Inverse problems for the study of climate-ecological processes in - - PowerPoint PPT Presentation

Inverse problems for the study of climate-ecological processes in the conditions of anthropogenic impacts

Пененко В.В. Цветова Е.А.

Report content:

2

• What are inverse problems?
• definitions
• differences from direct problems
• implementation specifics
• How can we summarize the formulations of inverse problems?
• variational approach
• What is the meaning of the term climate-ecological?
• How to take into account the different scales in climate & ecology?
• How to consider and evaluate anthropogenic effects in inverse problems?
• The main types of inverse problems on research topics

What are inverse problems?

3

• definitions
• differences from

direct tasks

• Specific

implementation:

• Conditional well-

posedness

• Ill-posednes
• interrelated models

and observational data

Direct problems are formulated as Initial- boundary- value problems. Goals defined; What is known? all model parameters; sources of impact; initial and boundary conditions. Problems are solved “forward” in time, they are mostly correct, sometimes conditionally correct.

“In essence, the inverse of this task is the task in which the desired and the given are reversed” (Wikipedia, 1st grade of elementary school). The inverse problem is a type of tasks, often arising in many sections of science, when the values of model parameters must be

• btained from observable data.

Examples of inverse problems can be found in the following areas: geophysics, astronomy, medical imaging, computed tomography, remote sensing of the Earth, spectral analysis, scattering theory and non- destructive testing tasks, etc.

What is the meaning of the term climate-ecological problems in the Earth system?

4

• Earth system climate
• climatic zones of the globe
• regional climate
• climate of urban

agglomerations

• meso-climate
• microclimate
• Man
• Environment
• Ecology

High uncertainty

• Unknown sources of impacts, initial and

boundary conditions, rates of transformations and other parameters of models, structure of models.

• Lack of required measurement data

The difficulty

• f direct

modeling

• Nonlinearity, a variety of time

scales and processes, high (up to 1012) dimension of the considered models

Real time

• peration
• The need to obtain and refine the

forecast as measurement data becomes available.

Features of mathematical modeling in environmental protection problems

5

• Ozone (O3), carbon monoxide

(CO), ammonia (NH3), hydrogen sulfide (H2S), sulfur dioxide (SO2), nitrogen oxides (NO, NO2), formaldehyde (HCHO), ...

• Aerosols: PM 10, PM 2.5, nano

...

• radioactive ...
• biologically active ...

Main pollutants:

The uniqueness of each situation

How can we present formulations of direct and inverse problems in a generalized form?

6

Variational principle

Formulation with weak constraints, combining

• mathematical models of processes (in the form of an integral identity);
• available observational data;
• target (goal) functionals.

For unification, we use

• the classical method of Lagrange multipliers (1762);
• adjoint functions
• the concept of adjoint integrating factors

General structure of modeling systems

 

( , , , ) ( , ) L L G t         X U Y f r U Y f r   

a

t      

 

, X U Y,f,r,ξ  

The vector of functional arguments of the modeling system Results and models of observations

( ) , 1,

m m m

H m M    %   

Target functionals of studies

( ) ( ) ( , ) ( , ) , 1, .

t t

k k k k k D D

F t dDdt F k K      



x  

7

 

I ,  X  Variational principle for combining all objects of the modeling system

* *(

)

t

Q D  φ

adjoint functions vector system energy balance equation

 

 

 

* *

I , L   X φ X ,φ

*

( , ) ,

t

D

G dDdt t            



U Y f r φ   Integral identity

8

9

Adjoint functions

• 1. "Global":

Distributed Lagrange multipliers for combining process models, observational data, and prediction target functionals within a variational principle;

• 2. "Local":

In the modeling technology in the decomposition and splitting mode in combination with the methods of finite elements / volumes: Solutions of homogeneous adjoint problems for the construction of numerical schemes with conservation laws (adjoint integrating factors)

System of convection-diffusion-reaction equations in models of hydrothermodynamics and atmospheric chemistry

10

( ) div div grad ( ) ( ) 0, 1,

i i

i i i i i

L S f r i n t

 

                X u φ ( ) ( ) ( ), 1,

i i i i

S P i n      φ φ φ

Extended functional of the variational principle with weak constraints

* *

( , , ) ( , )

h t

h h D

I        X η X  

 

2 2 3 3 1 1 4

0.5 ( , ) ( , ) 0.5 ( ) ( )

h h m t t

h h h k D D D

W W W          r r ξ ξ   ( 0)  r

11

Decomposition and splitting for the construction of numerical schemes and algorithms with the properties of the total approximation

1 1

; ( , ) ;

q q

G t t

   



 

            

 

φ φ U Y φ 

1 1 1 1

; ; 1;

q q q j j

t t t

     



   

    

  

f f r r 0 ; 1,q t

     

          φ φ f r

   

1 1 1 1 1 1

; ;

q q j j j j j j       

 

     

  

 

φ φ φ φ r r

12

Gateaux-variations for functionals. Algorithms for sensitivity relations

 

( ) , 1

p p p

p



   



             S S S

   

 

1

, 1 !

p p

O p

  

      



         



S S S ( ),  S

Definition of Gateaux derivatives for the functional For each structure of vectors at

 S Taylor series  

When p = 1: Euler-Lagrange equations: systems of basic and adjoint equations, equations for uncertainties; sensitivity relations for target functionals to parameter variations When p = 2: sensitivity ratios of the second order; Hessian

 

*

,  S X φ

13

* *

(,..., ) , , ,

h h h h k k k k

                                φ φ r φ r φ % % % % , , , ,

h h h h k k k k m m

                                        ξ Ψ Y U ξ Ψ Y U % % % %

Variations of extended functional ( p=1)

*

0,     φ % h

k

0,     φ % h

k

0,     r % h

k

    ξ % h

k

Stationarity conditions

,            Y Y   % %

h h k k

Sensitivity relations

,

h h k k

             U U % %

14

( , ,

h h k t

G



      φ φ U Y ) - f - r φ %

( ) ( , , ) 0,

h T T k t k k k

A

 

       φ φ U Y φ d φ %

h T k k 1 1

( ( ) 0.5 ( M )),      d φ φ   

( ) 0,

k t t

D

 

  φ x x

1 a 3 k

M ( , 0 ), t 0 ,

 

   φ φ φ x

1 * 2 k

( , t ) M ( , t ),



 r x φ x

h

( , , ) G ( , , ) , A



 



        φ U Y φ φ φ U Y  

Basic algorithm for forward / inverse modeling

approximation of time derivatives

• direct problem

adjoint problem Initial data with uncertainty model uncertainty function

15

16

Modeling technology based on variational principle

Five fundamental spaces for environmental forecasting and design:

• State functions of the direct problem
• Solutions of adjoint problems
• Uncertainty functions of process models
• Sensitivity functions of process models and target

functionals to variations of model parameters, initial data, functions of boundary conditions and sources of impacts

• Sensitivity functions of target functionals to variations

in observational data

How to take into account the different scales?

Identification of the main factors: methods of orthogonal decomposition of phase spaces of climate-ecological information and the results of mathematical modeling

• identifying elements of the long-term memory of the climate system;
• variability analysis of multicomponent 4D spaces;
• climate analysis as the implementation of the behavior of a dynamic

system;

• identification of centers of action in the climate-ecological system;
• formation of strategies for targeted monitoring;
• continuation problems;

Climate data analysis in terms of orthogonal subspaces makes it possible to identify the dominant structures in the overall system and use them to construct forecast scenarios.

17

Methods of orthogonal decomposition. Subspaces of informative basis

3 5 7 7 9 9 9 9 11 11 11 11 11 1 1 11 13 1 3 13 13 13 13 1 3 1 3 13 13 13 13 13 15 1 5 15 1 5 1 5 1 7

Longitude y 30 60 90 120 150 180 210 240 270 300 330 30 60 90 120 150 180 Level value: 1

• 0.82

3

• 0.68

5

• 0.54

7

• 0.39

9

• 0.25

11

• 0.10

13 0.04 15 0.18 17 0.33 19 0.47 21 0.61

January 15

Penenko V., Tsvetova E. Orthogonal decomposition methods for inclusion of climatic data into environmental studies//Ecol. Model. V. 217. P. 279–291. 2008.

Activity centers of the climatic system

18

• 0.49

• 0.39

• 0.29

• 0.18

• 0.08

April 15

Separation of processes scales. Main factors

Eigenvalues of Gram matrices, June, 500-hPa, 1950-2005 Global scale Regional scale

Longitude y 30 60 90 120 150 180 210 240 270 300 330 30 60 90 120 150 180 Level value: 1

• 0.64

3

• 0.49

5

• 0.35

7

• 0.20

9

• 0.06

11 0.09 13 0.23 15 0.38 17 0.52 19 0.66 21 0.81

July 15

19

How to take into account anthropogenic impacts in direct and inverse problems?

20

Scenario approach based on orthogonal expansions Meteorology scenario Basic+ Monte-Carlo Study of specific functional behavior Analysis of uncertainty functions Variations in relations “source-receptor” and “receptor-source”

PEEX-Modelling-Platform (PEEX-MP)

21

https://www.atm.helsinki.fi/peex/index.php/modelling-tools-demonstration.

• ATMES - Atmospheric Transport and dispersion Models of

global and regional scales for climatic Environmental Studies

• MMAD&IT - Mesoscale Models of Atmospheric Dynamics and

Impurity Transport in areas with complex terrain

• IMDAF - Inverse Modeling and Data Assimilation Framework

22

Acting and promising

• bservation systems
• In situ observations

Every 6 hours at altitudes of 1.5-2 m: temperature, pressure, wind, relative humidity; over 80,000 observations every day

• Terrestrial Remote Sensing Networks

Over 10 types of different observing systems Observations of the upper atmosphere (in situ): temperature, gas composition, aerosols, cloud components Satellite systems of remote observation More than 1,000,000 observations of the atmosphere and the surface of the Earth are received every day. ! Only about 20% of satellite data is assimilated in numerical models.

According to the review of WMO-No 1156

Calculation of spectral intensity measured by satellites

Remote observing models to calculate the images of measured values and solutions of inverse problems

1 1 1

( , , ) : ( (1)) (1, , ) ( ( )) ( ) P W J B T P B T d     



φ θ θ

    

      

 

 

3

( ) / exp / ( ) 1 B T a b kT



   

Planck function

   ;

m

J  θ





zenith angle of sounding

23

Comparative experiment

• n the recovery of meteorological fields

as a result of variational data assimilation: geopotential field at levels (a) 100 hGt, (b) 500 hGt, (s) 1000 hGt.

left - all available stations in the Northern Hemisphere. right - all available data on the continents; data over the oceans are not available.

Marchuk G.I., Penenko V.V. Application of perturbation theory to problems

• f simulation of atmospheric processes.
• Int. Symp on Monsoon dynamics, 1977, New Delhi,

published in book “Monsoon dynamics” (Eds. J. Lighthill , R.Pearce), Cambridge University Press, 1981, 639-655

Common types of inverse problems: Continuation problem Variational data assimilation

24

Common types of inverse problems: data assimilation

25

Sequential solving linked inverse problems

Left: "Exact" solution of the transport model with given sources (black triangles) and initial data. Right: Scenario

• f restoring the impurity fields by the transport model with the assimilation of the monitoring system data (red

dots). There are no sources (= 0) in the system of equations. Pollution begins to be displayed by the system only after it is fixed at the stations.

Common types of inverse problems: sensitivity studies of quality functionals

26

Baikal region Zoning of the Baikal region according to the pollution danger degree of the lake water area (inverse problem). Arctic Risk function of receiving pollution by the Arctic region from existing and potential sources in the Northern hemisphere of the Earth (inverse problem).

27

Conclusion

Currently, the field of interconnected climate-ecological research is expanding toward solving inverse problems that allow, for example,

• assimilate observational data;
• evaluate the parameters of process models and observation models;
• to evaluate uncertainty in models and processes;
• to investigate the sensitivity of target functionals;
• search for sources of influence,
• successive continuation problems,
• etc….

Classical theory and methods for solving inverse problems can be successfully applied in such studies.

28

1. Penenko V. V., Penenko A. V., Tsvetova E. A. Gochakov A.V. Methods for studying the sensitivity of air quality models and inverse problems of geophysical hydrothermodynamics // Journal of pplied Mechanics and Technical Physics, 2019, Vol.60, N 2, 392-399 2. Penenko V. V., Penenko A. V., Tsvetova E. A. Variational approach to the study of processes of geophysical hydro- thermodynamics with assimilation of observation data A// Journal of pplied Mechanics and Technical Physics. –– 2017. –– Vol. 58, no. 5. –– P. 771–778. 3. Penenko, V. V., Tsvetova, E. A., and Penenko, A. V. , “Development of variational approach for direct and inverse problems of atmospheric hydrothermal dynamics and chemistry,” Izvestiya, Atmospheric and Oceanic Physics, 51(3), 311–319 (2015). 4. Penenko, V.V., Tsvetova, E.A., Penenko, A.V., “ Methods based on the joint use of models and observational data in the framework of variational approach to forecasting weather and atmospheric composition quality,” Russ. Meteor.

• Hydr. 40(6), 365-373 (2015).

5.

• A. V. Penenko, V. V. Penenko, and E. A. Tsvetova Sequential data assimilation algorithms for air quality monitoring

models based on a weak-constraint variational principle // Numerical Analysis and Applications, 9(4):312–325, 2016. doi: 10.1134/s1995423916040054 6. V.V. Penenko and E.A. Tsvetova and A.V. Penenko Variational approach and Euler’s integrating factors for environmental studies // Computers and Mathematics with Applications (2014) v.67 №. 12 2240 - 2256 doi: 10.1016/j.camwa.2014.04.004

Спасибо за внимание!

В докладе использованы результаты работы неформального творческого коллектива: Пененко В.В., Пененко А.В., Пьянова Э.А., Цветова Е.А., Мукатова Ж.С., Антохин П.Н., Гочаков А.В., Колкер А.Б.

29

30

Direct and inverse problems

• Local, regional and global

problems of atmospheric pollution

• Environmental tasks in normal

and emergency situations

• Identification of sources of

anthropogenic impacts

• Identify prerequisites and

predict the consequences of environmental disasters

• Environmental risk and

vulnerability assessment

• Preparation of solutions for

environmental impact assessment

• Evaluating the effectiveness
• f environmental strategies

Models of processes

• Hydro-thermo-

dynamics of atmosphere and water bodies

• Transport &

transformation of gases and aerosols

• Turbulence
• Radiation transfer

Methods

• Variational principles for the

joint use of models and

• bservational data
• Methods of assimilation of
• bservational data
• Splitting and decomposition

methods

• Finite elements and volumes
• Discrete-analytical

approximations

• Adjoint integrating factors for

differential and integral equations

• Monotone schemes
• Parallel computations

Problems, models, measurement data, methods

Multifunctional complex