GATNDOR TopicCollocation Uncertainty in Vertical Pro les:statistical Functional Regression Approach

Transcription

1 GATNDOR Topic Collocation Uncertainty in Vertical Pro les: Statistical Functional Regression Approach Joint work with: B. Demoz, R. Ignaccolo, F.Madonna. - ˆ* University of Bergamo - DIIMM - alessandro.fasso@unibg.it De Bilt, ICM-5, 26 February 2013

2 Scienti c questions Scienti c questions We would like to answer the following questions: 1 is the co-location uncertainty related to environmental factors? 2 is the co-location uncertainty related to the paired trajectories distance? 3 Is the co-location uncertainty related to height? 4 Is uncertainty a static or dynamic concept? 5 Are above points valid for all ECV?

3 Scienti c questions Scienti c questions We would like to answer the following questions: 1 is the co-location uncertainty related to environmental factors? 2 is the co-location uncertainty related to the paired trajectories distance? 3 Is the co-location uncertainty related to height? 4 Is uncertainty a static or dynamic concept? 5 Are above points valid for all ECV?

4 Scienti c questions Scienti c questions We would like to answer the following questions: 1 is the co-location uncertainty related to environmental factors? 2 is the co-location uncertainty related to the paired trajectories distance? 3 Is the co-location uncertainty related to height? 4 Is uncertainty a static or dynamic concept? 5 Are above points valid for all ECV?

5 Scienti c questions Scienti c questions We would like to answer the following questions: 1 is the co-location uncertainty related to environmental factors? 2 is the co-location uncertainty related to the paired trajectories distance? 3 Is the co-location uncertainty related to height? 4 Is uncertainty a static or dynamic concept? 5 Are above points valid for all ECV?

6 Scienti c questions Scienti c questions We would like to answer the following questions: 1 is the co-location uncertainty related to environmental factors? 2 is the co-location uncertainty related to the paired trajectories distance? 3 Is the co-location uncertainty related to height? 4 Is uncertainty a static or dynamic concept? 5 Are above points valid for all ECV?

7 Introduction Introduction "... The primary goals of GRUAN are to provide vertical pro les of reference measurements suitable for reliably detecting changes in global and regional climate on decadal time scales" (GRUAN Manual V7). We consider here an empirical approach to uncertainty analysis with no reference to a priori metadata but relying on data by means of a statistical modelling approach. Stochastic structure of vertical pro les is important from the statistical point of view not only for facing the co-location problem but also for uncertainty analysis and decomposition of vertical pro les in general.

8 Introduction Introduction "... The primary goals of GRUAN are to provide vertical pro les of reference measurements suitable for reliably detecting changes in global and regional climate on decadal time scales" (GRUAN Manual V7). We consider here an empirical approach to uncertainty analysis with no reference to a priori metadata but relying on data by means of a statistical modelling approach. Stochastic structure of vertical pro les is important from the statistical point of view not only for facing the co-location problem but also for uncertainty analysis and decomposition of vertical pro les in general.

9 Outline Outline 1 Heteroskedastic functional regression co-location model 2 Application to relative humidity from Beltsville-Sterling radiosonde

10 The Co-location Model Let y denote the measurement of a physical quantity, e.g. an ECV, along a trajectory through the atmosphere. y (s, t), s = (lat, lon, h), h h 0, t t 0 The space-time vertical trajectory can then be described by the parametric representation h! (s h, t h ) for h 0 h h 1. y () = y (h)

11 We consider a vertical pro le as a single object described by a smooth function: µ (h), h h 0. and the standard measuerment error decomposition of observation pro le launched at time t j from place s j, j = 1,..., n, is given by a random function y j () = µ j () + ε j () where µ () is the "true" smooth pro le and ε () is the zero mean measurement error with variance σ 2 ε ().

12 Suppose we are comparing two instruments, e.g. radiosondes, at the same height and giving measurements y and y 0 respectively. Hence we have y () = y () y 0 () = µ () + ε () where µ = µ µ 0 is the co-location drift and ε = ε ε 0 is the co-location measurement error, with Var ( ε) = σ 2 ε + σ 2 ε 0 = 2σ 2 ε.

13 The Heteroskedastic Model The Heteroskedastic Functional Regression Model for Co-location µ () = β 0 + β () 0 x () + ω () where β 0 is the inter-site constant co-location bias. x () is a set of functional environmental factors. ω () is an heteroskedastic error component: namely σω 2 (jx) = Var (ω () jx) For example a log-linear component σω 2 (jx) = exp γ () 0 x ().

14 The Heteroskedastic Model The Global Uncertainty Pro le U () = Var ( y ()) Its decomposition extends the heteroskedastic uncertainty decomposition of Fassò et al. (2003) and is given by Var ( y ()) = U () = U µ () + σ 2 ε where, U µ () = Var (µ ()) de nes the apportionable co-location error.

15 Uncertainty Decomposition Uncertainty Decomposition where U µ () = E β 2 0 ( µ ()) 2 = β U x () + U ω () + U ˆβ (). is the inter-site error U x + U ω is the environmental error U ˆβ is the sampling error.

16 Uncertainty Decomposition In particular U x is the drift uncertainty and is given by U x () = ˆβ 0 () Σ x () ˆβ () The second component of the environmental error, namely U ω, cannot be reduced by observing x () and is given by U ω () = E x ˆσ ω 2 (jx) The sampling error is given by U ˆβ () = E x () 0 Σ ˆβ () x () where Σ ˆβ () = Var ˆβ () jx is the estimation functional variance covariance matrix of ˆβ ().

17 Uncertainty Decomposition Average Uncertainty In order to get a simple global uncertainty decomposition, we sumarize the above uncertainty pro les by integrating over the vertical pro le. This is given by Ū = β Ū x + Ū ω + Ū ˆβ + σ2 ε

18 The Beltsville Case Study n=49 trajectories Mutual departure trajectories

19 (mr) (rh) (vwind) (t) Figure: Data pro les

20 Modelling of relative humidity The response variable used here is relative humidity and all the data from the other radiosonda and all but relative humidity from the "co-located" radiosonda are used as explanatory factors. The resulting co-location error analysis corresponds understanding the variability of relative humidity for xed water vapour content in dry air mass. Information on location, distance and ECV s have been used as regressors for both µ and σ 2 and selected using suitable statistical model selection techniques.

21 Modelling of relative humidity The response variable used here is relative humidity and all the data from the other radiosonda and all but relative humidity from the "co-located" radiosonda are used as explanatory factors. The resulting co-location error analysis corresponds understanding the variability of relative humidity for xed water vapour content in dry air mass. Information on location, distance and ECV s have been used as regressors for both µ and σ 2 and selected using suitable statistical model selection techniques.

22 Modelling of relative humidity The response variable used here is relative humidity and all the data from the other radiosonda and all but relative humidity from the "co-located" radiosonda are used as explanatory factors. The resulting co-location error analysis corresponds understanding the variability of relative humidity for xed water vapour content in dry air mass. Information on location, distance and ECV s have been used as regressors for both µ and σ 2 and selected using suitable statistical model selection techniques.

23 Co-location model (Beltsville - Sterling) Mean component R 2 = 88% rh () = ˆβ 1 () rh 0 () + ˆβ 2 () mr 0 () + ˆβ 3 () mr () + ˆω () + ε () Skedastic component log σ 2 ω (jx) = 1.6 (0.26) + ˆγ 1 () p 0 () + ˆγ 2 () t 0 () + ˆγ 3 () rh 0 () + + ˆγ 4 () p () + ˆγ 5 () lon () + ˆγ 6 () mr () + + ˆγ 7 () u () + ˆγ 8 () v ()

24 (rh) (mr) ( mr) Figure: Beta functions for relative humidity co-location drift.

25 p 0 t 0 rh 0 ( p) Figure: Gamma functions for relative humidity co-location error ω 2, Figure 1 of 2.

26 ( lon) ( mr) ( u) ( v) Figure: Gamma functions for relative humidity co-location error ω 2, Figure 2 of 2.

27 Uncertainty Component Pro les p Figure: Square-root uncertainty budget U for relative humidity.

28 Average Uncertainty Components Source Ū Ū% pū Natural variability µ Inter-site error β % 3.37 Environmental Error (reducible) x % 16.4 Environmental Error (irreducible) ω % 6.1 Sampling error ˆβ % 0.7 Measurement error ε % 0.9

29 Conclusions This approach is a viable method for detaild uncertainty decomposition For example in our case study 1 is the co-location uncertainty related to environmental factors? These data show strong nonlinear correlation of rh on other variables 2 is the co-location uncertainty related to the paired trajectories distance? For these (limited spatial coverage and anisotropic) data, we found an EST-WEST e ect on the skedastic component. 3 Is the co-location uncertainty related to height? According to data the larger uncertainty is about 3 000m 4 Is uncertainty a static or dynamic concept? Yes and the skedastic function σω 2 (jx) may account for its variation 5 Are above points valid for all ECV?... future work...

30 Conclusions This approach is a viable method for detaild uncertainty decomposition For example in our case study 1 is the co-location uncertainty related to environmental factors? These data show strong nonlinear correlation of rh on other variables 2 is the co-location uncertainty related to the paired trajectories distance? For these (limited spatial coverage and anisotropic) data, we found an EST-WEST e ect on the skedastic component. 3 Is the co-location uncertainty related to height? According to data the larger uncertainty is about 3 000m 4 Is uncertainty a static or dynamic concept? Yes and the skedastic function σω 2 (jx) may account for its variation 5 Are above points valid for all ECV?... future work...

31 Conclusions This approach is a viable method for detaild uncertainty decomposition For example in our case study 1 is the co-location uncertainty related to environmental factors? These data show strong nonlinear correlation of rh on other variables 2 is the co-location uncertainty related to the paired trajectories distance? For these (limited spatial coverage and anisotropic) data, we found an EST-WEST e ect on the skedastic component. 3 Is the co-location uncertainty related to height? According to data the larger uncertainty is about 3 000m 4 Is uncertainty a static or dynamic concept? Yes and the skedastic function σω 2 (jx) may account for its variation 5 Are above points valid for all ECV?... future work...

32 Conclusions This approach is a viable method for detaild uncertainty decomposition For example in our case study 1 is the co-location uncertainty related to environmental factors? These data show strong nonlinear correlation of rh on other variables 2 is the co-location uncertainty related to the paired trajectories distance? For these (limited spatial coverage and anisotropic) data, we found an EST-WEST e ect on the skedastic component. 3 Is the co-location uncertainty related to height? According to data the larger uncertainty is about 3 000m 4 Is uncertainty a static or dynamic concept? Yes and the skedastic function σω 2 (jx) may account for its variation 5 Are above points valid for all ECV?... future work...

33 Conclusions This approach is a viable method for detaild uncertainty decomposition For example in our case study 1 is the co-location uncertainty related to environmental factors? These data show strong nonlinear correlation of rh on other variables 2 is the co-location uncertainty related to the paired trajectories distance? For these (limited spatial coverage and anisotropic) data, we found an EST-WEST e ect on the skedastic component. 3 Is the co-location uncertainty related to height? According to data the larger uncertainty is about 3 000m 4 Is uncertainty a static or dynamic concept? Yes and the skedastic function σω 2 (jx) may account for its variation 5 Are above points valid for all ECV?... future work...

34 Conclusions This approach is a viable method for detaild uncertainty decomposition For example in our case study 1 is the co-location uncertainty related to environmental factors? These data show strong nonlinear correlation of rh on other variables 2 is the co-location uncertainty related to the paired trajectories distance? For these (limited spatial coverage and anisotropic) data, we found an EST-WEST e ect on the skedastic component. 3 Is the co-location uncertainty related to height? According to data the larger uncertainty is about 3 000m 4 Is uncertainty a static or dynamic concept? Yes and the skedastic function σω 2 (jx) may account for its variation 5 Are above points valid for all ECV?... future work...

35 Conclusions This approach is a viable method for detaild uncertainty decomposition For example in our case study 1 is the co-location uncertainty related to environmental factors? These data show strong nonlinear correlation of rh on other variables 2 is the co-location uncertainty related to the paired trajectories distance? For these (limited spatial coverage and anisotropic) data, we found an EST-WEST e ect on the skedastic component. 3 Is the co-location uncertainty related to height? According to data the larger uncertainty is about 3 000m 4 Is uncertainty a static or dynamic concept? Yes and the skedastic function σω 2 (jx) may account for its variation 5 Are above points valid for all ECV?... future work...

36 Thanks for your attention