4 shows the rectified dawn to dusk solar wind electric field Ey. It is nonzero when GSM Bz in the solar wind is southward. It is fluctuations in the direction of Bz that is the primary cause of changes in the level of magnetic activity. The PC index for this interval is plotted in Fig. 4(b). It is obvious that increases in PC are related to increases in Ey.

The auroral electrojet indices AL and AU are respectively measures of the westward and eastward electrojets (Davis and Sugiura 1966; Newell and Gjerloev 2011). The standard indices are obtained from 12 stations distributed around the auroral oval. A station beneath the eastward electrojet will measure positive perturbations in the north component and beneath the westward electrojet it will measure a negative perturbation. These perturbations can be calculated by subtracting a background level from the traces to obtain fluctuations about zero. When all 12 stations are plotted relative to a single baseline some station will have the maximum positive perturbation and another will have the most negative. Because these are the largest disturbances these two stations are assumed to be closest to the overhead current. Assuming a current sheet of infinite width in the north-south direction shows the perturbation is directly proportional to the overhead current density. On the other hand assuming the current is a line current the perturbation is proportional to the total current attenuated by distance. Repeating this procedure for each time step produces the upper and lower envelopes to the overlapping traces. The sequence of values of the upper envelope is the AU (auroral upper) index and the lower envelope is the AL index. The difference between the two \(\mathrm{AE} = \mathrm{AU}-\mathrm{AL}\) is the auroral electrojet index. These indices are plotted in Fig. 4(c). In this case there is very little variation in AU, but the AL index displays four major decreases of about three hour duration called negative bays. Dotted vertical lines are times of expansion phase onsets derived from the MPB index which is described in this paper.

The Dst (disturbance storm time) index is an hourly measure of magnetic disturbance measured at low latitudes. It is calculated from a latitude weighted mean of the variations in the north component recorded by five stations around the Earth (Sugiura and Kamei 1991; Iyemori et al. 1994). This index depends on accurately calibrated data from which the secular variation and Sq variations have been removed. Dst is designed to measure the strength of a westward ring current symmetric around the Earth. However, it is also sensitive to changes in solar wind dynamic pressure moving the magnetopause relative to the Earth, and to the tail current growing and decaying during activity. Since the current is generally not symmetric a local time average is used. A one-minute resolution version of this index is Sym-H plotted in Fig. 4(d). The trace of Sym-H has few features that correlate with the AL index and in this interval is only weakly disturbed. If a magnetic storm were in progress Sym-H would be much more negative.

The degree of asymmetry of the ring current is measured by the Asym-H index plotted with a negative sign downward in Fig. 4(e). Asym-H is quite sensitive to perturbations caused by the substorm current wedge as seen by the peaks delineated by two of the vertical dotted lines. The Asym-D index is plotted upward in the top part of the panel. Both indices are defined as the difference between the maximum and minimum perturbation in local time profiles of the deviations of H or D from the quiet day.

The bottom trace Fig. 4(f) shows the midlatitude positive bay (MPB) index for this interval of time (Chu et al. 2015). The index series has been processed by a search algorithm to generate a list of current wedge onsets which on this day found four as indicated by vertical dotted lines. Three of the MPB onsets are associated with substantial decreases in the AL index but the event after 02:00 UT is not. A peak in MPB at the left edge of panel (f) actually began at 22:37 UT on the preceding day. There is a corresponding decrease in the AL index in panel (c). Also there are 4-5 weak pulses in MPB that were not selected. One of these after 07:00 UT was associated with a weak changes in AL but two others after 18:00 UT were not. MPB pulses often have a complex waveform such as the one after 04:00 UT that suggest the substorm has multiple onsets. Similar ambiguity is apparent in the AL trace. The difficulty in identifying the onset of a substorm is one of the major reasons the cause of the substorm still remains controversial. From this figure it is obvious that there was no quiet interval on this day. As a measure of how quiet a given day is we calculate daily mean MPB. We find than March 3, 2008 ranked 190th in 2008 by this indicator.

The MPB index is a new magnetic index originally developed by one of the authors (XC) to facilitate his dissertation research on the substorm current wedge. The index was briefly described in the publication (Chu et al. 2015). As a coauthor of this paper RLM found there was not sufficient detail to allow one to reproduce the index series and onset list. With the goal of understanding the procedure and validating the index RLM created new algorithms for index generation and onset determination. In this report we will refer to the two versions of index generation as the “Chu method” and the “McPherron method”. As we will show the two index series are not identical and the onset lists derived from them also differ. The Chu method has been applied to all data from 1980 through 2012 producing the Chu MPB index and simultaneously the Chu MPB onset list. The McPherron method for index generation has been applied only to the two years 2007-2008. The McPherron onset determination is a post processing of the index series so we have generated a list of onsets using the Chu index series called McPMPB onset list and a separate list using the McPherron index called mcpmpons78. The second list has been restricted by the area of MPB pulses to produce a third list ons78700.

## 2 Generation of the Midlatitude Positive Bay (MPB) Index

Stations used in the creation of the MPB index using the McPherron method, their identification codes, locations in geographic and geomagnetic coordinates, and the universal time of local midnight

Code | Station Name | Geo Lat | Geo Long | Mag Lat | Mag Long | UT Midn |
---|---|---|---|---|---|---|

MBO | Mbour | 14.38 | 343.03 | 20.10 | 57.50 | 0.29 |

ASC | Ascension Island | −7.95 | 345.62 | −2.40 | 56.60 | 0.96 |

KOU | Kourou | 5.10 | 307.30 | 11.90 | 19.50 | 3.51 |

PST | Port Stanley | −51.70 | 302.11 | −41.70 | 11.50 | 3.86 |

TRW | Trelew | −43.26 | 294.62 | −33.10 | 5.60 | 4.16 |

SJG | San Juan | 18.11 | 293.85 | 28.60 | 6.10 | 4.41 |

HUA | Huancayo | −12.05 | 284.67 | −1.80 | 356.50 | 5.02 |

FRD | Fredicksburg | 38.20 | 282.63 | 48.40 | 353.38 | 5.08 |

BSL | Bay St. Louis | 30.35 | 270.36 | 40.10 | 339.80 | 6.05 |

TEO | Teoloyucan | 19.75 | 260.81 | 28.80 | 330.40 | 6.61 |

BOU | Boulder | 40.14 | 254.76 | 48.40 | 320.59 | 7.32 |

TUC | Tucson | 32.18 | 249.27 | 39.90 | 316.00 | 7.70 |

FRN | Fresno | 37.09 | 240.28 | 43.50 | 305.30 | 8.42 |

PPT | Pamatai | −17.57 | 210.43 | −15.10 | 285.10 | 9.97 |

HON | Honolulu | 21.32 | 202.00 | 21.60 | 269.70 | 10.53 |

EYR | Eyrewell | −43.47 | 172.39 | −50.23 | 256.39 | 11.17 |

API | Apia | −13.81 | 188.22 | −15.40 | 262.70 | 11.45 |

GUA | Guam | 13.59 | 144.87 | 5.30 | 215.70 | 14.34 |

ASP | Alice Spring | −23.76 | 133.88 | −32.90 | 208.20 | 14.86 |

KAK | Kakioka | 36.23 | 140.18 | 27.40 | 208.80 | 15.05 |

KNY | Kanoya | 31.42 | 130.88 | 21.90 | 200.80 | 15.58 |

CNB | Canberra | −35.32 | 149.36 | −42.70 | 226.90 | 16.04 |

BMT | Beijing Ming Tombs | 40.06 | 116.18 | 30.10 | 187.00 | 16.45 |

GNA | Gnangara | −31.78 | 115.95 | −41.90 | 188.90 | 16.58 |

LRM | Learmonth | −22.22 | 114.10 | −32.40 | 186.50 | 16.72 |

PHU | Phuthuy | 21.03 | 105.95 | 10.80 | 177.90 | 17.02 |

IRT | Irkutsk | 52.17 | 104.45 | 41.90 | 176.90 | 17.13 |

GCK | Grocka | 44.30 | 102.30 | 43.30 | 102.40 | 17.24 |

AAA | Alma-Ata | 43.25 | 76.92 | 34.30 | 152.70 | 18.66 |

ABG | Alibag | 18.63 | 72.87 | 10.20 | 146.20 | 19.15 |

AAE | Addis Ababa | 9.02 | 38.77 | 5.30 | 111.80 | 21.42 |

IZN | Iznik | 40.50 | 29.73 | 37.70 | 109.60 | 21.43 |

TAM | Tamanrasset | 22.79 | 5.52 | 24.70 | 81.80 | 23.07 |

EBR | De L’Ebre | 40.82 | 0.49 | 43.20 | 81.30 | 23.38 |

SPT | San Pablo-Toledo | 39.55 | −4.35 | 42.80 | 76.00 | 23.62 |

HER | Hermanus | −34.43 | 19.23 | −34.00 | 84.00 | 23.78 |

Both implementations of the MPB index calculation and onset list are carried out in three main steps summarized here. Details of each step in the McPherron implementation are described in subsequent sections. The first step in this procedure is an estimation of the quiet day variation in the two horizontal components of the field at each station. We do this using principal component analysis (PCA) of a 23-data segment centered on the day of interest (Golovkov et al. 1978). This process is repeated for both X and Y at each station for an entire year. The resulting arrays are filtered to obtain lower harmonics of the diurnal and annual variations. These harmonics are stored in an annual quiet day file and used in the next step.

The second step loads all data in a year for a given station and subtracts the Sq variations from X and Y. In some cases the Sq variation estimated from the 23-day segment for a given component is a good fit to the day of data at the center of the window. We estimate the quality of this fit using prediction efficiency (PEF) defined as PEF – 1-var(data-Sq)/var(data). This measure of quality is later used to rank a given day on a scale from least to most disturbed. We next high pass filter the residual series (data-Sq) to obtain field perturbations related to substorms. The power in the filtered residuals of X and Y is summed giving horizontal power at this station in the substorm frequency range. A year-long array containing daily series of power in substorm perturbations is saved to a file.

The final step is to mask the power trace from each station keeping only night time data. The year-long vectors of station power are stored in the 35 columns of an array with \(366\times 1440\) rows. The column vector of row averages of this array is the midlatitude positive bay (MPB) index for the year. In the following paragraphs we illustrate some of the important steps in these calculations.

The adjusted ensemble array (\(\mathbf{A}\)) for a field component contains successive days as rows. Because the Sq variation changes slowly with season we expect that successive rows will be correlated. We can calculate the covariance matrix for this ensemble array by the matrix product \(\mathrm{cov} = \mathbf{A}^{T}\mathbf{A}\). This matrix can be diagonalized by an orthogonal transformation that provides its eigenvalues and eigenvectors. Although the vector of eigenvalues has length 1440 only the first 23 (number of rows in \(\mathbf{A}\)) are significant. All eigenvectors are linearly independent. A linear superposition of these weighted by the square root of the eigenvalues reproduces the original signal contained in the sequence of rows. If we use only a few of the eigenfunctions we will obtain an approximation to the original signal. This same operation is more simply achieved by the singular value (svd) decomposition of the original array (Press et al. 1986). This decomposition can be written as \(\mathbf{A} = \mathbf{U}\times\mathbf{S}\times\mathbf{W}^{\mathrm{T}}\) where \(\mathbf{U}(23,23)\) is column orthogonal, \(\mathbf{S}(23,1440)\) is diagonal, and \(\mathbf{W}(1440,1440)\) is fully orthogonal. The columns of \(\mathbf{W}\) are the same as the eigenvectors of \(\mathbf{A}^{T} \mathbf{A}\) and the singular values on the diagonal of \(\mathbf{S}\) are the square root of its eigenvalues ordered by decreasing magnitude. The matrix product \(\mathbf{Z} = (\mathbf{S}\times\mathbf{W}^{\mathrm{T}})/(\mathrm{sqrt}(\mathrm{nrows}))\) creates a matrix whose rows are the weighted eigenvectors. These rows are called the principal components of the covariance matrix. The first row contains most of the variance as it corresponds to the largest eigenvalue.

We assume that the Sq variations are periodic with local time and with day of year. We then perform an FFT to obtain the 2-D Fourier coefficients of each map. An examination of the amplitude of the coefficients reveals that five diurnal coefficients and ten annual coefficients are sufficient to represent most of the variation in both components. We then set all other coefficients to zero (2-D low pass filter). The coefficients in the filtered array are then multiplied by the appropriate complex number to time shift from local time to universal time. The resulting array has only 50 unique complex numbers which is a tremendous reduction in the amount of information written to external files. Subsequent analysis using the Sq variations is accomplished by loading the annual file of Fourier coefficients for a station and storing the coefficients in the proper locations. These matrices are inverse transformed to obtain the filtered Sq maps. An example for the Honolulu Sq-X map is plotted in Online Resource 6. The filtered map is considerably smoother than the original map presented in Fig. 9. Also the time shift to universal time places the peak disturbance for HON at 22.53 UT near the end of the UT day. This array is transposed and its columns written to a single column vector containing all quiet days for the year. It is this time series that is subtracted from a column vector containing all original data for the year to obtain the residual variation.

Chu procedure used 41 stations from 20 to 52° north and south magnetic latitude while here we used 35 stations from −45° to \(+45\)°

Chu procedure removed main field and secular variation by linear fits to X and Y while here a spline fit to midnight values of a station component was used

Chu procedure removed Sq variations by a superposed epoch average of 21 days while principal component analysis of 23 days was used here

Chu procedure removed residual low frequency variations with a high pass filter with cutoff at 12 hours while in this work a 3 hour cutoff was used

Chu procedure used stations \(\pm5\) hours of 23.5 LT while here all stations between dusk and dawn were used

## 3 Determination of midlatitude bay onset from the MPB Index

The purpose of the MPB index is to detect the effects of substorm associated field-aligned currents using midlatitude stations. The onset of these currents is closely related to the formation and expansion of the auroral bulge near midnight. Historically the onset of the midlatitude positive H-bay has been used as a proxy for this onset (Caan et al. 1978). We take the minimum in the MPB index preceding a rapid increase to a peak as a first approximation to the time of this onset. In Fig. 12 one can see at least four major increases above the background. The procedure to automatically detect such onsets is described in this section.

Four large negative bays occurred on this day. The onsets of three of these as listed in the SuperMag onset list are shown by dotted vertical red lines at 04:22, 09:37 and 15:45 UT. Panel (c) presents the MPB index series determined by the procedure described in the preceding section. Panel (d) shows the MPB index as determined by the Chu method (Chu et al. 2015). Pulses in the MPB index time series are primarily the consequence of substorm FACs. It is evident that both MPB index series have similar waveforms and that the start of pulses appear to correspond to the beginning of negative bays seen in the SML index. Panel (b) shows a standardized version of the MPB index series plotted in panel (c). To standardize the data a 4-hour running mean of the MPB index is subtracted from the data and the difference is divided by the 4-hour running standard deviation. This procedure shrinks large peaks and enhances small peaks in the index series. This facilitates the use of thresholding to detect peaks well above background. In this case we define a significant peak as all values exceeding a threshold of 1.0. In panel (b) there are eight peaks that pass this threshold. Annotation beneath each MPB peak shows the area under each peak.

A comparison of Fig. 13 panels (b) and (c) reveals that some standardized peaks are originally quite small in the raw data. For example a peak at 12:19 UT in panel (b) has an area of only \(194~\mbox{nT}^{2}\) as compared to the peak at 09:41 which has an area of \(1351~\mbox{nT}^{2}\). We find that peaks with area less than \(700~\mbox{nT}^{2}\) almost never correspond to a SuperMag onset in panel (a) so we have used this value as a threshold to select major onsets. In this case vertical dashed lines in panel (c) satisfy this criterion. It should be noted that the 12:19 UT MPB standardized peak appears to correspond to very weak disturbances in the SuperMag trace in panel (a) that were not identified as SuperMag substorm onsets. It is also evident that the two different MPB algorithms produce different onset times. The 02:18 UT event in panel (d) was not selected because of its small area but was selected by the Chu algorithm. Similarly the event at 16:05 UT in panel (c) was not selected by the Chu algorithm. Note in this case the SuperMag onset is earlier at 15:45 UT.

Only peaks in the original MPB index above \(25~\mbox{nT}^{2}\) were examined. This is similar to our standardization and rejection of small peaks.

When two peaks were closer than 30 minutes the onset of the larger peak was selected while we merged the peaks and determined the onset of the first.

The first approximation to the onset was done in the way described above, but a second step refined this onset time. Only stations that observed a change of more than 10 nT in 20 minutes were used in the refinement. For these stations a combined time rate of change \(( \partial X / \partial t )^{2} + ( \partial Y / \partial t )^{2}\) was calculated for each station, averaged, and the time at which it suddenly increased was taken as the onset.

## 4 Properties of the MPB Index

### 4.1 Occurrence Statistics

The Chu MPB index has been calculated for four solar cycles beginning in 1982 and ending December 2012 and some occurrence statistics are described in Chu et al. (2015). During this interval the Chu MPB list contains 40,562 MPB onsets corresponding to an average occurrence rate of 3.58 per day. This rate is modulated by the solar cycle with 6 per day in 2003 and 1.2 per day in 2009. The rate also depends on season with peaks at the equinoxes (4.2 and 3.8 per day) and minima at the solstices (3.2 and 2.9 per day). The occurrence rate depends on time relative to the stream interface at the center of a corotating interaction region with only 2 per day a day before the interface and 6.2 the day after. The average waiting time distribution between onsets is a function with a long tail to later times with a mode of 0.72 hours and a median of 3.35 hours. Probability distributions for rise and decay times of a peak in the MPB index have modes of 16 and 18 minutes, but the means are longer at 21 and 29 minutes. A comparison of MPB onsets with auroral expansion onsets determined by the IMAGE spacecraft has a sharp peak at 0.4 min with a width of 1.9 min. Since the spacecraft images have a cadence of ∼2 minutes we conclude the MPB onset is virtually simultaneous with the auroral expansion onset.

### 4.2 Point Process Cross Correlation Between MPB and SuperMag Onset Lists

The association number is defined as the number of times the time difference between events of opposite type occurs in a given bin. The units of the curve are number per unit time. In this case the bin width was one minute and we consider lags in the interval \(\mathrm{Noff} = \pm3000\) minutes. In Fig. 15 we have plotted only the interval from −400 to \(+400\) minutes since otherwise the central peak would collapse to a vertical line. Beyond the edges of the plot both traces asymptotically approach zero and show no other particular behavior. We have normalized the association number by dividing by the quantity \(2h \cdot \sqrt{\mathit{rateB} \cdot \mathit{rateC}}\). The peak of the normalized observation curve (blue) is \(\mathrm{max} =18.5\) per minute at \(\mathrm{xlag} = 0\). Points on the left (negative lag) correspond to the comparison event (SMag) occurring earlier than a base event (ChuMPB). The numbers Nl, Nc, Nr are respectively the number of events beyond the left side of the analysis interval (Time Delay \(<-3000~\mbox{min}\)), in the interval \([-3000, +3000]\), and beyond the right side (Time Delay \(>+3000~\mbox{min}\)). Very few events in either list are ever separated by more than 3000 minutes.

To estimate the number of chance associations we have used surrogate times series that have the same waiting time distributions as the MPB and SuperMag lists but with events occurring at random times. The procedure for the generation of surrogate time series is described on p. 203 of Press et al. (1986). The two artificial event lists had nearly 10 million events each. The smooth red curve is the output of applying the point process algorithm to the surrogate series for the two lists. The normalized association number at zero lag for random data is 1.0. Note the non-intuitive result that the rate of random association is highest at zero lag. Also the curve for the association of random data is asymmetric. The decay at negative lags is characteristic of the base list (ChuMPB) and at positive lags is characteristic of the SuperMag list. The more rapid decay of the SMag list is because its waiting time distribution is peaked closer to zero delay than the Chu MPB list as shown by the average occurrence rates for the two lists.

The normalized point process association number for the two lists (blue curve) is strongly peaked at zero lag far above the chance level (red line) indicating that many events in the two lists are associated. This applies in the range \([-226, 79]\) minutes outside of which the random association curve becomes dominant. We call this the “causal interval”. We estimate the fractional number of associations in this interval by integrating the normalized association number curve for the observed data obtaining 465 events. The random association number in this same interval has an area of 227 events leaving 238 events as the area corresponding to real associations. We conclude that the ratio \(238/465 \ge 51.2~\%\) of the observed associations in this interval are real causal associations.

The annotation in the upper right shows the total number of events and the number of associations within the limits of the correlation (Ncen). The values Lagl, Lpeak, and Lagr are respectively the locations of the left edge, the peak, and the right edge of the causal interval. The three pairs of quantities with names of the form Fxxx, and Axxx are respectively the fraction of the associations that are likely real, and the fraction of the total number of events within a given interval of time delay. The characters “xxx” refer to the causal interval “cau”, a reference interval “ref” (\(\pm30~\mbox{min}\)), and the simultaneous interval “sim” (\(\pm4~\mbox{min}\)). These numbers indicate that 51.1 % of the events in the causal interval are real events and that 62.8 % of all events lie in this interval.

### 4.3 Comparison of Additional Onset Lists

The Chu MPB onset list was determined from the Chu index by the Chu procedure and contains 40,562 events in the interval 1982-2012. Another list, McPMPB was determined from the Chu index using the McPherron onset procedure with the constraint that only peaks with area exceeding \(700~\mbox{nT}^{2}\) were included. This list contains 56,342 events in the same time interval as the Chu list. A third MPB onset list was determined by the McPherron procedure using the McPherron index for the years 2007-2008. This list (mcpmpbons78) has 6,384 events. A fourth list was created from this list by constraining its onsets to those with pulse area exceeding \(700~\mbox{nT}^{2}\). The final list (ons78700) has 2,141 events. Since we have no “correct list” with which to compare we have arbitrarily taken the list (SMag) determined from the SuperMag SML index as a comparison list.

*merit*defined as

*Lagl*and

*Lagr*define an interval on the abscissa of the point process association number graph,

*fraction of events*is the ratio of the number of associations between these lags to the total number of event in the combined lists, and

*fraction real*is the ratio of the number of real events to the total of real and chance events. How these fractions are calculated is explained next.

The point process cross correlation algorithm combines the two lists and sorts them into ascending order. It then sequentially calculates the separation between the current element and the next element from the other list and increments the association number stored in the corresponding bin. The curve generated in this fashion has units of number per unit time. If the number of lags used exceeds any observed separation the sum of the raw association number as a function of lag equals the total number of events in the combined list. The raw association number is normalized by dividing by the dimensionless factor \(2h \cdot \sqrt{\mathit{rateB} \cdot \mathit{rateC}}\) so that the number of associations at zero lag for random data is 1.0. If we sum the raw association number for interval [*Lagl, Lagr*] we obtain the total number of events within this range. This is equivalent to a definite integral of the association number curve with the increment \(2h\). The units of the increment cancel the units of the association number leaving a pure number. The ratio of this sum to the total number of events in the combined list is the quantity *fraction of events*.

The raw association number curve contains both real and chance associations. As explained in Sect. 4.2 we estimate chance associations by generating surrogate lists from random numbers with the same waiting time distributions as those of the observed lists. The normalized association number is calculated for the two surrogate lists. The normalization is chosen so the chance association at zero lag is always 1.0 and decreases away from zero lag. We subtract the normalized chance association curve from the observed curve to obtain a quantity proportional to the number of real associations. We then sum this difference curve between two lags to obtain a measure of the real associations over this interval. The ratio of this sum to the sum of the observed normalized association curve over the same interval is defined as *fraction real*. The product of the two fractions is the fraction of the total number of events that are real associations in a given interval.

In our case we find all lists have a peak association within 1-2 minutes of zero lag and that the association number curves are not symmetric about the origin indicating that some events in one list are delayed relative to events in the other list. Our measure *merit* uses sums from *Lagl* below zero to *Lagr* minutes above zero. Usually the observed association number curves have a deficit of events relative to chance at larger lags because many events are truly associated and appear in the curve near zero lag. Thus the observed curve drops below the chance curve at some negative and positive lags. Our first measure of quality uses sums between these two lags. The locations of these lags are identified in Fig. 8 with large black “X” and also shown in the figure annotation as Lagl and Lagr. We refer to this interval as the “causal” interval. Our second measure is the “reference interval” which we take as \(\pm30\) minutes. A third measure is the “simultaneous interval” which we take to be \(\pm4\) minutes.

*merit*of eight different list pairs and summarize the results in Table 2. In this table the base and comparison list names are entered in the first two rows of each column. Column 1 identifies each parameter characterizing the point process cross correlation curve. These are defined in the footnotes to the table. The sixth row shows the total number of events in the combined lists. The rows labeled Fxxx and Axxx respectively show the fraction of events in an interval that are real, and the fraction of all events that lie in the interval. The last three rows show the

*merit*for each of the three intervals used. The interpretation of

*merit*is the fraction of NTOT events that are real and associated in the interval. Higher

*merit*corresponds to more real events associated close to zero lag.

Parameters obtained from point process cross correlations between list pairs defined by first two rows of each column

Parm | Lists | |||||||
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

Sbase | ChuMPB | ChuMPB | ChuMPB | ChuMPB | McPMPB | McPMPB | McPMPB | ons78700 |

Scomp | MCPMPB | SMag | mcpmpbons78 | ons78700 | Smag | mcpmpbons78 | ons78700 | SMag |

Obase | 40562 | 20840 | 2265 | 2265 | 28212 | 2665 | 2665 | 2141 |

Ocomp | 56339 | 28356 | 6384 | 2141 | 28358 | 6384 | 2141 | 2651 |

Tdays | 11321 | 6208 | 731 | 731 | 6209 | 731 | 731 | 731 |

Ntot | 96901 | 49196 | 8649 | 4406 | 56570 | 9049 | 4806 | 4792 |

Ncen | 95870 | 48560 | 7899 | 4317 | 55770 | 8641 | 4697 | 4646 |

Lagl | −39 | −80 | −25 | −74 | −77 | −8 | −16 | −107 |

Lpeak | 2 | 0 | 2 | 2 | 0 | 1 | 2 | −4 |

Lagr | 14 | 70 | 15 | 26 | 44 | 20 | 43 | 80 |

FCAU | 0.749 | 0.618 | 0.633 | 0.727 | 0.582 | 0.738 | 0.810 | 0.643 |

Acau | 0.313 | 0.466 | 0.178 | 0.377 | 0.430 | 0.199 | 0.357 | 0.503 |

Fref | 0.749 | 0.618 | 0.633 | 0.727 | 0.582 | 0.738 | 0.810 | 0.643 |

Aref | 0.309 | 0.318 | 0.196 | 0.322 | 0.316 | 0.236 | 0.353 | 0.319 |

Fsim | 0.907 | 0.903 | 0.849 | 0.941 | 0.872 | 0.884 | 0.955 | 0.908 |

Asim | 0.141 | 0.125 | 0.095 | 0.160 | 0.112 | 0.140 | 0.228 | 0.111 |

Mcau | 0.235 | 0.288 | 0.113 | 0.274 | 0.250 | 0.147 | 0.289 | 0.324 |

Mref | 0.232 | 0.197 | 0.124 | 0.234 | 0.184 | 0.174 | 0.286 | 0.205 |

Msim | 0.128 | 0.113 | 0.081 | 0.150 | 0.098 | 0.123 | 0.217 | 0.101 |

To understand the significance of these results consider the row labeled ACAU. In column 1 we see that only 0.313 (31.3 %) of the events in the combined ChuMPB and McPMPB lists are associated within the causal interval (defined by rows Lagl and Lagr). In contrast the ChuMPB and SMag lists in column 2 are associated at the level of 46.6 %. This is better than the correlation of McPMPB list with SMag at 43.0 % in column 5. The *merit* values in row MCAU lead to a similar conclusion – the ChuMPB onsets are better associated with SMag onsets than are the McPMPB onsets. In contrast the *merit* values for the simultaneous interval is 12.8 % for the ChuMPB and McPMPB lists as compared to 11.3 % for the ChuMPB and SMag lists. This indicates that events in the two MPB lists are more nearly simultaneous than events in the ChuMPB – SMag lists. We emphasize that the McPMPB list is derived from the Chu index series by the McPherron onset procedure. Column 8 tells a different story. The list mcpmpbons78 derived from the McPherron MPB index series in 2007-2008 by the McPherron procedure was constrained by pulse area creating list ons78700. This constrained list is associated with SMag at the level of 50.3 % and has a *merit* of 0.324, i.e. 32.4 % of all associations in the causal interval are real associations. Note, however, that the fraction of simultaneous associations (0.101 in row MSIM column 8) is slightly smaller than the simultaneous real ChuMPB-SMag associations (0.128 in column 2). This suggests that McPherron procedure applied to the McPherron index creates the highest correlation with the SMag list, but the associations are more broadly dispersed in the causal interval. It is conceivable that the two years 2007-2008 were different from other years. Thus we also limited the ChuMPB list to just these two years and associated it with the SMag list. We find (not shown) that the causal *merit* is virtually the same whether calculated from all years or just these two years.

Table 2 also contains a comparison of the list mcpmpbons78 with the lists ChuMPB and McPMPB. By all measures the comparison is very poor. The reason is that this list is the initial list derived from the standardized version of the McPherron index in 2007-2008 without consideration of the size of MPB pulses. Many of the events in this list are small pulses that may have been produced by other causes than a substorm expansion. To determine what threshold to use in limiting this list we performed a sequence of calculations limiting the values of pulse area in the mcpmpbons78 list to values greater than a given threshold and correlating this subset to the SMag list. We find that all measures increase until an area of \(700~\mbox{nT}^{2}\) where simultaneous *merit* reaches a maximum. The fraction of events within the causal interval maximizes at \(1000~\mbox{nT}^{2}\) and the causal fraction of events maximizes at \(1200~\mbox{nT}^{2}\). We have chosen \(700~\mbox{nT}^{2}\) as the threshold. This value also makes the number of events in the SMag list and the new list ons78700 nearly the same. Table 2 shows that this new list correlates better with the McPMPB list (\(\mathrm{ACAU} =35.7~\%\)) derived from the Chu index series than does the ChuMPB list (31.3 %). More important this new list correlates with SMag with 50.3 % of all events within the causal interval as compared to ChuMPB at 46.6 %.

It is not clear how many of the events in any of the lists are true onsets of a substorm expansion. The events in the SML onset list may correspond to the onset of a growth phase, to a pseudo breakup, to an intensification of an on-going expansion, or to poleward boundary intensifications in the recovery phase (see discussion in McPherron 2015). In the MPB index a peak can be produced by a dynamic pressure pulse, short term deviations of the Sq variation from the calculated value, development of the partial ring current or inadequate distribution of stations in latitude and local time.

### 4.4 Local Time Profiles of Current Wedge Magnetic Perturbations

## 5 Discussion

The concept of the substorm current wedge was introduced more than 40 years ago to explain the pattern of magnetic field variations recorded on the ground during the expansion and recovery phases of a magnetospheric substorm (McPherron 1972). During the expansion phase the north component X of the midlatitude magnetic field increases reaching a peak at the center of the wedge as illustrated in Fig. 16. The east component Y increases to a maximum under the outward current and decreases to a minimum under the downward current. In the auroral zone the downward current closes both westward across the bulge of expanding aurora and equatorward to an outward current (Kepko et al. 2014b). At the western edge of the auroral bulge the directions of these currents are reversed. On the ground the observed effects are caused by the difference between the inner and outer loops so ground measurements are unable to determine the total current in the outer loop, but should be equal to the westward ionospheric current. The current wedge was originally modeled with a simple line current model of currents flowing on dipole field lines (Horning et al. 1974). Improved models now use sheet currents on field lines determined from empirical models (Chu et al. 2014; Sergeev et al. 2011).

The current wedge is an important tool in organizing ground and space observations of substorm phenomena. As initially demonstrated by Clauer and McPherron (1974b) the current wedge may be centered almost anywhere in the night sector although its most probable location is at 23 local time (McPherron et al. 2011; Frey et al. 2004). We have inverted ground data for most of the substorms detected with the MPB index and examined the relation of plasma flow bursts in the tail recorded by THEMIS spacecraft. It is found that over 85% of all flows occur inside the current wedge and that they establish a distribution of plasma pressure and flux tube volume consistent with the idea that the field-aligned currents are driven by gradients in these two quantities (Chu 2015).

The onset of the current wedge has been extensively used to time the onset of a substorm (Caan et al. 1978, 1975). Akasofu et al. (1965) demonstrated that the polar magnetic substorm is closely associated with the auroral substorm, and Akasofu and Meng (1969) established that it is also associated with positive bays at lower latitudes. More recently Slavin et al. (1992) and Nagai et al. (1997) have used the midlatitude positive bay onset to time plasmoid release in the tail. Chu et al. (2015) have shown that to the time resolution of auroral images on the IMAGE spacecraft that the MPB onset is simultaneous with the onset of the auroral expansion.

Remove the main field and a crude approximation of storm time field from X and Y

Remove the quiet day variations in X and Y

High pass filter the X and Y residuals to include only substorm variations

Sum the squared residual variations to obtain total power in horizontal field variations

Find the average power in all nighttime stations at each time step

Standardize the MPB index with a moving window to obtain a time series with roughly zero mean and unit standard deviation

Apply a threshold to the standardized data to identify possible peaks in MPB

Determine the time when the peak reaches its maximum value

Eliminate narrow peaks and merge closely spaced peaks into a single peak

Find the minima from which the peak grows out of background and eventually returns

Evaluate the peaks to determine the rise and decay times, the true maximum value, and the total area under the peak

Use criteria involving the peak properties to select a subset of all standardized peaks to select those thought to be true midlatitude positive bays associated with auroral zone disturbances.

In this work we have used principal component analysis (PCA) to determine the Sq variations in X and Y, and harmonic analysis to smooth and simplify their representation. The advantage of this approach is the ability of PCA to utilize the correlations between the variations on successive days to reject noise and determine a relatively smooth and slowly changing signal. The disadvantage is that both atmospheric and magnetic activity alters the Sq variation on a shorter time scale than allowed by a moving window of 23 days. An improved determination of the Sq would require modeling of the phenomena that change the strength and pattern of the Sq system.

It might seem that removal of the Sq variation is not important as most of the substorm disturbances are on the night side where the Sq current is weak or not present. We note, however, that quite frequently the substorm current wedge extends into the afternoon sector as was illustrated in Fig. 16 and it is important to minimize effects of Sq in the frequency band of substorm perturbations. In our calculations we use subtraction of Sq and high pass filtering of the residual to create the index. In evaluating an MPB peak we also use subtraction of a reference profile to study changes in disturbance. This eliminates any residual effects present at the onset of a peak. Of course, changes in the background field due to rotation of the Earth, the magnetopause, magnetotail, and ring current alter this reference profile and slightly distort the calculated profiles. In an extension of this work reported by Chu et al. (2014) we use a model of the current wedge, partial ring current and symmetric ring current to track changes in these other currents with time. We find that over the approximately one hour duration of a current wedge the changes due to other factors are very small compared to those caused by the current wedge.

One major difficulty encountered in this work is how to deal with several closely spaced peaks in the MPB index. Is each peak a separate substorm, or do they represent successive intensifications of a single substorm? We do not yet have an answer to this question. A second problem is how to refine the current wedge onset time. The McPherron procedure uses a minimum in the index prior to the rise to a peak as an approximation of this time. We have found that quite frequently there are delays in the onset times between widely separated stations that are lost in the averaging procedure. The Chu MPB list of 40,562 MPB onsets is based on a procedure that performs an additional step to refine the onset time. This procedure takes the subset of stations producing the largest contribution to the index and only use them to determine the onset time.

The MPB index is potentially useful in identifying quiet days. In many studies investigators use as reference the international quiet days located at http://www.gfz-potsdam.de/en/section/earths-magnetic-field/data-products-services/kp-index/qd-days/ which lists the 10 quietest and 5 most disturbed days in each month. There are considerable differences between months in the level of disturbance present in the “quietest day”. An alternative is to use the daily sum of Kp to sort days throughout the year into ascending order of disturbance. The MPB index can be used in the same way by taking day averages in universal time and sorting. We have calculated and sorted sum Kp for the year 2008 finding that Dec. 1, Dec. 2, Nov. 22, Sept. 13 and Oct. 9 are the five quietest with values 0, 0, 0.3, 0.6, 0.6. In comparison the five quietest by the MPB index are Nov. 5, Dec. 1, Oct. 9, Oct. 24, and Nov. 4 with values of average MPB of 0.94, 0.96, 0.99, 1.07, and 1.16. The days Sept. 13, Oct. 9, Nov. 22, and Dec. 1 were selected as the quietest days in the last 4 months of 2008. It is apparent that there is considerable overlap in these different methods of defining quiet days. During quiet times the MPB index assumes value of order \(1~\mbox{nT}^{2}\) with peaks less than \(5~\mbox{nT}^{2}\) and so can be used to select quiet intervals shorter than a day. We believe that the best way to select times without magnetic activity it to use the joint probability that sum Kp, AL, and MPB fall below appropriate thresholds.

In Sect. 4.3 we correlated the different MPB onset lists with each other and with the SMag onset list. For pairs of MPB onset lists we found that lists McPMPB and onset78700 have the highest causal *merit* at 28.9 %. These two lists were derived using the McPherron onset procedure applied to the Chu MPB index and McPherron index series in the years 2007-2008. This indicates that slight differences in the index series can lead to significant differences in the onset lists. It is surprising that the ChuMPB onset list correlates better with the SMag list than it does with the McPMPB onset list derived from the same index series. The best association between any pair of lists (32.4 %) was between ons78700 and SMag. This pair had the highest fraction of events in the causal interval and the highest *merit*. However, this should not be surprising as the list was selected by pulse area to maximize the correlation with SMag.

It is disturbing that different implementations of the same procedure produce onset lists that have as little correspondence as demonstrated here. At best only about 10 % of all associations occur within \(\pm4\) minutes of zero lag and no more than 32 % within the causal interval. The reason for this is clear from our discussion of Fig. 13 where we compared the SML index, the McPherron MPB index and the Chu MPB index for a single day. On a day where it is visually obvious that there were at least five negative bays only one was selected by the automatic procedure for selecting negative bay onsets. Similarly the McPherron onset procedure applied to the McPherron index series selected only two MPB onsets neither of which corresponded to the SML onset. Finally the Chu procedure applied to the Chu index series also selected only two onsets, neither of which corresponded to the SML onset, and only one to a McPherron onset with a delay of 8 minutes. The situation for other lists of substorm onsets is even worse than shown here. We have examined the correspondence of four lists of AL onsets and two lists derived from satellite auroral images. The figures of *merit* for these lists are lower than the values presented here.

The McPherron onset determination procedure was designed to maximize the correlation with the SMag list. If the SML procedure misses events then we expect MPB events will be missing as well. This can be seen in panel (b) of Fig. 13 where analysis of the standardized MPB series identified seven possible MPB onsets of which three are visually associated with negative bays and the other three correspond to very weak SML perturbations (\(\sim-100~\mbox{nT}\)). All but three MPB pulses were rejected as being too small. Another problem is how the onset algorithms deal with closely spaced onsets. It appears likely that the substorm at 04:22 UT has a double onset. The SML onset is at 04:22 UT and is associated with a weak MPB pulse, However it is followed 32 minutes later by a strong MPB pulse at 04:54 UT. It is also clear that slight differences in the index series and onset determination procedures lead to differences between two MPB onset lists. For example the 02:13 UT MPB pulse in the McPherron index series (panel b) was rejected by the McPherron procedure but kept by the Chu procedure. The opposite occurred at 16:05 UT These may be consequence of insufficient data creating problems with the indices, inadequate onset selection constraints, or they may be indications of failure of the substorm current wedge to properly model the magnetic variations during some substorms.

Currently no definitive list of substorm onsets exists so there is no obvious way to optimize a procedure to detect them. In fact different researchers do not agree on the definition of substorm. In the examination of Fig. 1 we found that a better selection of onsets can be accomplished by using the SML and MPB index series together. We have developed a new algorithm which selects SML onsets in a manner similar to what we have done with the MPB index. We then use the unabridged SML list to determine if there is an MPB onset within \(\pm30\) minutes. If not we adjust the MPB selection parameters to see if a slightly weaker threshold on the standardized index will reveal a weak peak. We then invert the comparison using MPB onsets to detect weaker SML onsets. If other data such as Pi 2 pulsations, auroral luminosity, or auroral kilometric radiation (AKR) it should be possible to further refine the onset time.

## 6 Conclusions

The midlatitude positive bay (MPB) index is a new magnetic index designed to respond to the field-aligned currents that are part of the substorm current wedge. It is based on the observation that the onset of an auroral expansion and sharp increase in magnetic activity in the auroral zone is accompanied by perturbations in X and Y at midlatitudes. Historically visual selection of the start of an increase in the midlatitude X component has been used as a proxy for the expansion onset in many studies of magnetic activity. The MPB index improves on this technique by utilizing both the X and Y components increasing the signal strength and decreasing the probability that the increase is caused by some process other than a substorm. The procedure described here automates the creation of the index and the selection of onsets.

Our initial implementation of this procedure (Chu et al. 2014; Chu 2015) uses all available midlatitude data from 1982 to the end of 2012 to calculate the index and produce a list of 40,562 onsets. The implementation described in this paper has only been applied to data in 2007 and 2008. We have used the list created by the initial implementation to study the statistical properties of substorm occurrence as a function of universal time, season, solar cycle, and time in a corotating interaction region. In addition we have used this list as a driver for a program that inverts the midlatitude perturbations with a realistic model of the substorm current wedge. The parameters obtained in this inversion allow us to organize satellite observations of disturbances in the magnetotail in a coordinate system based on the location and extent of the current wedge. The MPB index is currently available on request from either of the authors. The onset lists are provided as supplementary information attached to this paper.

## Notes

### Acknowledgements

The principal author is grateful to the International Space Science Institute (ISSI) for inviting him to take part in the Workshop on “Earth’s Magnetic Field” held in Bern in May 2015. This work was supported by ISSI, NSF GEM 1003854, and NASA NESSF NNX14AO02H. We gratefully acknowledge THEMIS (themis. ssl.berkeley.edu), INTERMAGNET (www.intermagnet.org), GOES, OMNI database (omniweb.gsfc.nasa.gov), SuperMAG (supermag.jhuapl.edu), and their data providers. The authors would also like to thank the NASA NSSDC and NASA VMO centers.

## Supplementary material

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