SEARCH

SEARCH BY CITATION

Keywords:

  • data assimilation;
  • electrodynamics;
  • high-latitude ionosphere;
  • plasma dynamics;
  • space weather

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Brief Model Description
  5. 3. Defining the Model Drivers
  6. 4. Data Ingestions and Model Outputs
  7. 5. Ionospheric Space Weather Variability Driven by the Optimal Model Drivers With Various Real Data
  8. 6. Validation and Comparison
  9. 7. Summary and Discussion
  10. Acknowledgments
  11. References

[1] The high-latitude ionosphere is a dynamic region in the solar-terrestrial system. The disturbances in this region can adversely affect numerous military and civilian systems, and the accurate specification and forecast of its plasma and electrodynamic structures are important for space weather research. Presently, most of the space weather models use limited observations and/or indices to define a set of empirical drivers for physical models forward in time. The empirical drivers have a “climatological” nature, and there are significant physical inconsistencies among various empirical drivers. Therefore, the specifications of high-latitude environment from these space weather models cannot truthfully reflect the weather features. Utah State University (USU) has developed a data assimilation model for the high-latitude ionosphere plasma dynamics and electrodynamics to overcome these hurdles. With a set of physical models and an ensemble Kalman filter, the model can define the drivers that are most truthful to the real space environment by ingesting data from multiple observations. In this paper, we will provide the details on how the model drivers truthful to real space weather are defined in the developed USU data assimilation model and show the space weather variability of the model outputs driven by these model drivers for various seasonal and geomagnetic conditions. Also, we will present preliminary results of validation and comparison studies to demonstrate that the model results with the optimal magnetospheric drivers determined by data assimilation are the better representations of real space environment.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Brief Model Description
  5. 3. Defining the Model Drivers
  6. 4. Data Ingestions and Model Outputs
  7. 5. Ionospheric Space Weather Variability Driven by the Optimal Model Drivers With Various Real Data
  8. 6. Validation and Comparison
  9. 7. Summary and Discussion
  10. Acknowledgments
  11. References

[2] Different from the low latitudes and midlatitudes, the high-latitude ionosphere and the electrodynamics in the region are directly driven by the magnetospheric convection, precipitation, and the field-aligned currents. So, to specify and forecast the weather conditions in this region, the magnetospheric drivers with realistic weather features are of fundamental importance.

[3] Presently, in most of the space weather models for the high-latitude ionosphere, there are basically two schools of magnetospheric drivers [Anderson et al., 1998]. One is to use empirical convection and precipitation models [e.g., Weimer, 1996; Hardy et al., 1985, 1989] to drive the ionosphere [e.g., Fuller-Rowell et al., 1994]. Since these empirical models are climatological models, the outputs of these ionospheric models using empirical magnetospheric drivers cannot accurately specify and forecast the weather features in the region. Furthermore, these empirical models were developed by using different observational data, different numerical fittings, and were even defined by different indices and physical parameters. So, simply adding them together into an ionospheric model as drivers can lead to severe physical inconsistencies in the model, which can cause the developments of unrealistic plasma and electrodynamic structures in the high-latitude ionosphere. The other school of the ionospheric models uses the outputs of global magnetospheric models as drivers [e.g.,Slinker et al., 1998]. As we know, global magnetospheric models need the information of the solar wind as inputs. The solar wind data are extremely sparse in space and the satellite measurements at just a few locations cannot catch most of the plasma and electrodynamic structures in the interplanetary space. This leads to the lack of weather features in the global magnetospheric models and in the resulted drivers for the high-latitude ionosphere.

[4] In contrast to the sparse measurements in the magnetosphere and solar wind, there are many more ground-based and in situ measurements for the ionosphere. Our physics-based data assimilation model for the high-latitude ionosphere takes the advantage of this and uses the ionospheric measurements and an ensemble Kalman filter to define the magnetospheric drivers with weather features, thus leading to more accurate space weather specifications and forecastings for the high-latitude ionosphere.

2. Brief Model Description

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Brief Model Description
  5. 3. Defining the Model Drivers
  6. 4. Data Ingestions and Model Outputs
  7. 5. Ionospheric Space Weather Variability Driven by the Optimal Model Drivers With Various Real Data
  8. 6. Validation and Comparison
  9. 7. Summary and Discussion
  10. Acknowledgments
  11. References

[5] There are three major components in our physics-based data assimilation model for the high-latitude ionosphere, which are a coupled first principle physical model for the high-latitude ionosphere plasma dynamics and electrodynamics, an ensemble Kalman filter, and the measurements and data ingestions. The details of these model components are described bySchunk et al. [2006]. In this paper, after a brief overall model description, we will focus on how the model drivers with space weather features are defined via a data assimilation technique, the space weather variability of the model outputs driven by these model drivers for various seasonal and geomagnetic conditions, and preliminary results of validation and comparison studies.

[6] The coupled first principle model for the ionosphere dynamics and electrodynamics is a 3-D physics-based time-dependent model that can account for rapid time variations (∼min) and small spatial scales (horizontal, ∼10 s km, vertical, 4∼10 s km). The model consists of three parts, a global ionosphere model [Schunk, 1988; Sojka, 1989], an M-I (magnetosphere-ionosphere) electrodynamics model [Zhu et al., 1993, 2000, 2005], and a 3-D magnetic inversion model [Zhu et al., 1999]. The global ionosphere model is a three-dimensional, high-resolution, multi-ion model of the global ionosphere that covers the altitude range from 80 to 1500 km. The model is based on a numerical solution of the continuity, momentum, and energy equations. The equations are solved along B for individual convecting flux tubes of plasma, and the 3-D nature of the model is obtained by following a large number of plasma flux tubes. The ionosphere model takes into account the field-aligned diffusion, cross-field electrodynamic drifts, thermospheric winds, neutral composition changes, energy-dependent chemical reactions, ion production due to solar UV/EUV radiation and auroral precipitation, thermal conduction, diffusion thermal heat flow, ion temperature anisotropies, and a myriad of local heating and cooling processes. The M-I electrodynamics model is based on a numerical solution of the MHD transport equations and Ohm's law, with the height-integrated Hall and Pedersen conductivities. A unique feature of this model is an Alfven wave approach for the M-I electrodynamic coupling, which allows the ionosphere to play an active role in the coupled M-I system, thus providing a fully self-consistent and more accurate description of the high-latitude electrodynamics. The magnetic inversion model takes any current system in the ionosphere (the field-aligned current, Hall and Pedersen currents) as an input and calculates the magnetic disturbances in space and on the ground caused by the current system. The current system can be represented in either geographic or geomagnetic coordinates, it can be global or local, and it can be located at high, middle or low latitudes. The model includes the Earth's induction effect and uses spherical coordinates. Therefore, the inferred magnetic disturbances are realistic.

[7] Figures 1 and 2 in the following show the sample results from the coupled physical model.

image

Figure 1. The field-aligned current distributions for (top left) quiet time and (top right) substorm time, (bottom left) the Hall current contribution to the field-aligned currents, and (bottom right) the magnetic disturbances on the ground.

Download figure to PowerPoint

image

Figure 2. HmF2 for (top left) quiet time and (top right) substorm time and the electron density at 800 km for (bottom left) quiet time and (bottom right) substorm time.

Download figure to PowerPoint

[8] For the field-aligned current plots inFigure 1, the negative values mean the upward field-aligned currents. It can be seen that during the substorm time, in addition to the significant enhancement of both the large-scale region 1 and region 2 currents, there is a clear substorm wedge and many small-scale current structures appear in the region. These small-scale structures have fundamental importance for the specification of the high-latitude ionosphere and the development of these structures in the model is due to the adoption of an Alfven wave M-I coupling approach in the coupled model in which the active role of the ionosphere is included. It is conventionally believed that the region 1 and region 2 currents are converted to the Pedersen current. The model results inFigure 1indicate that during substorm times, due to the development of small-scale current structures, the Hall current can make significant contribution to the field-aligned currents, especially in the region of substorm current wedge.

[9] The model results in Figure 2 show that during substorms (Figure 2, right), the HmF2 in the auroral oval decreases significantly due a lowering of the ionosphere. Correspondingly, we can see a significant decrease of electron density at 800 km in the auroral oval during substorms.

[10] The second component of the data assimilation model is an ensemble Kalman filter [Evensen, 2009]. The Kalman filter is a well-documented recursive algorithm that minimizes the error and finds the best estimate of the state at a time t based on all information prior to this time [Daley, 1991]. For the data assimilation of the high-latitude ionosphere, directly using the full Kalman filter poses a number of difficulties. First of all, due to the complexity of the electrodynamics and plasma dynamics in the high-latitude ionosphere, the propagation of the model error covariance matrix puts enormous computational requirements on both data storage and CPU time. In addition, the required linearization in the full Kalman filter may lead to significant additional source of error due to the complexity of governing physical equations. Therefore, we use an ensemble Kalman filter in which an approximation of the state error covariance matrix is adopted [Evensen, 2009]. This approximation leads to a dramatic reduction in the computational requirements and also eliminates the linearization of the model, thus the full nonlinear physical model can be used. The ensemble Kalman filter is basically a sequential data assimilation method that uses a Monte Carlo or ensemble integration [Evensen, 2003]. The details of the mathematical descriptions of these two Kalman filters and the difficulties associated with the full Kalman filter are further discussed by Schunk et al. [2006]. The third component of the data assimilation model is the measurements and data ingestions, which will be described in a following section.

3. Defining the Model Drivers

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Brief Model Description
  5. 3. Defining the Model Drivers
  6. 4. Data Ingestions and Model Outputs
  7. 5. Ionospheric Space Weather Variability Driven by the Optimal Model Drivers With Various Real Data
  8. 6. Validation and Comparison
  9. 7. Summary and Discussion
  10. Acknowledgments
  11. References

[11] Our data assimilation model uses an ensemble Kalman filter and the ionospheric measurements to define the magnetospheric drivers. Figure 3 is a diagram showing the details of how the model drivers are defined.

image

Figure 3. A diagram showing how the magnetospheric drivers are defined.

Download figure to PowerPoint

[12] In the data assimilation model, the optimal model drivers can be defined via ensemble runs of physical models and the Kalman filter, in which the observational data are assimilated. Specifically, we use the ACE (Advanced Composition Explorer) [e.g., Stone et al., 1998] data and empirical models to define the magnetospheric drivers on the 0th order. Then, we add statistical variability to the 0th-order magnetospheric drivers to produce an ensemble of initial drivers. The exact number of ensemble members that are needed for high-latitude ionosphere data assimilation is not well defined. What we use is a trial-and-error approach in which various numbers of ensemble members are used to see how the convergence of the results goes. We find that 20–30 ensemble members can be an optimal number. We then use these initial drivers with statistical variability to make 20–30 physical model runs in parallel. We call these parallel physical model runs as “slave runs.” It is important to note here that the ensemble Kalman filter is naturally parallel and these individual physical model runs can be performed on separate computer processors to significantly reduce the computational time.

[13] The outputs from these “slave runs” form an ensemble of new states at a later time. Based on the variability and correlations of these new states, an error covariance matrix is constructed that is used to calculate the Kalman gain. Then the ionospheric measurements, and the Kalman gain are combined to define the optimal magnetospheric drivers. Finally, we use these optimal magnetospheric drivers to make a “master run” to specify the ionospheric status for next time step.

[14] At the new time step, the 0th-order changes of the magnetospheric drivers are the differences between the empirical drivers determined by the ACE data at new time step and the optimal drivers for the previous time step. We apply statistical variability to these 0th-order changes of the magnetospheric drivers and then add them to the optimal drivers for the previous time step to form an ensemble of initial drivers for the new time step. These new initial drivers are then used to drive physical model forward in time for 20–30 “slave runs.” While this process continues in time, we can define a time sequence of optimal magnetospheric drivers that are constrained by ionospheric measurements and are most truthful to real space weather.

[15] One important thing needs to be noted here. The optimal drivers for the “master run” are different from any of the 20–30 sets of initial drivers for the “slave runs.” They include the Kalman gain that combines the information from all the “slave runs” and the ionospheric measurements. As a result, the optimal drivers have the realistic weather features that are consistent to the ionospheric measurements.

4. Data Ingestions and Model Outputs

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Brief Model Description
  5. 3. Defining the Model Drivers
  6. 4. Data Ingestions and Model Outputs
  7. 5. Ionospheric Space Weather Variability Driven by the Optimal Model Drivers With Various Real Data
  8. 6. Validation and Comparison
  9. 7. Summary and Discussion
  10. Acknowledgments
  11. References

[16] In the data assimilation model, we use the solar wind and IMF (interplanetary magnetic field) measurements from the ACE satellite to define the initial 0th-order magnetospheric drivers. We have archived the ACE data from 1997 to the present for historical study and are continuously collecting the real-time ACE data for the real-time space weather study. For data assimilation, presently the model is able to assimilate the following four sets of data: (1) magnetic field measurements from ground stations (specifically, we have been using the ground-based magnetometer data from the INTERMAGNET (International Real-Time Magnetic Observatory Network) [e.g.,Kerridge, 2001] and the World Data Center), (2) DMSP cross-track ion drift measurements [e.g.,Drayton et al., 2005], (3) SuperDARN line-of-sight velocity [e.g.,Greenwald et al., 1995], and (4) IRIDIUM in situ magnetometer measurements [e.g., Anderson et al., 2000, 2002].

[17] These four types of data can be assimilated into the model individually or in any combination. With minimal modifications, the model is capable of assimilating other types of observational data relevant to the high-latitude ionosphere.

[18] By assimilating the ionospheric data from multiple observational resources, a full set of self-consistent distributions of electrodynamics and plasma parameters that more accurately represent the space weather of the high-latitude ionosphere can be created continuously as a function of time from the USU data assimilation model. The model outputs include the 2-D distributions of the electric potential and convection electric field, energy flux and average energy of precipitation, field-aligned and horizontal currents, Hall and Pedersen conductances, and Joule heating rate with a horizontal spatial resolution of 10 s km. The outputs also include the 3-D distributions of the density and temperature for electrons and ions with a vertical spatial resolution of 4–10 s km for the regions from 80 to 1500 km. The data assimilation model can also provide the time-dependent distributions of a number of conventional ionospheric parameters, including TEC, NmF2, HmF2, NmE, and HmE. Another set of the useful outputs from the model are the magnetic disturbances on the ground and in space, which are calculated by using the high-latitude time-dependent current system (the field-aligned and horizontal currents as well as the underground induced currents) in the model, and these magnetic disturbance outputs are useful for model-observation comparison and validation studies.

5. Ionospheric Space Weather Variability Driven by the Optimal Model Drivers With Various Real Data

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Brief Model Description
  5. 3. Defining the Model Drivers
  6. 4. Data Ingestions and Model Outputs
  7. 5. Ionospheric Space Weather Variability Driven by the Optimal Model Drivers With Various Real Data
  8. 6. Validation and Comparison
  9. 7. Summary and Discussion
  10. Acknowledgments
  11. References

[19] We have assimilated four types of data into our high-latitude ionosphere model for various geomagnetic and seasonal conditions, including DMSP plasma drift velocities (4 satellites), ground magnetometer data (40 stations), IRIDIUM cross-track magnetic measurements (66 satellites), SuperDARN line-of-sight velocities (9 radars in the northern hemisphere). These measurements can be assimilated individually into the model, or simultaneously with any combination.

5.1. Geomagnetic Variability

[20] Figures 4 and 5show the distributions of the electrodynamic parameters for the high-latitude ionosphere for quiet time and storm time respectively. In these two runs, the ground-based magnetometer data were assimilated into the model. The Kp values for the quiet period were around 1 and those for the storm period were around 6.

image

Figure 4. The distributions of the (top left) Pedersen and (top right) Hall conductance, (bottom left) cross polar cap potential, and (bottom right) energy flux during quiet time.

Download figure to PowerPoint

image

Figure 5. The distributions of the (top left) Pedersen and (top right) Hall conductance, (bottom left) cross polar cap potential, and (bottom right) energy flux during storm time.

Download figure to PowerPoint

[21] It can be seen from the figures that during quiet time, the polar cap potential drop was only about 30 Kv and it increased to about 90 Kv during the storm time. The maximum energy flux in the auroral oval was only about 1.6 erg/cm2 for the quiet time and it was about 8 erg/cm2 for the storm time. Also, during the storm time, a strong substorm current wedge developed in the poleward region of the auroral oval. During the quiet time, in the auroral oval, the precipitation was soft and the average energy of the precipitation was less than l KeV (not shown here). This feature of soft precipitation during quiet time is reflected in Figure 4 where the Hall conductance was slightly smaller than the Pedersen conductance. But during the storm time, the situation changed dramatically. The Hall conductance became much larger than the Pedersen conductance due to hard precipitations as shown in Figure 5and the average energy of the precipitations increased to about 4 KeV (not shown here). These are important space weather variations in the high-latitude ionosphere derived from the physics-based data assimilation model with the real observational data as inputs.

5.2. Seasonal Variability

[22] Figures 6 and 7show the distributions of the electrodynamic parameters for the high-latitude ionosphere for winter and equinox respectively. For the periods of these two runs, the Kp had medium values with a range from 3 to 4, and there were no substorm activities. Cross polar cap potentials were about 50 Kv. One interesting space weather feature shown in the plots is that when Kp reached 3 or 4, the Hall conductance started to be larger than the Pedersen conductance in the auroral oval. The seasonal effect is very clear in the plots of these two figures. During the equinox, the dayside auroral oval merged into the daylight region, but the enhancement due to the auroral precipitation was still visible.

image

Figure 6. The distributions of the (top left) Pedersen and (top right) Hall conductance, (bottom left) cross polar cap potential, and (bottom right) energy flux for winter condition.

Download figure to PowerPoint

image

Figure 7. The distributions of the (top left) Pedersen and (top right) Hall conductance, (bottom left) cross polar cap potential, and (bottom right) energy for equinox condition.

Download figure to PowerPoint

[23] For each of the above cases, we ran the model continuously for multiple days. The model results show the clear UT variability of the space weather of the high-latitude ionosphere, which is not discussed in this paper.

6. Validation and Comparison

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Brief Model Description
  5. 3. Defining the Model Drivers
  6. 4. Data Ingestions and Model Outputs
  7. 5. Ionospheric Space Weather Variability Driven by the Optimal Model Drivers With Various Real Data
  8. 6. Validation and Comparison
  9. 7. Summary and Discussion
  10. Acknowledgments
  11. References

[24] A number of validation and comparison study runs have been conducted to seek the exploratory answers for the following two questions: (1) With the ionospheric measurements as constrains, can the data assimilation procedure adopted in this model really produce the magnetospheric drivers that are realistic and most truthful to the real space weather in the high-latitude ionosphere? (2) How much do the optimal drivers determined by data assimilation improve the model results?

[25] One way to answer the first question is to validate the results of the data assimilation model with independent ionospheric measurements. In the following, we show one example in which the ground magnetometer data, DMSP plasma drift velocity, IRIDIUM cross-track magnetic measurements were assimilated in the model simultaneously. Then we used the SuperDARN line-of-sight velocity data to validate the model results.

[26] In this validation, the SuperDARN data were totally independent of the ingested ground magnetometer data and the DMSP and IRIDIUM measurements. It can be seen from the plots that the model-produced drift velocity pattern is in overall consistent to the observed SuperDARN drift velocity pattern. This consistency between the two persists in the entire multiday run, which indicates that the optimal drivers determined by data assimilation are realistic and largely truthful to the real space weather during the period.

[27] To answer the second question, one example of comparison studies is shown in the following. In this case, we made a multiday run for the same period as that shown in Figure 8. But instead of treating the SuperDARN line-of-sight drift velocity as independent data for validation, we ingested them into the model in this case. Other ingested data were the ground magnetometer measurements from about 40 stations. We then compare the model results driven by the empirical drivers that used ACE data as inputs, the models results driven by the optimal drivers determined by data assimilation, and the DMSP measurements. The comparison results are shown inFigure 9.

image

Figure 8. (top right) Observed SuperDARN drift velocity and (bottom left) those determined from the Kalman filter run.

Download figure to PowerPoint

image

Figure 9. (left) Potential pattern obtained from the data assimilation model and (right) the comparison among the cross-track drift velocities from the DMSP satellite (black line), the model results with optimal drivers (red line), and the model results with empirical drivers (green line).

Download figure to PowerPoint

[28] Figure 9(left) shows the high-latitude potential pattern obtained by our data assimilation model for day 93, 2000 at 0740 UT. The ground track of the DMSP F13 overflights and the locations of the ground magnetometers are superposed on the plot.Figure 9(right) shows a comparison of the DMSP cross-track drift velocity to those obtained with the optimal drivers determined by data assimilation and with the empirical drivers using ACE data as inputs. During this period, we only have the data from DMSP F13, so there are three blank panels inFigure 9.

[29] It can be seen from Figure 9(right) that the modeled cross-track drift velocity with the empirical drivers deviates significantly from the DMSP measurements. In contrast, the results with the optimal drivers determined by data assimilation are in overall consistent with the DMSP cross-track drift velocity measurements, except small-scale fluctuations. Since both model runs used the same physical models, the results shown inFigure 9 indicate that the optimal drivers determined by data assimilation can improve model results significantly.

[30] To have a systematic quantitative assessment on how the ensemble Kalman filter and ionospheric measurements help define the optimal model drivers, large number of model runs with different combinations of ingested observational data for various solar, geomagnetic, and seasonal conditions need to be conducted and the results from these runs then need to be compared to various independent observations.

7. Summary and Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Brief Model Description
  5. 3. Defining the Model Drivers
  6. 4. Data Ingestions and Model Outputs
  7. 5. Ionospheric Space Weather Variability Driven by the Optimal Model Drivers With Various Real Data
  8. 6. Validation and Comparison
  9. 7. Summary and Discussion
  10. Acknowledgments
  11. References

[31] As was found in meteorology and oceanography, the most accurate specification and forecast models are those that assimilate real-time (or near-real-time) measurements into a physics-based numerical model [Atlas, 1997]. For the specification and forecast of the space weather, the same philosophy applies. Specifically in this work, we use the data assimilation technique of ensemble Kalman filter and the measurements from multiple observations to define model drivers and produce more accurate representations of the space weather for the high-latitude ionosphere. This approach overcomes the problems of lacking weather features in most of the ionosphere space weather models, which are either due to the use of the empirical model drivers or due to the sparse measurements in the solar wind and interplanetary space.

[32] With the optimal model drivers determined by an ensemble Kalman filter and real ionospheric measurements, the results of our data assimilation model display the space weather variability for various geomagnetic and seasonal conditions. Our results show that the optimal model drivers from data assimilation are more realistic and more truthful to the real space environment than the empirical model drivers. As a result, the specifications of the high-latitude ionosphere from a data assimilation model are more accurate and more consistent to the real space weather observations.

[33] Our data assimilation model for the high-latitude ionosphere plasma dynamics and electrodynamics has several unique strengths. One of the unique strengths is its high spatial (∼10 s km) and temporal (∼min) resolutions. These high resolutions are very crucial for describing the plasma dynamics and electrodynamics in the high-latitude ionosphere, which has quite different spatial and temporal scales from those in the low-latitude and midlatitude ionosphere. Also these higher resolutions could better meet the needs of space weather users. In addition to the high resolutions, the model has several other unique strengths. First of all, the active role of the ionosphere is taken into account in the physical models and the small-scale structures are included. Second, the model drivers are defined by data assimilation of various ionospheric measurements, thus having more accurate and realistic weather features. Third, the model outputs are a full set of self-consistent 2-D/3-D electrodynamic and plasma parameters for the high-latitude ionosphere and most of these parameters can be directly measured by the in situ and ground observations. Last, the model can be run autonomously, thus having a strong potential to be used in real-time space weather specifications and forecastings.

[34] With the preliminary model results, their comparisons to ionospheric measurements, and the unique model strengths, the USU physics-based data assimilation model for the high-latitude ionosphere could have significant contribution to the space weather community in future.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Brief Model Description
  5. 3. Defining the Model Drivers
  6. 4. Data Ingestions and Model Outputs
  7. 5. Ionospheric Space Weather Variability Driven by the Optimal Model Drivers With Various Real Data
  8. 6. Validation and Comparison
  9. 7. Summary and Discussion
  10. Acknowledgments
  11. References