S. C. Wofsy, Harvard University, 25 April 2009

(1) Particle dispersion models;

(2) Adjoint implementation at Harvard: Stochastic Time-Inverted Lagrangian Transport (STILT) Model;

(3) Application to assessment of sources of greenhouse gases to the atmosphere.

Lagrangian tracer transport models:Measuring concentrations, assessing emissions

FormulationAdjointMass Conservation and other complications

GV aircraft approaches the terminator over the Alaska range in January, 2009, in HIPPO_1 experiment to define transport rates and global sources of greenhouse gases

Global and regional studies of CO2 , other greenhouse gases, ozone destroying species, etc, use observations of

concentrations at the ground, and from aircraft and satellites, to infer emission rates from surface sources.

An inverse problem.Why do we want to know?• Emissions of these gases drive changes in radiative forcing, ozone

loss, atmospheric chemistry• Processes regulating emissions, and overall budgets are much less

certain than needed to predict future atmospheric composition• Natural vs. human—caused balances is debated• Future monetization of CO2 emissions requires verification of

source reductions

NOAA Greenhouse Gas records

55% of annual emissions

The advection of a particle or puff is computed from the average of the three-dimensional velocity vectors for the initial-position P(t) and the first-guess position P'(t+∆t). The velocity vectors are typically linearly interpolated in both space and time. The first guess is

P'(t+∆t) = P(t) + V(P,t) ∆t

and the final position is

P(t+∆t) = P(t) + 0.5 [ V(P,t) + V(P',t+∆t) ] ∆t,

Basic Lagrangian Equation: Trajectories

If V is the mean wind vector from an assimilated meteorological field, the equation provides a trajectory model. It is fully reversible in time, because particles (“air parcels”) do not spread. Trajectory models cannot compute concentrations because the particles have zero measure.

Puff model: the source releases pollutant puffs at regular intervals. Each puff is advected according to the trajectory of its center of mass, as it expands (both horizontally and vertically) to simulate dispersion in a turbulent atmosphere. Useful for short term simulations.

Lagrangian particle dispersion model: the source is simulated by releasing many particles at the same time. A random component to the motion is added at each step according to atmospheric turbulence ( unresolved motions). Useful for longer term simulations.

Both require that stability and mixing coefficients be computed from meteorological data.

Puff and Dispersion Models

•Air parcels are represented by particles advected with the mean wind plus stochastic velocities that simulate turbulent transport, convection, and other unresolved/"non-conservative" motions of the atmosphere.•Particle positions are resolved to arbitrary accuracy, reducing grid-averaging errors such as representation error (for observations) and source spreading (for emissions).•Particles may be followed backwards in time, providing a direct adjoint model.

Lagrangian approach to atmospheric chemistry models : Particle Dispersion Modeling

Lagrangian particle dispersion models, and the analogous puff dispersion models, look superficially like trajectory models (same basic equation), but there are fundamental differences. Reversibility follows a statistical criterion.

Time reversal and the advection-diffusion equation for mass transport

n c/t = (Unc) + FTime reversal velocity reversal: t´ -t ; U´ - U ; F = <U'c'>, does not change sign !

Diffusion does not reverse—puffs or ensembles of particles expand, forward or backward in time!

The direction of the velocity in a random walk varies randomly, thus u' -u changes nothing.

Receptor-oriented model: time-inverted (adjoint) model

Dispersion Random walk

X2/t n2 steps to move nL

Particles tell where air is arriving from and link observations with upstream sources/sinks

Receptor-oriented model: time-inverted (adjoint) model

Backward Lagrangian transport

receptor

source

(x, y, z)

(xi’,yi’,zi’)

Forward = Backward?

receptor

source

forwardBackward

(x, y, z)

(xi’,yi’,zi’)

STILT (Stochastic Time Inverted Lagrangian Transport Model)

Based on HYSPLIT (Hybrid Single Particle Lagrangian Integrated Trajectory) model code [Draxler and Hess, 1998] Driven by ETA, AVN (forecasts) or EDAS, GDAS, NGM (assimilations)Improved turbulence parameterization

•TLw (vertical) and w after Hanna [1982]•reflection/transmission scheme at interfaces between high and low turbulence after Thomson [1997]

mean advection

stochastic (turbulent)advection

Receptorlocationu'u

u’ depends on: 1) TL (decorrelation timescale)2) w (fluctuations in turbulent velocity)

(“off-line” model)

Framework: Lagrangian Particle Dispersion Model Framework: A particle ensemble is released at the receptor and transported backward in time, tracking air parcels that arrive (in the forward-time sense) at the receptor at a given time.

The density of STILT particles is used to calculate the influence function I(xr,tr|x,t) and the footprint f(xr, tr| x,t). I(xr,tr|x,t) and f(xr, tr|

x,t) link concentration measurements at the receptor, C(xr, tr), to the

sum of all upstream contributions:

C(xr, tr) =

Units: C is in ppm; I is vol-1 (density); S is ppm s-1;

I(xr,tr|x,t) is represented by the density (xr,tr|x,t) of particles at (x,t)

which were transported backward in time from (xr,tr), normalized by

the total particle number Ntot:

Influence Function, I

p=1

( , | , ) 1( , | , ) ( ( ) )

totNr r

r r ptot tot

t tI t t t

N N

x xx x x x

The fields of I(xr,tr|x,t) and S(x, t), continuous in space and time, are in

practice resolved only at finite resolution with a discrete volume (x, y, z) and finite time interval (). Integrating over the finite volume and time elements to get the time/volume integrated I:

, , , ,1

1 ( , | , ) ( , | , )

1

j jm i k m i k

m i j k m i j k

tot

y y y yt x x z z t x x z z

r r r rtott x y z t x y z

N

p m i j kptot

dt dx dy dz I t t dt dx dy dz t tN

tN

x x x x

(finite small volume, time interval…)

The time- and volume-integrated influence function is simply quantified by tallying tp,m,i,j,k, the total amount of time each particle

p spends in a volume element i,j,k over timestep m. The source-receptor matrix elements that link sources at finite temporal and spatial resolutions directly to receptor concentrations is:

Source-receptor matrix elements in I (tr, xr, yr, zr | ts, xs, ys, zs )

I (m´, i ´, j ´, k ´ | m, i, j, k ) /(V t)

Model should run equally well forwards or backwards… now we should check it!

( , , ) for

( , , )( , )

0 for

airF x y t mz h

h x y tS t

z h

x

Representing Surface Sources; mass conservation (Part 1)

F is surface flux (e.g. mole m-2 s-1), h a near-surface mixing depth (m), the mean density of dry air in h (kg m-3), mair

the molecular mass of dry air (kg mole-1), giving S in ppm s-1.

Particles are effectively tagged with a mole fraction of pollutant, and the mole fraction is conserved as the particle is transported to different pressure levels. Since S is defined consistently in the interior or at the boundary, and each receptor is at a single pressure, particles are counted with the correct weight (mass is conserved) regardless of where they are emitted or observed.

_

S(x,t) is integrated over discrete volume and time elements, defining the footprint function, f :

  

, ,

0

( , ) ( , | , ) ( , , )( , , )

( , | , , ) ( , , )

jm i

m i j

y yt x x hair

m i j r r r r i j mi j m t x y

r r i j m i j m

mC t dt dx dy dz I t t F x y t

h x y t

f t x y t F x y t

x x x

x

, , ,1

1( , | , , )

( , , )

totNair

r r i j m p i j kpi j m tot

mf t x y t t

h x y t N

x

In our application, f(xr, tr| xi, yj, tm) is in units of [ppm/(mole/m2/s)], i.e.

given a unit surface flux of 1 mole/m2/s at (xi, yj, tm) persisting over a

time interval , f(xr, tr| xi, yj, tm) is the concentration change C in ppm at

the receptor.

Footprint function, f: surface influence

-8 -6 -4 -2

-24 hrs

-12 hrs

-6 hrs

Footprint of Vertical Profile over WLEF from LT1600 Particle Release on June 8th, 1999

log10(footprint)

log10[ppmv/(mole/m2/s)]

Detailed Linkage between Observation and Upstream Sources/Sinks (“Footprint”) from STILT

Simulations of transport and upstream influence by particle models must satisfy the following criteria:

1) maintenance of an initially well-mixed state

2) simulation of the close interaction between wind shear and vertical turbulence

3) high temporal resolution to resolve the decay in the autocorrelation of u'

4) consistent representation of particles as air parcels of equal mass in both the mean and turbulent transport components of the model.

Physical Requirements for Realistic Simulation of Source-Receptor Relationships

Turbulent transport: 1st order Markov process

Hanna [1979] showed from Eulerian and Lagrangian measurements that a Markov assumption is reasonable, i.e. that the particle velocity vector u can be decomposed into a mean component and a turbulent component u', with the turbulent component following the relation:

u' (t+t)=R(t)u'(t)+u'' (t)

u'' is a random vector, R an autocorrelation coefficient: R(t) = exp(–t/TLi), where TLi is the Lagrangian time scale (decorrelation

time) (i=u: horizontal; i=w: vertical). The random velocity u'' is:  

u'' = [1-R2(t)]1/2  

with drawn from a Gaussian distribution, and standard deviation i, is derived from the meteorological field (e.g. from TKE, or

mixed-layer height+surface fluxes of heat, momentum).

•Wind coming out of ground

•Failure to incorporate vertical density gradients

•Outliers from random Gaussian generator

•Operator splitting

•Particles trapped in low-turbulence regions

In this house we obey the Laws of Thermodynamics! -- Homer Simpson

Beware of Looking Under the Hood:Issues Addressed in Time-Reversibility

Comparisons

w

more turbulent

altitude

Problem: Particles become trapped in low-turbulence regions!

(spurious entropy loss)

Particles accumulate here.

This problem is closely analogous to the problem of detailed balance in the kinetic theory of gases.

ww

zaltitude

where( ) ( ) ( )

, =( ) ( ) ( )

w i w i it i

w i w i i

z z zw w

z z z

Satisfying the Well-Mixed Criterionby Treating Turbulence

Profiles as Separate Layers

Problem: Particles become trapped in regions of spurious

mass loss!

Actual simulation of particle densities for a set of winds from EDAS-80.

Violates 2nd law of Thermodynamics…

0.0 0.2 0.4 0.6 0.8

0.0

0.5

1.0

1.5

2.0

NGM Mass-violating Windsslope=1.852 R2=0.805

Day=15

1:1 Line

orthogonal regression

NGM Mass-violating Windsfor one starting time

Particle Density in Receptor Box [#/km3]

Par

ticle

Den

sity

in S

ourc

e B

ox [

#/km

3]

1st order mass correction

Backward- vs Forward-time Comparison

(Forward)

(Bac

kwar

d)

source receptor

creation ofmass

source

receptor

destruction of mass

source receptor

Implications of Mass Violation

•Affects lots of atmospheric models, especially ‘off-line’ models

•Careful attention needed in using wind fields from other models!

•Forward-time not necessarily better than backward-time results

•Resulting time-irreversibility may be an issue for ‘adjoint’ models as well

0.0 0.1 0.2 0.3 0.4

0.0

0.1

0.2

0.3

0.4

Synthetic Wind Fieldslope=0.979 R2=0.969

Mass-conserving, Simplified Winds

1:1 Line

R2=0.97

orthogonal regression

slope=0.98±0.03

Particle Density in Receptor Box [#/km3]

Par

ticle

Den

sity

in S

ourc

e B

ox [

#/km

3]

(Forward)

(Bac

kwar

d)

Backward- vs Forward-time ComparisonMass-Conserving Winds

Applications of STILT

1. CH4, N2O, CO at ground stations, from aircraft.

2. CO2 sources and sinks globally

Quantifying Anthropogenic Combustion Contributions to C Budget

7 8 9 10Date (August 1999)

150

200

250model

obs

CO

ppb

log influenceppm/(mol-2s-1)

log influenceppm/(mol-2s-1)

7

8

910

date (Aug. 1999)100

200300

time back [h]

05

01

00

cum

ula

tive

CO

fro

m

em

issi

ons

[p

pb]

Example: CO at Harvard Forest, 1999. Courtesy C. Gerbig, J. Lin

STILT shows that: Accurate measurements from a single tall tower "cover" a remarkably large area.

Influence: ppm per unit flux (mole m-2s-1) at WLEF 396 m

log10

Scott M. Miller (2008) STILT

Methane and Nitrous Oxidein North America: Using an LPDM to Constrain

Emissions

Eric Kortkort@fas.harvard.eduNon-CO2 Workshop

October 23, 2008

Case Study- COBRA-NA 2003

• ~300 flasks measured @ NOAA/Boulder, UND Citation II, 23 May to 28 June 2003

LPDM Model: STILTEmissions: EDGAR-2000Met fields: WRF (AER, 35 km, LPDM outputs, Grell-2)

WRF

June 16, 2003 36.72N,96.94W 609 m AGL

1600

170

0 1

800

190

0 2

000

CH4

0 50 100 150 200 250 300

Flask number

Footprint * Prior Emission Field

(EDGAR)

Result = Enhancement of gas at measurement point due to source

*

Prior Emissions Fields• Methane

– Anthropogenic- EDGAR32FT2000

– Biogenic- Jed Kaplan wetland inventory

• Nitrous Oxide– Anthropogenic-

EDGAR32FT2000– Anthropogenic & Biogenic-

GEIAFIRES

Measurements- Footprint

Results- Methane

Slope: 0.924 ± 0.13

Scaling Factor: 1.08 ± 0.15Note: Prior Emissions Field EDGAR32FT 2000 & JK wetland

STILT LPDM provides directly applicable error estimates for Bayesian inverse.

PDF for model CH4 concentrations for 100 particles from one receptor:

Mean: 1790 ppb; 2 = 38 ppbv

CH

4 (p

pb)

(mod

el,

each

par

ticle

)1

76

0

17

80

18

00

1

82

0

18

40

1

86

0

Measurement Error: Atmospheric Variability in a correlation length (170 km) derived from CO measurement & CO:CH4 ratio:2 = 23 ppbv

WRF/STILT/EDGAR model vs data, with gray and green.

Errors used in fitting are + 38 ppbv for the model, and + 23 ppbv for the measurements

Slope = 0.9

slope = 0.1

EDGAR—2000 confirmed ±10% for CH4 !

This result pertains to urban-industrial sources, which dominate the flight region

N2O Observed vs Model (EDGAR—2000 )

COBRA-2003

Observed N2O (ppbv)

Mo

de

l S

TIL

T/

N2O

(p

pb

v)

US sources of N2O are ~2.5x higher than EDGAR Kort et al., 2008

Carbon Monoxide

EPA Inventory 3!

CO, 3!

International dateline (-180 Longitude)

Summary—Lagrangian particle dispersion models for source assessment• Lagrangian particle dispersion models provide

powerful tools to utilize atmospheric observations to study sources and sinks of atmospheric greenhouse gases.

• Their unique capability to run forward and backward in a consistent numerical framework permits checks on model transport not available any other way.

• Considerable effort is needed to ensure proper formulation of this type of model. Rigorously mass-conserving wind fields are required.

• Applications to aircraft measurements over North America reveal important aspects of greenhouse gas emissions on continental scale.

R component exercise: make a movie; visualization of a random walk#generate a random walk movie; N particles in a random walk along the x axisN=500 ; M = 1000 # 100 particles, up to 1000 time stepsX=rep(0,N) #N particles start at location 0x=matrix( sample(c(-1,1),N*M,replace=T) , ncol=N) # matrix w/ each row random steps of +/-1 X1=rbind( X, apply(x, 2, cumsum)) ; dimnames(X1)= list(as.character(0:M),as.character(1:N))# is this a Markov process ? what about staying in place?#matplot(t(X1)) ##very simple animationX11(); plot(0:1000,apply(X1,1,var)) ### the variance grows linearly with the number of steps (time)X11(); plot(0:1000,apply(X1,1,function(x){sqrt(var(x))}))X11(); plot(sqrt(0:1000),apply(X1,1,function(x){sqrt(var(x))}))##make the moviefor(i in 0:M){ s.n=(paste("000",i,sep="")); ser=substring(s.n,nchar(s.n)-3,nchar(s.n)) if(i%%100==0){yll=range(hist(X1[i+1,],breaks= seq( - 3.5*sqrt(M),3.5*sqrt(M),length.out=120))$counts)} png(file=paste("random_walk_movie/Fig.random_walk_",ser,".png",sep="") )hist(X1[i+1,],breaks= seq(-3.5*sqrt(M),3.5*sqrt(M),length.out=120),xlim=c(-3.5,3.5)*sqrt(M),ylim=yll)dev.off()}

0 200 400 600 800 1000

05

10

15

20

25

30

0:1000

ap

ply

(X1

, 1, f

un

ctio

n(x

) {

s

qrt

(va

r(x)

)})

vs time (#steps)

Histogram of X1[i + 1, ]

X1[i + 1, ]

Fre

qu

en

cy

-100 -50 0 50 100

05

10

15

distance distance

0 200 400 600 800 1000

020

040

060

080

010

00

0:1000

appl

y(X

1, 1

, var

)

distance (#steps)

varia

nce

(d

N )2