Fusing point and areal level space-time data. data with application to wet deposition

Transcription

1 Fusing point and areal level space-time data with application to wet deposition Alan Gelfand Duke University Joint work with Sujit Sahu and David Holland

2 Chemical Deposition Combustion of fossil fuel produces various chemicals including sulfate and nitrate gases. In the eastern U.S., most SO 2, and NO x release attributed to power plants. Emitted to the air; wet deposition and dry deposition; interest in total deposition. Deposition means return to the earth s surface by means of precipitation (rain or snow) for example. Wet Deposition = Precipitation Concentration. Wet deposition is responsible for damage to lakes, forests, and streams.

3 Chemical Deposition Combustion of fossil fuel produces various chemicals including sulfate and nitrate gases. In the eastern U.S., most SO 2, and NO x release attributed to power plants. Emitted to the air; wet deposition and dry deposition; interest in total deposition. Deposition means return to the earth s surface by means of precipitation (rain or snow) for example. Wet Deposition = Precipitation Concentration. Wet deposition is responsible for damage to lakes, forests, and streams.

4 Chemical Deposition Combustion of fossil fuel produces various chemicals including sulfate and nitrate gases. In the eastern U.S., most SO 2, and NO x release attributed to power plants. Emitted to the air; wet deposition and dry deposition; interest in total deposition. Deposition means return to the earth s surface by means of precipitation (rain or snow) for example. Wet Deposition = Precipitation Concentration. Wet deposition is responsible for damage to lakes, forests, and streams.

5 Chemical Deposition Combustion of fossil fuel produces various chemicals including sulfate and nitrate gases. In the eastern U.S., most SO 2, and NO x release attributed to power plants. Emitted to the air; wet deposition and dry deposition; interest in total deposition. Deposition means return to the earth s surface by means of precipitation (rain or snow) for example. Wet Deposition = Precipitation Concentration. Wet deposition is responsible for damage to lakes, forests, and streams.

6 Chemical Deposition Combustion of fossil fuel produces various chemicals including sulfate and nitrate gases. In the eastern U.S., most SO 2, and NO x release attributed to power plants. Emitted to the air; wet deposition and dry deposition; interest in total deposition. Deposition means return to the earth s surface by means of precipitation (rain or snow) for example. Wet Deposition = Precipitation Concentration. Wet deposition is responsible for damage to lakes, forests, and streams.

7 Chemical Deposition Combustion of fossil fuel produces various chemicals including sulfate and nitrate gases. In the eastern U.S., most SO 2, and NO x release attributed to power plants. Emitted to the air; wet deposition and dry deposition; interest in total deposition. Deposition means return to the earth s surface by means of precipitation (rain or snow) for example. Wet Deposition = Precipitation Concentration. Wet deposition is responsible for damage to lakes, forests, and streams.

8 The first stage likelihood f (P, Z, Q U, Y, X, V, Ṽ) = f (P U, V) f (Z Y, V) f (Q X, Ṽ) which takes the form [ { } T n t=1 i=1 1 exp u(si, t) 1 exp y(si, t) I (v(s i, t) > 0) { }] 1 exp x(aj, t) I (ṽ(a j, t) > 0) J j=1 where 1 x denotes a degenerate distribution with point mass at x and I( ) is the indicator function.

9 Deposition Model Y (s i, t) = β 0 + β 1 U(s i, t) + β 2 V (s i, t) + (b 0 + b(s i )) X(A ki, t) +η(s i, t) + ɛ(s i, t). Spatially varying coefficients, b = (b(s 1 ),..., b(s n )) is a Gaussian process (GP). Spatio-temporal intercept η t = (η(s 1, t),..., η(s n, t)) is a GP independent in time. Allow for spatially varying calibration of CMAQ. Could imagine common η(s i ). ɛ(s i, t) N(0, σ 2 ɛ ), provides the nugget effect.

10 The second stage models... Precipitation U(s i, t) = α 0 + α 1 V (s i, t) + δ(s i, t), where δ t = (δ(s 1, t),..., δ(s n, t)) is a GP independent in time. CMAQ output X(A j, t) = γ 0 + γ 1 Ṽ (A j, t) + ψ(a j, t), j = 1,..., J. Assume ψ(a j, t) N(0, σψ 2 ), independently.

11 Specification of latent processes Measurement Error Model (MEM) V (s i, t) N(Ṽ (A k i, t), σv 2 ), i = 1,..., n, t = 1,..., T. The process Ṽ (A j, t) is AR in time and CAR in space Ṽ (A j, t) = ρṽ (A j, t 1) + ζ(a j, t), ( J ) ζ(a j, t) N h ji ζ(a i, t), σ2 ζ, m j i=1 Let j define the m j neighboring grid cells of the cell A j. h ji = { 1 m j if i j 0 otherwise.

12 AR in time and CAR in Space Assume the initial condition for Ṽ0: Ṽ (A j, 0) = 1 T T X(A j, t), t=1 giving Ṽ0. Now we can write the CAR in closed form: f (Ṽt Ṽt 1, ρ, σ 2 ζ ) { exp 1 (Ṽt ρṽt 1) D 1 (I H) (Ṽt ρṽt 1) }, 2 D is a diagonal matrix with entries σ 2 ζ /m j. Note that this is an improper CAR.

13 National Atmospheric Deposition Program (NADP) NADP collects point-referenced data at several sites. They then use simple interpolation to produce maps

14 The CMAQ model Community Multi-scale Air Quality Model (CMAQ) A computer simulation model which produces averaged output on 36km, 12 km(used here), and now 4 km grid cells Uses variables such as power station emission volumes, meteorological data, land-use, etc. with atmospheric science (appropriate differential equations) to predict deposition levels. Not driven by monitoring station data. Predictions are biased but no missing data; monitoring data provide more accurate deposition but missingness

15 The CMAQ model Community Multi-scale Air Quality Model (CMAQ) A computer simulation model which produces averaged output on 36km, 12 km(used here), and now 4 km grid cells Uses variables such as power station emission volumes, meteorological data, land-use, etc. with atmospheric science (appropriate differential equations) to predict deposition levels. Not driven by monitoring station data. Predictions are biased but no missing data; monitoring data provide more accurate deposition but missingness

16 The CMAQ model Community Multi-scale Air Quality Model (CMAQ) A computer simulation model which produces averaged output on 36km, 12 km(used here), and now 4 km grid cells Uses variables such as power station emission volumes, meteorological data, land-use, etc. with atmospheric science (appropriate differential equations) to predict deposition levels. Not driven by monitoring station data. Predictions are biased but no missing data; monitoring data provide more accurate deposition but missingness

17 The CMAQ model Community Multi-scale Air Quality Model (CMAQ) A computer simulation model which produces averaged output on 36km, 12 km(used here), and now 4 km grid cells Uses variables such as power station emission volumes, meteorological data, land-use, etc. with atmospheric science (appropriate differential equations) to predict deposition levels. Not driven by monitoring station data. Predictions are biased but no missing data; monitoring data provide more accurate deposition but missingness

18 Our Contribution A fully model-based framework for fusing the NADP and CMAQ wet deposition data To accommodate the point masses at 0, i.e., no wet deposition if no precipitation To accommodate misalignment between NADP data at points and CMAQ data at grid cells in a computational feasible way across space and time To provide spatial interpolation and temporal aggregation Both sulfate and nitrate deposition

19 Our Contribution A fully model-based framework for fusing the NADP and CMAQ wet deposition data To accommodate the point masses at 0, i.e., no wet deposition if no precipitation To accommodate misalignment between NADP data at points and CMAQ data at grid cells in a computational feasible way across space and time To provide spatial interpolation and temporal aggregation Both sulfate and nitrate deposition

20 Our Contribution A fully model-based framework for fusing the NADP and CMAQ wet deposition data To accommodate the point masses at 0, i.e., no wet deposition if no precipitation To accommodate misalignment between NADP data at points and CMAQ data at grid cells in a computational feasible way across space and time To provide spatial interpolation and temporal aggregation Both sulfate and nitrate deposition

21 Our Contribution A fully model-based framework for fusing the NADP and CMAQ wet deposition data To accommodate the point masses at 0, i.e., no wet deposition if no precipitation To accommodate misalignment between NADP data at points and CMAQ data at grid cells in a computational feasible way across space and time To provide spatial interpolation and temporal aggregation Both sulfate and nitrate deposition

22 Our Contribution A fully model-based framework for fusing the NADP and CMAQ wet deposition data To accommodate the point masses at 0, i.e., no wet deposition if no precipitation To accommodate misalignment between NADP data at points and CMAQ data at grid cells in a computational feasible way across space and time To provide spatial interpolation and temporal aggregation Both sulfate and nitrate deposition

23 Inverse Distance weighting (IDW) Poor person s methodology Value at a new site = weighted mean of observations, Weights inversely proportional to the square of the distance. Problems Not model based! Unable to accommodate known covariate - precipitation! Unable to handle 0 s, unable to handle missing data Can t fuse with model output data With dynamic data, can only do independent weekly or aggregated annually No associated uncertainty maps!

24 Inverse Distance weighting (IDW) Poor person s methodology Value at a new site = weighted mean of observations, Weights inversely proportional to the square of the distance. Problems Not model based! Unable to accommodate known covariate - precipitation! Unable to handle 0 s, unable to handle missing data Can t fuse with model output data With dynamic data, can only do independent weekly or aggregated annually No associated uncertainty maps!

25 Inverse Distance weighting (IDW) Poor person s methodology Value at a new site = weighted mean of observations, Weights inversely proportional to the square of the distance. Problems Not model based! Unable to accommodate known covariate - precipitation! Unable to handle 0 s, unable to handle missing data Can t fuse with model output data With dynamic data, can only do independent weekly or aggregated annually No associated uncertainty maps!

26 Change of support problem Fuentes and Raftery, 2005 Need to upscale (block-average) point level Z (s, t) to obtain grid level Z (A j, t). Z (A j, t) = 1 Z (s, t) ds, (1) A j A j PSfrag replacements A ki si Many more A s than s s Use MEM (measurement error model) at point level centred around grid level values Make inference at the point level by downscaling Huge computational advantages, NO integration (1).

27 Change of support problem Fuentes and Raftery, 2005 Need to upscale (block-average) point level Z (s, t) to obtain grid level Z (A j, t). Z (A j, t) = 1 Z (s, t) ds, (1) A j A j PSfrag replacements A ki si Many more A s than s s Use MEM (measurement error model) at point level centred around grid level values Make inference at the point level by downscaling Huge computational advantages, NO integration (1).

28 Hierarchical Bayesian Model Model data weekly. Fuse with gridded weekly CMAQ output. Use weekly precipitation information, available from other monitoring networks. Interpolate in space, predict in time Obtain quarterly and annual maps. Reveal spatial pattern in deposition. Results are illustrative, not definitive.

29 Our data set Data Weekly deposition data from 128 sites in the eastern U.S. for the year Use 120 sites to estimate, remaining 8 to validate. Weekly CMAQ output from J = 33, 390 grid cells (about 1.8 million values!) Weekly precipitation data from 2827 predictive sites.

30 Location of the NADP A B C D E F G H Figure: A map of the study region; points denote the NADP sites for fitting and A-H denote the eight validation sites.

31 Annual Precipitation Figure: Map of annual total precipitation in 2001.

32 Exploratory Analyses wetso4 2 wetno (a) (b) Figure: Boxplot of weekly depositions: (a) sulfate and (b) nitrate.

33 Exploratory Analyses... log precipitation log sulfate (a) log precipitation log nitrate (b) Figure: Deposition against precipitation (both on the log scale): (a) sulfate and (b) nitrate.

34 Exploratory Analyses... log cmaq vals log sulfate (a) log cmaq vals log nitrate (b) Figure: Deposition at the NADP sites against the CMAQ values in the grid cell covering the corresponding NADP site on the log scale: (a) sulfate and (b) nitrate.

35 Modeling Requirements No deposition, Z (s i, t), without precipitation, P(s i, t); enforced by a latent atmospheric space-time process V (s i, t) below, i = 1,..., n = 120, and for each week t, t = 1,..., 52. Similarly, model CMAQ output, Q(A j, t) for each grid cell A j, j = 1,..., J = 33, 390 and for each week t, modeled using a latent atmospheric areal process Ṽ (A j, t). Model everything on the log-scale; latent processes take care of point masses at zero. Avoid log(0) problems.

36 First stage models Precipitation model { exp (U(si, t)) if V (s P(s i, t) = i, t) > 0 0 otherwise, Deposition model Z (s i, t) = { exp (Y (si, t)) if V (s i, t) > 0 0 otherwise. Model for CMAQ output { exp (X(Aj, t)) if Q(A j, t) = Ṽ (A j, t) > 0 0 otherwise.

37 Clarification P s, Z s, and Q s are the observed precipitation, NADP deposition, and CMAQ deposition, respectively. V (s, t) is a conceptual point level latent atmospheric process which drives P(s, t) and Z (s, t). P(s, t) and Z (s, t) = 0 if V (s, t) 0. U(s, t) and Y (s, t) are log precipitation and deposition, respectively. Models below will specify their values when V (s, t) 0 or if P(s, t) or Z (s, t) are missing. Ṽ (A j, t) is a conceptual areal level latent atmospheric process which drives Q(A j, t). X(A j, t) is log CMAQ output where modeling below will specify its values when Ṽ (A j, t) 0.

38 Clarification P s, Z s, and Q s are the observed precipitation, NADP deposition, and CMAQ deposition, respectively. V (s, t) is a conceptual point level latent atmospheric process which drives P(s, t) and Z (s, t). P(s, t) and Z (s, t) = 0 if V (s, t) 0. U(s, t) and Y (s, t) are log precipitation and deposition, respectively. Models below will specify their values when V (s, t) 0 or if P(s, t) or Z (s, t) are missing. Ṽ (A j, t) is a conceptual areal level latent atmospheric process which drives Q(A j, t). X(A j, t) is log CMAQ output where modeling below will specify its values when Ṽ (A j, t) 0.

39 Clarification P s, Z s, and Q s are the observed precipitation, NADP deposition, and CMAQ deposition, respectively. V (s, t) is a conceptual point level latent atmospheric process which drives P(s, t) and Z (s, t). P(s, t) and Z (s, t) = 0 if V (s, t) 0. U(s, t) and Y (s, t) are log precipitation and deposition, respectively. Models below will specify their values when V (s, t) 0 or if P(s, t) or Z (s, t) are missing. Ṽ (A j, t) is a conceptual areal level latent atmospheric process which drives Q(A j, t). X(A j, t) is log CMAQ output where modeling below will specify its values when Ṽ (A j, t) 0.

40 Clarification P s, Z s, and Q s are the observed precipitation, NADP deposition, and CMAQ deposition, respectively. V (s, t) is a conceptual point level latent atmospheric process which drives P(s, t) and Z (s, t). P(s, t) and Z (s, t) = 0 if V (s, t) 0. U(s, t) and Y (s, t) are log precipitation and deposition, respectively. Models below will specify their values when V (s, t) 0 or if P(s, t) or Z (s, t) are missing. Ṽ (A j, t) is a conceptual areal level latent atmospheric process which drives Q(A j, t). X(A j, t) is log CMAQ output where modeling below will specify its values when Ṽ (A j, t) 0.

41 Clarification P s, Z s, and Q s are the observed precipitation, NADP deposition, and CMAQ deposition, respectively. V (s, t) is a conceptual point level latent atmospheric process which drives P(s, t) and Z (s, t). P(s, t) and Z (s, t) = 0 if V (s, t) 0. U(s, t) and Y (s, t) are log precipitation and deposition, respectively. Models below will specify their values when V (s, t) 0 or if P(s, t) or Z (s, t) are missing. Ṽ (A j, t) is a conceptual areal level latent atmospheric process which drives Q(A j, t). X(A j, t) is log CMAQ output where modeling below will specify its values when Ṽ (A j, t) 0.

42 Clarification P s, Z s, and Q s are the observed precipitation, NADP deposition, and CMAQ deposition, respectively. V (s, t) is a conceptual point level latent atmospheric process which drives P(s, t) and Z (s, t). P(s, t) and Z (s, t) = 0 if V (s, t) 0. U(s, t) and Y (s, t) are log precipitation and deposition, respectively. Models below will specify their values when V (s, t) 0 or if P(s, t) or Z (s, t) are missing. Ṽ (A j, t) is a conceptual areal level latent atmospheric process which drives Q(A j, t). X(A j, t) is log CMAQ output where modeling below will specify its values when Ṽ (A j, t) 0.

43 Clarification P s, Z s, and Q s are the observed precipitation, NADP deposition, and CMAQ deposition, respectively. V (s, t) is a conceptual point level latent atmospheric process which drives P(s, t) and Z (s, t). P(s, t) and Z (s, t) = 0 if V (s, t) 0. U(s, t) and Y (s, t) are log precipitation and deposition, respectively. Models below will specify their values when V (s, t) 0 or if P(s, t) or Z (s, t) are missing. Ṽ (A j, t) is a conceptual areal level latent atmospheric process which drives Q(A j, t). X(A j, t) is log CMAQ output where modeling below will specify its values when Ṽ (A j, t) 0.

44 The first stage likelihood f (P, Z, Q U, Y, X, V, Ṽ) = f (P U, V) f (Z Y, V) f (Q X, Ṽ) which takes the form [ { } T n t=1 i=1 1 exp u(si, t) 1 exp y(si, t) I (v(s i, t) > 0) { }] 1 exp x(aj, t) I (ṽ(a j, t) > 0) J j=1 where 1 x denotes a degenerate distribution with point mass at x and I( ) is the indicator function.

45 Deposition Model Y (s i, t) = β 0 + β 1 U(s i, t) + β 2 V (s i, t) + (b 0 + b(s i )) X(A ki, t) +η(s i, t) + ɛ(s i, t). Spatially varying coefficients, b = (b(s 1 ),..., b(s n )) is a Gaussian process (GP). Spatio-temporal intercept η t = (η(s 1, t),..., η(s n, t)) is a GP independent in time. Allow for spatially varying calibration of CMAQ. Could imagine common η(s i ). ɛ(s i, t) N(0, σ 2 ɛ ), provides the nugget effect.

46 The second stage models... Precipitation U(s i, t) = α 0 + α 1 V (s i, t) + δ(s i, t), where δ t = (δ(s 1, t),..., δ(s n, t)) is a GP independent in time. CMAQ output X(A j, t) = γ 0 + γ 1 Ṽ (A j, t) + ψ(a j, t), j = 1,..., J. Assume ψ(a j, t) N(0, σψ 2 ), independently.

47 Specification of latent processes Measurement Error Model (MEM) V (s i, t) N(Ṽ (A k i, t), σv 2 ), i = 1,..., n, t = 1,..., T. The process Ṽ (A j, t) is AR in time and CAR in space Ṽ (A j, t) = ρṽ (A j, t 1) + ζ(a j, t), ( J ) ζ(a j, t) N h ji ζ(a i, t), σ2 ζ, m j i=1 Let j define the m j neighboring grid cells of the cell A j. h ji = { 1 m j if i j 0 otherwise.

48 AR in time and CAR in Space Assume the initial condition for Ṽ0: Ṽ (A j, 0) = 1 T T X(A j, t), t=1 giving Ṽ0. Now we can write the CAR in closed form: f (Ṽt Ṽt 1, ρ, σ 2 ζ ) { exp 1 (Ṽt ρṽt 1) D 1 (I H) (Ṽt ρṽt 1) }, 2 D is a diagonal matrix with entries σ 2 ζ /m j. Note that this is an improper CAR.

49 Clarification Note that we can have Z > 0, Q = 0 and vice versa. Therefore V and Ṽ can have opposite signs This arises because we are modeling at two different scales - need processes at two different scales. We can view V (s, t) Ṽ (A, t)as a deviation from the areal average. We assume these realized deviations are independent across space and time We have a conditional model for V and X given Ṽ. The resulting marginal model for U and Y given Ṽ is multiscale - additive random effects at two scales

50 Clarification Note that we can have Z > 0, Q = 0 and vice versa. Therefore V and Ṽ can have opposite signs This arises because we are modeling at two different scales - need processes at two different scales. We can view V (s, t) Ṽ (A, t)as a deviation from the areal average. We assume these realized deviations are independent across space and time We have a conditional model for V and X given Ṽ. The resulting marginal model for U and Y given Ṽ is multiscale - additive random effects at two scales

51 Clarification Note that we can have Z > 0, Q = 0 and vice versa. Therefore V and Ṽ can have opposite signs This arises because we are modeling at two different scales - need processes at two different scales. We can view V (s, t) Ṽ (A, t)as a deviation from the areal average. We assume these realized deviations are independent across space and time We have a conditional model for V and X given Ṽ. The resulting marginal model for U and Y given Ṽ is multiscale - additive random effects at two scales

52 Clarification Note that we can have Z > 0, Q = 0 and vice versa. Therefore V and Ṽ can have opposite signs This arises because we are modeling at two different scales - need processes at two different scales. We can view V (s, t) Ṽ (A, t)as a deviation from the areal average. We assume these realized deviations are independent across space and time We have a conditional model for V and X given Ṽ. The resulting marginal model for U and Y given Ṽ is multiscale - additive random effects at two scales

53 The second stage specification Deposition model: f (Y t U t, V t, X t, η t, b, θ). Space-time intercept: f (η t θ). Precipitation model: f (U t V t, θ). Measurement Error model: f (V t Ṽ(1) t, θ) CMAQ Model: f (X t Ṽt, θ) AR and CAR Model: f (Ṽt Ṽt 1, θ). T t=1 [ f (Yt U t, V t, X t, η t, b, θ) f (η t θ) f (U t V ] t, θ) f (V t Ṽ(1) t, θ) f (X t Ṽt, θ)f (Ṽt Ṽt 1, θ) f (b θ). θ = (α 0, α 1, β 0, β 1, β 2, b 0, γ 0, γ 1, ρ, σ 2 δ, σ2 b, σ2 η, σ2 ɛ, σ2 ψ, σ2 v, σ 2 ζ ).

54 The second stage specification Deposition model: f (Y t U t, V t, X t, η t, b, θ). Space-time intercept: f (η t θ). Precipitation model: f (U t V t, θ). Measurement Error model: f (V t Ṽ(1) t, θ) CMAQ Model: f (X t Ṽt, θ) AR and CAR Model: f (Ṽt Ṽt 1, θ). T t=1 [ f (Yt U t, V t, X t, η t, b, θ) f (η t θ) f (U t V ] t, θ) f (V t Ṽ(1) t, θ) f (X t Ṽt, θ)f (Ṽt Ṽt 1, θ) f (b θ). θ = (α 0, α 1, β 0, β 1, β 2, b 0, γ 0, γ 1, ρ, σ 2 δ, σ2 b, σ2 η, σ2 ɛ, σ2 ψ, σ2 v, σ 2 ζ ).

55 The second stage specification Deposition model: f (Y t U t, V t, X t, η t, b, θ). Space-time intercept: f (η t θ). Precipitation model: f (U t V t, θ). Measurement Error model: f (V t Ṽ(1) t, θ) CMAQ Model: f (X t Ṽt, θ) AR and CAR Model: f (Ṽt Ṽt 1, θ). T t=1 [ f (Yt U t, V t, X t, η t, b, θ) f (η t θ) f (U t V ] t, θ) f (V t Ṽ(1) t, θ) f (X t Ṽt, θ)f (Ṽt Ṽt 1, θ) f (b θ). θ = (α 0, α 1, β 0, β 1, β 2, b 0, γ 0, γ 1, ρ, σ 2 δ, σ2 b, σ2 η, σ2 ɛ, σ2 ψ, σ2 v, σ 2 ζ ).

56 The second stage specification Deposition model: f (Y t U t, V t, X t, η t, b, θ). Space-time intercept: f (η t θ). Precipitation model: f (U t V t, θ). Measurement Error model: f (V t Ṽ(1) t, θ) CMAQ Model: f (X t Ṽt, θ) AR and CAR Model: f (Ṽt Ṽt 1, θ). T t=1 [ f (Yt U t, V t, X t, η t, b, θ) f (η t θ) f (U t V ] t, θ) f (V t Ṽ(1) t, θ) f (X t Ṽt, θ)f (Ṽt Ṽt 1, θ) f (b θ). θ = (α 0, α 1, β 0, β 1, β 2, b 0, γ 0, γ 1, ρ, σ 2 δ, σ2 b, σ2 η, σ2 ɛ, σ2 ψ, σ2 v, σ 2 ζ ).

57 The second stage specification Deposition model: f (Y t U t, V t, X t, η t, b, θ). Space-time intercept: f (η t θ). Precipitation model: f (U t V t, θ). Measurement Error model: f (V t Ṽ(1) t, θ) CMAQ Model: f (X t Ṽt, θ) AR and CAR Model: f (Ṽt Ṽt 1, θ). T t=1 [ f (Yt U t, V t, X t, η t, b, θ) f (η t θ) f (U t V ] t, θ) f (V t Ṽ(1) t, θ) f (X t Ṽt, θ)f (Ṽt Ṽt 1, θ) f (b θ). θ = (α 0, α 1, β 0, β 1, β 2, b 0, γ 0, γ 1, ρ, σ 2 δ, σ2 b, σ2 η, σ2 ɛ, σ2 ψ, σ2 v, σ 2 ζ ).

58 The second stage specification Deposition model: f (Y t U t, V t, X t, η t, b, θ). Space-time intercept: f (η t θ). Precipitation model: f (U t V t, θ). Measurement Error model: f (V t Ṽ(1) t, θ) CMAQ Model: f (X t Ṽt, θ) AR and CAR Model: f (Ṽt Ṽt 1, θ). T t=1 [ f (Yt U t, V t, X t, η t, b, θ) f (η t θ) f (U t V ] t, θ) f (V t Ṽ(1) t, θ) f (X t Ṽt, θ)f (Ṽt Ṽt 1, θ) f (b θ). θ = (α 0, α 1, β 0, β 1, β 2, b 0, γ 0, γ 1, ρ, σ 2 δ, σ2 b, σ2 η, σ2 ɛ, σ2 ψ, σ2 v, σ 2 ζ ).

59 The second stage specification Deposition model: f (Y t U t, V t, X t, η t, b, θ). Space-time intercept: f (η t θ). Precipitation model: f (U t V t, θ). Measurement Error model: f (V t Ṽ(1) t, θ) CMAQ Model: f (X t Ṽt, θ) AR and CAR Model: f (Ṽt Ṽt 1, θ). T t=1 [ f (Yt U t, V t, X t, η t, b, θ) f (η t θ) f (U t V ] t, θ) f (V t Ṽ(1) t, θ) f (X t Ṽt, θ)f (Ṽt Ṽt 1, θ) f (b θ). θ = (α 0, α 1, β 0, β 1, β 2, b 0, γ 0, γ 1, ρ, σ 2 δ, σ2 b, σ2 η, σ2 ɛ, σ2 ψ, σ2 v, σ 2 ζ ).

60 Graphical representation of our model. Regional atmospheric driver (point) Regional atmospheric centering process (areal) replacements δ(s i, t) V (s i, t) MEM Ṽ (A ki, t) b(s i ) η(s i, t) U(s i, t) Y (s i, t) X(A ki, t) Ṽ (A j, t 1) P(s i, t) (observed precipitation) Z(s i, t) (observed deposition) Q(A ki, t) (CMAQ model output)

61 Graphical representation of a fusion model. replacements Regional atmospheric driver (point) δ(s i, t) η(s i, t) V (s i, t) block average V (A ki, t) a(s i ) U(s i, t) Y (true) (s i, t) block average X(A ki, t) Q(A ki, t) Z (true) (s i, t) true P(s i, t) (observed precipitation) Y (s i, t) Q(A ki, t) Z(s i, t) (observed deposition) (CMAQ model output)

62 Predictions at new locations At a new site s and time t we need Z (s, t ) which depends on Y (s, t ). If P(s, t ) = 0 then Z (s, t ) = 0. Suppose otherwise. Bayesian predictive distributions: π(z pred z obs ) = π(z pred par) π(par z obs )dpar. Need to simulate Y (s, t ). V (s, t ) N(Ṽ (A, t ), σ 2 v). U(s, t ), η(s, t ) and b(s ) are simulated from the conditional distributions at s given s 1,..., s n. Kriging. X(A, t ) =logq(a, t ) if Q(A, t ) > 0, otherwise updated in the MCMC. More details in the paper.

63 Predictions at new locations At a new site s and time t we need Z (s, t ) which depends on Y (s, t ). If P(s, t ) = 0 then Z (s, t ) = 0. Suppose otherwise. Bayesian predictive distributions: π(z pred z obs ) = π(z pred par) π(par z obs )dpar. Need to simulate Y (s, t ). V (s, t ) N(Ṽ (A, t ), σ 2 v). U(s, t ), η(s, t ) and b(s ) are simulated from the conditional distributions at s given s 1,..., s n. Kriging. X(A, t ) =logq(a, t ) if Q(A, t ) > 0, otherwise updated in the MCMC. More details in the paper.

64 Predictions at new locations At a new site s and time t we need Z (s, t ) which depends on Y (s, t ). If P(s, t ) = 0 then Z (s, t ) = 0. Suppose otherwise. Bayesian predictive distributions: π(z pred z obs ) = π(z pred par) π(par z obs )dpar. Need to simulate Y (s, t ). V (s, t ) N(Ṽ (A, t ), σ 2 v). U(s, t ), η(s, t ) and b(s ) are simulated from the conditional distributions at s given s 1,..., s n. Kriging. X(A, t ) =logq(a, t ) if Q(A, t ) > 0, otherwise updated in the MCMC. More details in the paper.

65 Predictions at new locations At a new site s and time t we need Z (s, t ) which depends on Y (s, t ). If P(s, t ) = 0 then Z (s, t ) = 0. Suppose otherwise. Bayesian predictive distributions: π(z pred z obs ) = π(z pred par) π(par z obs )dpar. Need to simulate Y (s, t ). V (s, t ) N(Ṽ (A, t ), σ 2 v). U(s, t ), η(s, t ) and b(s ) are simulated from the conditional distributions at s given s 1,..., s n. Kriging. X(A, t ) =logq(a, t ) if Q(A, t ) > 0, otherwise updated in the MCMC. More details in the paper.

66 Choosing the spatial decay parameters Estimation is challenging due to weak identifiability of variances and ranges. Inconsistent see Stein (1999) and Zhang (2004). So, we fix φ η, φ δ and φ b Use validation mean-square error VMSE = 1 n v 8 52 i=1 t=1 ( Z (s i, t) Ẑ (s i, t) ) 2 I(observed) n v = total number of validation observed (=407, here). Optimal ranges were 500, 1000 and 500 kilometers. VMSE is not sensitive near these values.

67 Model Validation Prediction 3 Prediction 2 O O O O O O O O O O O O OOO OO OO OO OO OO OO O O O OO O O O O O O O OO O O O O O O O O O O O 1 0 O O O O O O O O O O O O O O O O O O OO OO OO OO OO OO OOO O O O OO O OO O O O OO O O OO O O O O O O O O O OO O O O O O O O O O O O O Sulfate Observation (a) Nitrate Observation (b) Figure: Validation versus the observed values at the 8 reserved sites. Validation prediction intervals are plotted as vertical lines. (a) sulfate and (b) nitrate.

68 b(s) Surfaces (a) (b) (c) (d) Figure: (a) Sulfate. (b) Nitrate. (c) The s.d. for sulfate. (d) The s.d. for nitrate.

69 Spatially varying slopes? Do we need the spatially varying b(s) s? Only a few of the b(s i ) are significant; they are small relative to their standard errors. Importance of precipitation and the spatially varying intercept makes it difficult to find spatially varying contribution of CMAQ Fusion approaches also have not found spatially varying intercepts Still can see space-time bias in CMAQ by comparing model predictions with CMAQ output.

70 Spatially varying slopes? Do we need the spatially varying b(s) s? Only a few of the b(s i ) are significant; they are small relative to their standard errors. Importance of precipitation and the spatially varying intercept makes it difficult to find spatially varying contribution of CMAQ Fusion approaches also have not found spatially varying intercepts Still can see space-time bias in CMAQ by comparing model predictions with CMAQ output.

71 Spatially varying slopes? Do we need the spatially varying b(s) s? Only a few of the b(s i ) are significant; they are small relative to their standard errors. Importance of precipitation and the spatially varying intercept makes it difficult to find spatially varying contribution of CMAQ Fusion approaches also have not found spatially varying intercepts Still can see space-time bias in CMAQ by comparing model predictions with CMAQ output.

72 Spatially varying slopes? Do we need the spatially varying b(s) s? Only a few of the b(s i ) are significant; they are small relative to their standard errors. Importance of precipitation and the spatially varying intercept makes it difficult to find spatially varying contribution of CMAQ Fusion approaches also have not found spatially varying intercepts Still can see space-time bias in CMAQ by comparing model predictions with CMAQ output.

73 Spatially varying slopes? Do we need the spatially varying b(s) s? Only a few of the b(s i ) are significant; they are small relative to their standard errors. Importance of precipitation and the spatially varying intercept makes it difficult to find spatially varying contribution of CMAQ Fusion approaches also have not found spatially varying intercepts Still can see space-time bias in CMAQ by comparing model predictions with CMAQ output.

74 Parameter Estimates Sulfate Nitrate mean sd 95%CI mean sd 95%CI α ( , ) ( , ) α (0.1017, ) (0.0843, ) β ( , ) ( , ) β (0.8972, ) (0.8274, ) β ( , ) ( , ) b (0.0386, ) (0.0409, ) γ ( , ) ( , ) γ (0.8891, ) (0.7303, ) ρ (0.7664, ) (0.7468, ) σδ (2.5254, ) (1.7942, ) ση (0.2616, ) (0.3149, ) σɛ (0.0607, ) (0.0588, ) σψ (2.4997, ) (2.2092, ) σv (0.7601, ) (0.7366, ) σζ (0.4322, ) (0.4316, )

75 IDW vs. our model Validation PMSE for the annual totals using the 8 holdout sites For sulfates, IDW PMSE is 20.4, our model PMSE is 8.1 For nitrates, IDW PMSE is 3.5, our model PMSE is 1.3 Would expect improvement given the complexity of our model. However 60% is substantial and perhaps justifies the effort

76 IDW vs. our model Validation PMSE for the annual totals using the 8 holdout sites For sulfates, IDW PMSE is 20.4, our model PMSE is 8.1 For nitrates, IDW PMSE is 3.5, our model PMSE is 1.3 Would expect improvement given the complexity of our model. However 60% is substantial and perhaps justifies the effort

77 IDW vs. our model Validation PMSE for the annual totals using the 8 holdout sites For sulfates, IDW PMSE is 20.4, our model PMSE is 8.1 For nitrates, IDW PMSE is 3.5, our model PMSE is 1.3 Would expect improvement given the complexity of our model. However 60% is substantial and perhaps justifies the effort

78 IDW vs. our model Validation PMSE for the annual totals using the 8 holdout sites For sulfates, IDW PMSE is 20.4, our model PMSE is 8.1 For nitrates, IDW PMSE is 3.5, our model PMSE is 1.3 Would expect improvement given the complexity of our model. However 60% is substantial and perhaps justifies the effort

79 Annual Sulfate Deposition Figure: Model predicted map of annual sulfate deposition in The observed annual totals are labeled; a larger font size is used for the validation sites.

80 Annual Nitrate Deposition Figure: Model predicted map of annual nitrate deposition in The observed annual totals are labeled; a larger font size is used for the validation sites.

81 Map of the length of 95% prediction intervals Figure: Uncertainty map of annual sulfate deposition Figure: Uncertainty map of annual nitrate deposition.

82 Quarterly Sulfate Deposition (a) (b) (c) (d) Figure: (a) Jan Mar, (b) Apr-Jun, (c) Jul Sep, (d) Oct-Dec.

83 Quarterly Nitrate Deposition (a) (b) (c) (d) Figure: (a) Jan Mar, (b) Apr-Jun,(c) Jul Sep, (d) Oct-Dec.

84 Discussion Novel spatio-temporal model for fusing point and areal data which validates well. Inference can be provided for any spatial or temporal aggregation. With yearly data can study trends in deposition with regard to regulatory assessment There are models in between IDW and ours but may sacrifice the features we accommodate. Preferable to fusion using block averaging since number of modeled grid cells much greater than number of monitoring sites, even worse if across time. Develop model for dry deposition, hence total deposition.