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WAMI Project : Wavelet Analysis of Magnetosphere and Ionosphere (WAMI)

WAMI is the acronym for Wavelet Analysis of Magnetosphere and Ionosphere, a research project funded by NSF award DMS-0413653 Statistical Wavelet Analysis and Indices Development of the Magnetosphere-Ionosphere Current System Observed by the Terrestrial Magnetometers (mathematical geosciences).Currents flowing in the magnetosphere-ionosphere (M-I) form a complex multiscale system in which a number of individual currents connect and influence each other. The variabilities of these currents are closely connected to various nonlinear dynamic M-I processes, such as magnetic storms and substorms, driven by the solar energy, Fig 1.

Among the various observational means, the global network of ground-based magnetometers stands out with unique strengths of global spacial coverage at all time scales. About a hundred terrestrial geomagnetic observatories form a network, INTERMAGNET, designed to monitor the variations of the M-I current system. Modern digital magnetometers record three components of the magnetic field in five second resolution, but the INTERMAGNET's data we use consist of one minute averages, i.e. 1440 data points per day per component per observatory.

An example of a one day long magnetometer record is shown in Fig. 2. It shows a signature of an event known as a substorm. Substorms occur over polar regions, and are associated with the Auroras.

MagnetosphereFigure 1: Artist's rendition of the impact of the enhanced solar wind on the magnetosphere.
Substorm SignatureFigure 2: Substorm signature recorded at College, Alaska

The main focus of our research has been the development of a cleaner index of storm activity. We have done extensive work on deconvoluting the signature of the symmetric ring current, depicted in Fig. 4, which is the largest storm time feature. This is a difficult task because magnetometer records contain nonlinearly superimposed signatures of many currents. We developed new statistical techniques, based on wavelet and functional data analysis, that have brought us closer to this goal.

AuroraFigure 3: The Aurora Borealis, or Northern Lights, shines above Bear Lake, Eielson Air Force Base, Alaska
Symmetric Ring CurrentFigure 4: Artist's rendition of the symmetric ring current.

WAMI Team

The team currently consists of three space physicists (Sojka, Zhu, Xu) and three statisticians (Gabrys, Kokoszka, Maslova).

Physicists

Jan Sojka is Professor and Department Head in the Department of Physics and a member of the Center for Atmospheric and Space Sciences (CASS), a research unit of the College of Science. He is the president of the Space Science and Aeronomy Section of the American Geophysical Union. In 2005-2007, he chaired the National Science Foundation CEDAR Science Steering Committee. In 2002, he was selected as the Utah Professor of the Year by the Carnegie Foundation for the Advancement of Teaching. His research interest include many areas of ionospheric and magnetospheric physics.

Lie Zhu is Research Associate Professor in the Center for Atmospheric and Space Sciences. He served on NSF and NASA review panels. His research focuses on magnetospheric physics and M-I coupling.

Zhonghua Xu is a PhD student in the Department of Physics. He works on improving the statistical techniques resulting from the WAMI project to obtain precise insights into the structure of selected magnetospheric currents.

Statisticians

Piotr Kokoszka is Professor in the Department of Mathematics and Statistics. He is an Associate Editor of the journal "Statistical Modelling". His research focuses on statistical modeling of nonstandard data sets exhibiting complex multiscale dependence and nonlinearities.

Agnieszka Jach received PhD in Statistics from Utah State University in 2005. She is now Assistant Professor in the Department of Statistics at Universidad Carlos III de Madrid Spain. Her research focuses on applications of wavelet and resampling methods.

Inga Maslowa received PhD in Statistics from Utah State University in 2009, and is now Assistant Professor in the Department of Mathematics at American University, DC. Her research has focused on applications of wavelet and functional data analysis to magnetometer data.

Robertas Gabrys is a PhD candidate in the Department of Mathematics and Statistics. He works on applications of the statistical techniques developed in the course of the WAMI project to data sets arising in climatology and finance.

WAMI Team MembersLeft to right: top: Z. Xu, R. Gabrys, I. Maslova Bottom: P. Kokoszka, J. Sojka, L. Zhu
A. JachA. Jach, now in Madrid.

Publications

Selected publications generated by the WAMI project.

(Statistical methodology and applications to magnetogram data)

A. Aue, R. Gabrys, L. Horvath, P. Kokoszka, Estimation of a change--point in the mean function of functional data Journal of Multivariate Analysis , Forthcoming

I. Berkes, R. Gabrys, L. Horvath P. Kokoszka, Detecting changes in the mean of functional observations, Journal of the Royal Statistical Society , Forthcoming

L. Horvath, M. Huskova, P. Kokoszka, Testing the stability of the functional autoregressive process, Journal of Multivariate Analysis , Forthcoming

I. Maslova, P. Kokoszka, J. Sojka, and L. Zhu, Removal of nonconstant daily variation by means of wavelet and functional data analysis, Journal of Geophysical Research , 114, A03202, doi:10.1029/2008JA013685, 2009

L. Horvath, M. Huskova and P. Kokoszka, Testing the stability of the functional autoregressive process, Journal of Multivariate Analysis, 00 , 000-000, 2008

A. Jach and P. Kokoszka, Robust wavelet domain estimation of the fractional difference parameter in heavy--tailed time series: an empirical study, Methodology and Computing in Applied Probability DOI: 10.1007/s11009-008-9105-3, 2008

A.Jach and P. Kokoszka, Wavelet based confidence intervals for the self-similarity parameter, Journal of Statistical Computation and Simulation, 78, 1179-1198, 2008

Z. Xu, L. Zhu, J. Sojka, P. Kokoszka, A. Jach, An Assessment Study of the Wavelet-Based Index of Magnetic Storm Activity (WISA) and Its Comparison to the Dst Index, Journal of Atmospheric and Solar-Terrestrial Physics DOI: 10.1016/j.jastp.2008.05.007, 2008

P. Kokoszka, I. Maslova, J. Sojka, L. Zhu, Testing for lack of dependence in the functional linear model, Canadian Journal of Statistics, Vol. 36, No. 2, 207-222, 2008

R. Gabrys and P. Kokoszka, Portmanteau test of independence for functional observations Journal of the American Statistical Association, 102 , 1338-1348, 2007

L. Horvath, P. Kokoszka and J. Steinebach, On sequential detection of parameter changes in linear regression, Statistics and Probability Letters, 77 , 885-895, 2007

R. Bhansali, L. Giraitis and P. Kokoszka, Approximations and limit theory for quadratic forms of linear variables Stochastic Processes and their Applications, 117 , 71-95, 2007

P. Kokoszka, I. Maslova, J. Sojka, L. Zhu, Probability tails of wavelet coefficients of magnetometer records. Journal of Geophysical Research, Vol. 111, No. A6, A06202, 10.1029/2005JA011486, 2006

R. Bhansali, L. Giraitis and P. Kokoszka, Estimation of the memory parameter by fitting fractionally differenced autoregressive models Journal of Multivaiate Analysis, 97 Issue 10, 2101-2130, 2006

I. Berkes, L. Horvath, P. Kokoszka, Q. Shao, On discriminating between long-range dependence and changes in mean The Annals of Statistics, 34, 1140-1165, 2006

P. Kokoszka, I. Maslova, J. Sojka, L. Zhu, Probability tails of wavelet coefficients of magnetometer records. Journal of Geophysical Research, Vol. 111, No. A6, A06202, 10.1029/2005JA011486, 2006

A. Jach, P. Kokoszka, J. Sojka, L. Zhu, Wavelet--based index of magnetic storm activity. Journal of Geophysical Research, 111 , A09215, 2006

Software

We present here the code for an R-package which will soon become available at the Comprehensive R Archive Network (CRAN). The code is now available from this site as a zip file Rpackage.zip. Please address any questions to inga.maslova@gmail.com.

The main functions are "SAI" (in the file index.R) and "SQ.new".

The first one computes the storm activity index associated with the ring current. It is an automated procedure that requires the user to input the raw magnetometer data and the coordinates of the stations used. The second major function of this package, "SQ", computes the estimate of the Solar quiet daily variation. Both functions are easy to use. The only requirement is that the data are provided in the matrix form. Below we provide the details on the use of these functions.

First, we describe basic requirements for function "SAI". In order to compute a storm index one must input the raw magnetometer records and the coordinates of the stations used. One can use any number of roughly equispaced equatorial stations.The input data should be provided as a matrix. Each column of this data matrix contains the records for each individual station. The coordinates should be written in form of a matrix as well. Each column corresponds to the station, the first row must contain the latitude and the second row -- longitude. This function returns the global storm index estimate.

In order to estimate the daily nonconstant variation, the "SQ" function can be used. First, the storm index must be calculated using the function "SAI". The resulting index is one of the "SQ.new" inputs. We recommend to use the data from at least two stations to estimate the daily variation. The data should be provided in the matrix form. Each column must contain records from different stations. This functions returns Sq estimated for each station individually.