The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

The International Reference Ionosphere (IRI) is the international standard for the specification of ionospheric densities and temperatures. It was developed and is being improved-updated by a joint working group of the International Union of Radio Science (URSI) and the Committee on Space Research (COSPAR). A new version of IRI is scheduled for release in the year 2000. This paper describes the most important changes compared to the current version of IRI: (1) an improved representation of the electron density in the region from the F peak down to the E peak including a better description of the F1 layer occurrence statistics and a more realistic description of the low-latitude bottomside thickness, (2) inclusion of a model for storm-time conditions, (3) inclusion of an ion drift model, (4) two new options for the electron density in the D region, and (5) an improved model for the topside electron temperatures. The outcome of the most recent IRI Workshops (Kuhlungsborn, 1997, and Nagoya, 1998) will be reviewed, and the status of several ongoing task force activities (e.g., efforts to improve the representation of electron and ion densities in the topside ionosphere and the inclusion of a plasmaspheric extension) will be discussed. A few typical IRI applications will be highlighted in section 6.

/pdf/international-reference-ionosphere-2000-zhazhxmwus.pdf

International Reference Ionosphere 2000

A worldwide network of receivers tracking the transmissions of Global Positioning System (GPS) satellites represents a new source of ionospheric data that is globally distributed and continuously available. We describe a technique for retrieving the global distribution of vertical total electron content (TEC) from GPS-based measurements. The approach is based on interpolating TEC within triangular tiles that tessellate the ionosphere modeled as a thin spherical shell. The high spatial resolution of pixel-based methods, where widely separated regions can be retrieved independently of each other, is combined with the efficient retrieval of gradients characteristic of polynomial fitting. TEC predictions from climatological models are incorporated as simulated data to bridge significant gaps between measurements. Time sequences of global TEC maps are formed by incrementally updating the most recent retrieval with the newest data as it becomes available. This Kalman filtering approach smooths the maps in time, and provides time-resolved covariance information, useful for mapping the formal error of each global TEC retrieval. Preliminary comparisons with independent vertical TEC data, available from the TOPEX dual-frequency altimeter, suggest that the maps can accurately reproduce spatial and temporal ionospheric variations over latitudes ranging from equatorial to about ±65°.

https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/97RS02707

A global mapping technique for GPS‐derived ionospheric total electron content measurements

https://iri.gsfc.nasa.gov/docs/iri_2007_asr.pdf

International Reference Ionosphere 2007: Improvements and new parameters

In this paper, our current understanding and recent advances in the study of ionospheric storms is reviewed, with emphasis on the F2-region. Ionospheric storms represent an extreme form of space weather with important effects on ground- and space-based technological systems. These phenomena are driven by highly variable solar and magnetospheric energy inputs to the Earth's upper atmosphere, which continue to provide a major difficulty for attempts now being made to simulate the detailed storm response of the coupled neutral and ionized upper atmospheric constituents using increasingly sophisticated global first principle physical models. Several major programs for coordinated theoretical and experimental study of these storms are now underway. These are beginning to bear fruit in the form of improved physical understanding and prediction of ionospheric storm effects at high, middle, and low latitude.

Ionospheric Storms — A Review

1. Introduction 2. Space environment 3. Transport equations 4. Collisions 5. Simplfied transport equations 6. Wave phenomena 7. Magnetohydrodynamic formulation 8. Chemical processes 9. Ionization and energy exchange processes 10. Neutral atmospheres 11. The terrestrial ionosphere at middle and low latitudes 12. The terrestrial ionosphere at high latitudes 13. Planetary ionospheres 14. Ionospheric measurement techniques Appendices.

Ionospheres: Physics, Plasma Physics, and Chemistry

The theory and observations relating to electron temperatures in the F region of the ionosphere are reviewed. The review is divided into three basic parts. In the first part the theory concerning electron heating, cooling, and energy transport processes is reviewed, and all the relevant expressions are updated. In the second part the behavior of F region electron temperatures, as measured by satellites, rockets, and incoherent scatter radars, is discussed. This portion covers electron temperature variations with altitude, latitude, local time, season, geomagnetic activity, and solar cycle. The third part is primarily devoted to a discussion of the various attempts to compare measured and calculated F region electron temperatures.

Electron temperatures in the F region of the ionosphere - Theory and observations

In this review we attempt to present a unified picture of transport in multispecies gas mixtures. To accomplish this task, it is necessary to outline the mathematical structure of the transport equations. Starting from Boltzmann's equation, we derive a general system of transport equations using an approach that is valid for flow situations in which there are large temperature and drift velocity differences between the interacting species. However, this system of equations, which is obtained by taking velocity moments of the Boltzmann equation, does not constitute a closed set, since the equation governing the velocity moment of order r contains the velocity moment of order r + 1. To close the system of transport equations, it is therefore necessary to adopt an approximate expression for the species velocity distribution function. For near-equilibrium flows, various levels of approximation are considered, including the 5-, 8-, 10-, 13-, and 20-moment approximations. The procedure for obtaining closed sets of transport equations for far-from-equilibrium flows is also discussed. When the transport equations are ordered with respect to the collisional mean free path, the result is the Euler, Navier-Stokes, or extended Navier-Stokes equations depending upon whether terms proportional to the zeroth, first, or second power of the mean free path are retained. For a collisionless plasma the analogous expansion using the Larmor radius yields the Chew-Goldberger-Low (CGL) equations to zeroth order and the extended CGL equations to first order.

Mathematical structure of transport equations for multispecies flows

We have obtained solutions of the coupled continuity, momentum, and energy equations for NO(+), O(+), and O2(+) ions for conditions appropriate to the daytime high-latitude E and F regions. Owing to the rapid increase of the reaction O(+) + N2 yielding NO(+) + N with ion energy, high-latitude electric fields and consequent perpendicular-E x B drifts deplete O(+) in favor of NO(+). For electric field strengths less than about 10 mV/m the depletion of O(+) is small, and the altitude profiles of ion density are similar to those found at mid-latitudes. However, for moderate electric field strengths (50 mV/m), NO(+) is substantially increased in relation to O(+) and becomes an important ion throughout the F region. For large electric fields (200 mV/m), NO(+) completely dominates the ion composition to at least 600 km, decreasing at high altitudes with a diffusive equilibrium scale height. Since the overall F region electron density decreases markedly with increasing electric field strength, it appears that high-latitude, daytime electron density troughs are directly related to the presence of ionospheric electric fields.

Effect of electric fields on the daytime high-latitude E and F regions

: Our primary goal is to construct a real-time data assimilation model for the ionosphere-plasmasphere system that will provide reliable specifications and forecasts. A secondary goal is to validate the model for a wide range of geophysical conditions, including different solar cycle, seasonal, storm, and substorm conditions.

/pdf/global-assimilation-of-ionospheric-measurements-gaim-nlw138bhrk.pdf

Robert W. Schunk

Papers

Ionospheres: Physics, Plasma Physics, and Chemistry

Electron temperatures in the F region of the ionosphere - Theory and observations

Mathematical structure of transport equations for multispecies flows

Effect of electric fields on the daytime high-latitude E and F regions

Global Assimilation of Ionospheric Measurements (GAIM)