Influence Functions

Influence functions describe how a unit change in any given element of the state vector (often, a unit change in the flux in a certain area) will change all of the observations. For the instance where the state vector is the fluxes in each grid cell, one might calculate this as a number of observation sites by number of observation times by number of flux times by number of grid cells in the y direction by number of grid cells in the x direction array. Many of the elements of this array will be zero, as no flux can affect a measurement before it. Similarly, if the observation time is much later than the flux time, the influence function will not vary much between sites 1.

There are two approaches to calculating influence functions: one using a forward model and one using an adjoint model.

Foreward Model

The most straightforward method to obtain influence functions is to do many runs of the tracer transport model, one for each element of the state vector. The surface fluxes are those corresponding to a state vector with one element set to one and all the rest zero. Sampling the simulated concentration fields at the time and place of the measurements then produces the influence functions.

It is very easy to run experiments with different observation networks using this method, if the simulated concentration fields are saved. However, the cost of a separate run for each element of the state vector can become quickly prohibitive for high-resolution inversions.

Adjoint Model

Another method is to find the adjoint of the tracer transport model, then do adjoint runs back from each observation. The adjoint model gives the sensitivity of a quantity of interest (a measurement at a particular place and time) to conditions before that (in particular, to the preceeding fluxes). This method requires one run for each measurment at each location, and works well when the total number of measurements is much smaller than the total number of elements in the state vector.

Running experiments with different observation networks with this method is somewhat difficult, since each new measurement requires a new run of the adjoint model.

One method used to avoid finding the adjoint of the full tracer transport model is to release many imaginary particles at the time and place of each measurement, then use the winds from the tracer transport model to move the particles backward in time, finding where the particles would have to have been at each prior time to become part of the measurement. Tracking where and how often these particles interact with the ground then gives the influence functions. This is the approach taken by Lagrangian Particle Dispersion Models, such as the [LPDM] of Uliasz (1994), the Stochastic Time-Inverted Lagrangian Transport [STILT] model, the HYbrid Single-Particle Lagrangian Integrated Trajectory [HYSPLIT] model, and the FLEXible PARTicle dispersion model [FLEXPART].

This method requires relating the change in mole fraction for one of these particles to the change in mole fraction of the measurement: if all the particles composing the measurement experience an increase in mole fraction of 1 ppm, then the measurement will be 1 ppm higher. If only half the particles experience this 1 ppm increase and the other half are unaffected, the measurement will only be 0.5 ppm higher. Extending this, if a single particle experiences an increase in mole fraction of 1 (in arbitrary units), the measurement will be larger by the reciprocal of the number of particles composing the measurement (in the same units).

1

If all the measurements are in the troposphere, “much later” is two to three years. If all the measurements are additionally in the same hemisphere, “much later” is on the order of a month, perhaps less. If the inversion is regional, with a spatial domain that does not cover the whole globe, “much later” is related to how long it takes air parcels to leave the domain.

References

LPDM

Uliasz, M. (1993). “The atmospheric Mesoscale Dispersion Modeling System”. Journal of Applied Meteorology, 32 (1), 139–149. Retrieved 2016-07-15, from https://journals.ametsoc.org/doi/abs/10.1175/1520-0450%281993%29032%3C0139%3ATAMDMS%3E2.0.CO%3B2 doi:10.1175/1520-0450(1993)032<0139:TAMDMS>2.0.CO;2

Uliasz, M. (1994). “Lagrangian particle dispersion modeling in mesoscale applications”. Environ Model: Comput Methods and Softw for Simulat Environ Pollut and Its Adverse Effects (CMP), 2 , 71–102. Retrieved from http://indico.ictp.it/event/a02274/contribution/22/material/0/0.pdf

STILT

http://stilt-model.org/index.php/Main/HomePage

Lin, J.C., C. Gerbig, S.C. Wofsy, A.E. Andrews, B.C. Daube, K.J. Davis, and C.A. Grainger, “A near-field tool for simulating the upstream influence of atmospheric observations: The Stochastic Time-Inverted Lagrangian Transport (STILT) model”. Journal of Geophysical Research-Atmospheres, (2003) 108(D16): 4493, doi:10.1029/2002JD003161.

HYSPLIT

https://ready.arl.noaa.gov/HYSPLIT.php

Stein, A.F., R.R. Draxler, G.D. Rolph, B.J. Stunder, M.D. Cohen, and F. Ngan, 2015: “NOAA’s HYSPLIT Atmospheric Transport and Dispersion Modeling System”. Bull. Amer. Meteor. Soc., 96, 2059–2077, doi:10.1175/BAMS-D-14-00110.1

FLEXPART

https://www.flexpart.eu/

Ignacio Pisso, Espen Sollum, Henrik Grythe, Nina I. Kristiansen, Massimo Cassiani, Sabine Eckhardt, Delia Arnold, Don Morton, Rona L. Thompson, Christine D. Groot Zwaaftink, Nikolaos Evangeliou, Harald Sodemann, Leopold Haimberger, Stephan Henne, Dominik Brunner, John F. Burkhart, Anne Fouilloux, Jerome Brioude, Anne Philipp, Petra Seibert, and Andreas Stohl (2019), “The Lagrangian particle dispersion model FLEXPART version 10.4”. Geosci. Model Dev., 12, 4955–4997, doi:10.5194/gmd-12-4955-2019.

Seibert, P., & Frank, A. (2004, 1). “Source-receptor matrix calculation with a lagrangian particle dispersion model in backward mode.” Atmospheric Chemistry and Physics, 4 (1), 51–63. doi:10.5194/acp-4-51-2004