1. Multi-Scale Probabilistic Modeling in Geospace Science Zach Thomas The Ohio State University Mentors: Tomoko Matsuo, Doug Nychka Ellen Cousins, Mike Wiltberger August 1, 2014

2. Outline of Summer Work.,. Application: Modeling high-latitude ionospheric convection from sparse radar observations

3. Outline of Summer Work.,. Application: Modeling high-latitude ionospheric convection from sparse radar observations.,. Statistical Methodology: Technique for spatial modeling on the sphere

4. Scientific Background Motivation.,. Scientific: Obtain better understanding of complex interaction between solar wind and Earth’s magnetic field by studying various ionospheric processes

5. Scientific Background Motivation.,. Scientific: Obtain better understanding of complex interaction between solar wind and Earth’s magnetic field by studying various ionospheric processes.., Numerical modeling

6. Scientific Background Motivation.,. Scientific: Obtain better understanding of complex interaction between solar wind and Earth’s magnetic field by studying various ionospheric processes.., Numerical modeling.., Data analysis

7. Scientific Background Motivation.,. Scientific: Obtain better understanding of complex interaction between solar wind and Earth’s magnetic field by studying various ionospheric processes.., Numerical modeling.., Data analysis.,. Practical/Societal: Understand changes in electromagnetic energy associated with auroras

8. Scientific Background Motivation.,. Scientific: Obtain better understanding of complex interaction between solar wind and Earth’s magnetic field by studying various ionospheric processes.., Numerical modeling.., Data analysis.,. Practical/Societal: Understand changes in electromagnetic energy associated with auroras.., Disturbances in telecommunication

9. Scientific Background Motivation.,. Scientific: Obtain better understanding of complex interaction between solar wind and Earth’s magnetic field by studying various ionospheric processes.., Numerical modeling.., Data analysis.,. Practical/Societal: Understand changes in electromagnetic energy associated with auroras.., Disturbances in telecommunication.., Disturbances in power grids

10. Scientific Background Interaction between Solar Wind and Earth’s Magnetic Field Figure: Output from the Lyon-Fedder-Mobarry model capturing complex interaction between solar wind and Earth’s magnetic field. Image courtesy of https://www.dartmouth.edu/physics/cism/science/lfmmodel.htmlhttps://www.dartmouth.edu/ps/cism/science/lfmmodel.html

11. Scientific Background Ionospheric Electric Potential and Plasma Convection Patterns Figure: IMF effect on ionospheric convection. Image courtesy of Cousins et al. (2010)

12. Modification of LatticeKrig for the Sphere What is LatticeKrig?.,. LatticeKrig is a new R package for fast/flexible spatial modeling based on a statistical methodology in Nychka et al. (2014).

13. Modification of LatticeKrig for the Sphere What is LatticeKrig?.,. LatticeKrig is a new R package for fast/flexible spatial modeling based on a statistical methodology in Nychka et al. (2014)..,. Key idea is to express the spatial process as a mulitresolution basis function expansion with random coefficients

14. Modification of LatticeKrig for the Sphere What is LatticeKrig?.,. LatticeKrig is a new R package for fast/flexible spatial modeling based on a statistical methodology in Nychka et al. (2014)..,. Key idea is to express the spatial process as a mulitresolution basis function expansion with random coefficients.,. Random coefficients modeled via a certain Markov random field model called a simultaneous autoregression (SAR) model

15. Modification of LatticeKrig for the Sphere What is LatticeKrig?.,. Observe a process at n locations within a spatial domain D...call them y ( s 1 ),..., y ( s n ).

16. Modification of LatticeKrig for the Sphere What is LatticeKrig?.,. Observe a process at n locations within a spatial domain D...call them y ( s 1 ),..., y ( s n )..,. For an arbitrary s ∈ D, we would like to make inference about the process y ( s ) from the observations.

17. Modification of LatticeKrig for the Sphere What is LatticeKrig?.,. Observe a process at n locations within a spatial domain D...call them y ( s 1 ),..., y ( s n )..,. For an arbitrary s ∈ D, we would like to make inference about the process y ( s ) from the observations..,. Common spatial model: For any s ∈ D (observed or not): y ( s ) = m ( s ) + g ( s ) + e ( s ) Process = Mean + Spatial Process + Ind. Error

18. Modification of LatticeKrig for the Sphere What is LatticeKrig?.,. Observe a process at n locations within a spatial domain D...call them y ( s 1 ),..., y ( s n )..,. For an arbitrary s ∈ D, we would like to make inference about the process y ( s ) from the observations..,. Common spatial model: For any s ∈ D (observed or not): y ( s ) = m ( s ) + g ( s ) + e ( s ) Process = Mean + Spatial Process + Ind. Error.,. LatticeKrig differs from other methods in its construction of the spatially dependent process g ( s ).

19. Modification of LatticeKrig for the Sphere What is LatticeKrig?.,. Center basis functions at grid locations on L grids over the observation region.

20. Modification of LatticeKrig for the Sphere What is LatticeKrig?.,. Center basis functions at grid locations on L grids over the observation region..,. The L grids are obtained by sequentially doubling the resolution of the previous grid.

21. Modification of LatticeKrig for the Sphere What is LatticeKrig?.,. Center basis functions at grid locations on L grids over the observation region..,. The L grids are obtained by sequentially doubling the resolution of the previous grid..,. The process g ( s ) is then expressed as a basis function expansion: g (s) =g (s) = L n l ) l =1 j =1 c W [ d ( s, s )] l,jl,jl,jl,jl,jl,j ∗

22. Modification of LatticeKrig for the Sphere What is LatticeKrig?.,. Center basis functions at grid locations on L grids over the observation region..,. The L grids are obtained by sequentially doubling the resolution of the previous grid..,. The process g ( s ) is then expressed as a basis function expansion: g (s) =g (s) = L n l ) l =1 j =1 c W [ d ( s, s )] l,jl,jl,jl,jl,jl,j ∗.,. The c l, j ’s are random coefficients following a certain Markov random field model called a Simultaneous Autoregression (SAR) model.

23. Modification of LatticeKrig for the Sphere The Geodesic Grid.,. Modifying LatticeKrig for use on the sphere was done in two steps:

24. Modification of LatticeKrig for the Sphere The Geodesic Grid.,. Modifying LatticeKrig for use on the sphere was done in two steps: 1. Instead of locating basis functions on a rectangular grid over a plane, we center them on a specialized grid on the sphere...called the geodesic grid

25. Modification of LatticeKrig for the Sphere The Geodesic Grid.,. Modifying LatticeKrig for use on the sphere was done in two steps: 1.Instead of locating basis functions on a rectangular grid over a plane, we center them on a specialized grid on the sphere...called the geodesic grid 2.Modify the construction of the SAR model for the random coefficients (which now live on the geodesic grid)

26. Modification of LatticeKrig for the Sphere The Geodesic Grid.,. Modifying LatticeKrig for use on the sphere was done in two steps: 1.Instead of locating basis functions on a rectangular grid over a plane, we center them on a specialized grid on the sphere...called the geodesic grid 2.Modify the construction of the SAR model for the random coefficients (which now live on the geodesic grid).,. Result is a tool for spatial modeling over general regions on the sphere...or the entire sphere...to be included in future versions of LatticeKrig.

27. Modification of LatticeKrig for the Sphere The Geodesic Grid.,. Modifying LatticeKrig for use on the sphere was done in two steps: 1.Instead of locating basis functions on a rectangular grid over a plane, we center them on a specialized grid on the sphere...called the geodesic grid 2.Modify the construction of the SAR model for the random coefficients (which now live on the geodesic grid).,. Result is a tool for spatial modeling over general regions on the sphere...or the entire sphere...to be included in future versions of LatticeKrig..,. This summer, focus on using the procedure for modeling electromagnetic processes in the ionosphere.

28. Figure: Low Resolution Basis Functions: Capturing Large-Scale Dependence

29. Figure: Add Medium Resolution Basis Functions: Capturing Medium-Scale Dependence

30. Figure: Add High Resolution Basis Functions: Capturing Small-Scale Dependence

31. Try Some Spatial Modeling Spatial Interpolation of Electric Potential from LFM-MIX Model Ouput.,. Key modification for ionosphere problem: variance of the electric potential is clearly nonstationary. We can embed information from numerical model output into the statistical model. Figure: (Left) Region of highest variability; courtesy of Minjie Fan, UC Davis (Right) Weights used to induce nonstationary variance in spatial model.

32. Try Some Spatial Modeling Spatial Interpolation of Electric Potential from LFM-MIX Model Ouput.,. Experiment: Use LFM-MIX model output of electric potential to study predictive skill of our model

33. Try Some Spatial Modeling Spatial Interpolation of Electric Potential from LFM-MIX Model Ouput.,. Experiment: Use LFM-MIX model output of electric potential to study predictive skill of our model 1. Randomly select sample of 1000 points (out of 16920) uniformly over the polar region.

34. Try Some Spatial Modeling Spatial Interpolation of Electric Potential from LFM-MIX Model Ouput.,. Experiment: Use LFM-MIX model output of electric potential to study predictive skill of our model 1.Randomly select sample of 1000 points (out of 16920) uniformly over the polar region. 2.Treat these points as ’The Data’...try to get back the full 16920 points using our spatial model on the sphere

35. Figure: Full electric potential process from model output

36. Figure: 1000 randomly sampled locations used to inform the interpolation

37. "True" Electric Potential Process (From LFM-MIX) 12 40 32 24 4040 1616 30° 20° 8 > e- 1818 06 0 -8 -16 -24 -32 -40 min: -25.37 max: 41.50 00

38. Spatial Interpolation of Electric Potential 12 40 32 24 4040 1616 30° 20° 8 c0c0 1818 06 0 B -0 -8 0. -16 -24 -32 -40 min: -25.19 max: 41.55 00

39. 1818 Error ("True" Process Minus Predicted Process) 12 2.0 1.6 1.2 00 10·10· 4040 0.80.8 30° 20° 0.4 "C 06 0.0 -0 0. -0.4 -0.8 -1.2 -1.6 -2.0 min: -2.19 max: 0.76

40. 1818 Error ("True" Process Minus Predicted Process) 12.......·:·.·.......·:·.· ···.···. ·..... ·...... :. ·... :... :. ·... :. :-· ·........,...·..·..,...·..·.,.'·..,.'·..... ·. ··....... ··.:...........,.·....,.·...,....,............ ·.··.·.....· --:. ·....·.... ::.:....::.:.... :-.:-. ·.·................ ·.. ·... :.···· : :.·.......:::.....:::......... :'-.... · 00... ·.·. ·.. ··..·.,..·..·.,..·..·..· ·:30°:..:. '1 :::·:....::.·:"·.,;;.... o I,-,- :. ·.·:. ·.·.:.: ·::.···.·...... :·.:·.. -.. -..,..,.. -.... min: -2.19 max: 0.76.·:.·: -2.0 2.02.0 1.6 1.2 0.80.8 ··:··: 06 0.4 "' 0.0 -0 0. -0.4-0.4 ::..:.: -0.8 -1.2-1.2 -1.6

41. Next Step: Modifications for Radar Observations Getting from Observation Space to Electric Potential Space.,. In practice, the electric potential can only be inferred from sparse observations of other (related) processes.

42. Next Step: Modifications for Radar Observations Getting from Observation Space to Electric Potential Space.,. In practice, the electric potential can only be inferred from sparse observations of other (related) processes..,. We use the methodology in Richmond and Kamide (1988)...transform basis functions into observation space

43. Next Step: Modifications for Radar Observations Getting from Observation Space to Electric Potential Space.,. In practice, the electric potential can only be inferred from sparse observations of other (related) processes..,. We use the methodology in Richmond and Kamide (1988)...transform basis functions into observation space.,. The radars measure projections of ionospheric plasma drift velocities onto the line-sight-direction: v LOS ( s ) = ✈ ( s ) · ❛ LOS.

44. Next Step: Modifications for Radar Observations Getting from Observation Space to Electric Potential Space.,. In practice, the electric potential can only be inferred from sparse observations of other (related) processes..,. We use the methodology in Richmond and Kamide (1988)...transform basis functions into observation space.,. The radars measure projections of ionospheric plasma drift velocities onto the line-sight-direction: v LOS ( s ) = ✈ ( s ) · ❛ LOS..,. These LOS velocites are related to the electric potential by the following: v LOS ( s ) = |❇(s)|∂θ|❇(s)|∂θ · 1 ∂ Φ( s ) ∂ Φ( s ), −, − ∂φ∂φ · ❛ LOS θ =Latitude, φ =Longitude, ❇ ( s )=Magnetic Field at s

45. Next Step: Modifications for Radar Observations Transforming Basis Functions.,. This function L : Φ( s ) 1→ v LOS ( s ) is a linear operator

46. Next Step: Modifications for Radar Observations Transforming Basis Functions.,. This function L : Φ( s ) 1→ v LOS ( s ) is a linear operator.,. Before: Φ( s ) = ) ) c i, j W i, j [ d ( s, s ∗ j )] l =1 j =1 LnlLnl

47. Next Step: Modifications for Radar Observations Transforming Basis Functions.,. This function L : Φ( s ) 1→ v LOS ( s ) is a linear operator.,. Before: Φ( s ) = ) ) c i, j W i, j [ d ( s, s ∗ j )] l =1 j =1.,. For Radar Data: LnlLnl L { Φ( s ) } = v LOS (s) =(s) = LnlLnl ) l =1 j =1 c L W [ d ( s, s )] i,ji,j {�{� i,ji,j ∗ j

48. Next Step: Modifications for Radar Observations Transforming Basis Functions.,. This function L : Φ( s ) 1→ v LOS ( s ) is a linear operator.,. Before: Φ( s ) = ) ) c i, j W i, j [ d ( s, s ∗ j )] l =1 j =1.,. For Radar Data: LnlLnl L { Φ( s ) } = v LOS (s) =(s) = LnlLnl ) l =1 j =1 c L W [ d ( s, s )] i,ji,j {�{� i,ji,j ∗ j.,. The coefficient process is the same...use same estimation procedure but with transformed basis functions

49. Figure: Full electric potential field from LFM-MIX model output.

50. Figure: Randomly sampled LOS directions and corresponding velocities used in numerical study.

51. "True" Electric Potential Process (From LFM-MIX) 12 40 32 24 1616 40 30° 20· 181806 " -8 -16 -24 -32 min: -25.37 max: 41.50 -40 00

52. Spatia l Interpolation of Electric Potential from LOS Velocities 12 40 32 24 1818 00 10'10' min: -24.60 max: 41.72 20· 1616 40 30'. 0 0 B 'C 06 J:: "' -8 -16 -24 -32 -40-40

53. 1818 Error (True Electric Potential Minus Predictions) 12 3.50 2.75 00 10' min:-4.02 max: 3.55 20' 2.00 1.25 4040 30' 0.50 'wC'wC 06 -0.25 £ w -i.ooiE -1.75 -2.50 -3.25 -4.00

54. Some Real-Data Examples.,. Data available from Super Dual Auroral Radar Network (SuperDARN). Network of high-frequency coherent backscatter radars providing measurements of line-of-sight velocity of ionospheric plasma Figure: SuperDARN radar coverage a. Northern Hemisphere b. Southern Hemisphere. Image courtesy of Cousins et al. (2012)

55. Figure: Spatial prediction for high-latitude ionospheric electric potential from sparse radar observations. October 09, 2011, 02:08-02:12.

56. Figure: Comparison with current methods: LatticeKrig on sphere (LEFT) SuperDARN Assimilative Mapping (CENTER) Map-Potential Algorithm (RIGHT)

57. Figure: Spatial prediction for high-latitude ionospheric electric potential from sparse radar observations. October 09, 2011, 07:012-07:16.

58. Figure: Comparison with current methods: LatticeKrig on sphere (LEFT) SuperDARN Assimilative Mapping (CENTER) Map-Potential Algorithm (RIGHT)

59. Conclusions/Future Work.,. We developed an extension of LatticeKrig model for use on the sphere

60. Conclusions/Future Work.,. We developed an extension of LatticeKrig model for use on the sphere.., Need to sort out theoretical properties of induced spatial process

61. Conclusions/Future Work.,. We developed an extension of LatticeKrig model for use on the sphere.., Need to sort out theoretical properties of induced spatial process.., Writing R scripts for inclusion in LatticeKrig package

62. Conclusions/Future Work.,. We developed an extension of LatticeKrig model for use on the sphere.., Need to sort out theoretical properties of induced spatial process.., Writing R scripts for inclusion in LatticeKrig package.,. Initial performance of model for ionospheric electric potential very promising. Much work left to do:

63. Conclusions/Future Work.,. We developed an extension of LatticeKrig model for use on the sphere.., Need to sort out theoretical properties of induced spatial process.., Writing R scripts for inclusion in LatticeKrig package.,. Initial performance of model for ionospheric electric potential very promising. Much work left to do:.., Underlying distributions are not gaussian

64. Conclusions/Future Work.,. We developed an extension of LatticeKrig model for use on the sphere.., Need to sort out theoretical properties of induced spatial process.., Writing R scripts for inclusion in LatticeKrig package.,. Initial performance of model for ionospheric electric potential very promising. Much work left to do:.., Underlying distributions are not gaussian.., Refinements of induced covariance (e.g. allow for stronger latitudinal dependence)

65. Conclusions/Future Work.,. We developed an extension of LatticeKrig model for use on the sphere.., Need to sort out theoretical properties of induced spatial process.., Writing R scripts for inclusion in LatticeKrig package.,. Initial performance of model for ionospheric electric potential very promising. Much work left to do:.., Underlying distributions are not gaussian.., Refinements of induced covariance (e.g. allow for stronger latitudinal dependence).., Later: Bayesian hierarchical model for combining observations of related processes...improve coverage/identifiability

66. THANKS!