Adjoint and Data Assimilation Working Group
- 1 Contact information
- 2 Historical Development
- 3 Forward Model Code
- 4 Code Versions, Bug Fixes and Developments
- 5 Summary of Main Adjoint Code Supported Features
- 6 Resources
- 7 Distribution and Use
- 8 Crediting GEOS-Chem adjoint developers
- 9 Current GEOS-Chem Adjoint Research Projects (please add yours!)
- 10 Publications
|Adjoint Working Group Co-Chairs||Kevin Bowman and Dylan Jones|
|Adjoint Model Scientist||Daven Henze|
|GC adjoint support team||Yanko Davila, Nicolas Bousserez|
|Adjoint Working Group email email@example.com|
|To subscribe to email list||Send email to firstname.lastname@example.org|
|To unsubscribe from email list||Send email to email@example.com|
Original work on the adjoint of GEOS-Chem v6 began in 2003, focusing on the adjoint of the offline aerosol simulation. By 2005, the adjoint was expanded to include a tagged CO simulation and a full chemistry simulation; an adjoint of GEOS-Chem v7 was also developed in the following years. Each of these branches of the adjoint code were been constructed in a hybrid fashion using a combination of automatic differentiation software (TAMC, KPP) and manual coding of both discrete and continuous adjoints. They shared many common elements yet had unique features for different applications.
During the summer of 2009, the existing branches were merged and updated to bring the adjoint into alignment with the latest release of GEOS-Chem, v8-02-01. This merged adjoint model is now the standard adjoint code into which all further development efforts will be placed.
Forward Model Code
The forward model on which the adjoint is based originally corresponded to GEOS-Chem v8-02-01. It was subsequently updated as follows:
- KPP solver for gas-phase chemistry (as in GCv8-02-03)
- Implement Bond 2007 BC/OC emissions (as in GCv8-02-02)
- Apply bug fixes from GCv8-02-02 listed here
- Apply bug fixes from GCv8-02-03 listed here
- Apply bug fixes from GCv8-02-04 listed here
All bug fixes and model updates were previous listed at the top of inverse_driver.f. We have now switched to documenting the code development cycle here in the wiki, see the following section.
Code Versions, Bug Fixes and Developments
Current GEOS-Chem adjoint version released
- GEOS-Chem_Adjoint_v35 (You will download this version when you check out.)
Previous GEOS-Chem adjoint versions released
Summary of Main Adjoint Code Supported Features
- Meteorological fields
- GEOS-3 needs testing
- model resolution
- 4 x 5
- 2 x 2.5
- Nested Asia and NA
- Forward model processes
- PBL mixing
- dry deposition
- wet deposition
- strat / trop exchange with LINOZ and new GMI strat chem (v9-01-03)
- NOy up fluxes (now replaced with new GMI strat chem)
- inorganic aerosol thermodynamics with RPMARES
- inorganic aerosol thermodynamics with ISORROPIA in progress
- sulfate chemistry
- SOA, Dust, sea salt needs doing
- aerosol surface area feedbacks needs updating
- aerosol optical feedbacks needs doing
- all standard emissions included
- Simulation modes
- full chemistry
- tagged CO
- tagged Ox
- offline aerosols (for BC and dust only)
- Observational Operators
- MOPITT CO column
- SCIAMACHY CO column
- AIRS CO column
- IMPROVE BC
- CASTNet (NH4+) needs updating
- GOME / SCIAMACHY NO2 column needs updating
- using KNMI retrieval (Henze)
- using Dalhousie retrieval (Shim)
- SCIAMACHY/OMI NO2
- using Dalhousie retrieval (Bousserez, Padmanabhan)
- TES NH3
- TES O3
- GOSAT CO2
- MLS O3 and TES CO2 in progress
- Control parameters
- Initial Conditions scaling factors (linear or log)
- Emissions scaling factors (linear or log)
- NH3, primary BC/OC, SO2: anthropogenic, natural, bioburn, biomass, ship
- NOx: soil, aircraft, anthropogenic, biofuel, bioburn
- Lightning NOx: injection height, yield in progress
- all other gas-phase tracers: anthropogenic, biofuel, bioburn
- Adjoint sensitivities
- w.r.t. all implemented control parameters
- w.r.t Reaction Rate Parameters
- w.r.t all emissions
- of AQ attainment metrics needs updating
- of spatiotemporally averaged species concentrations (e.g., arctic O3)
- Inverse Hessian approximation
- off-diagonal covariance matrices needs updating
- 3D-Var needs updating
Features may be qualified as:
- needs testing: an implemented feature that we haven't fully used yet
- needs updating: a feature developed with a previous branch that has yet to be updated to GEOS-Chem v8 and the merged adjoint
- needs doing: a feature nobody has tackled the adjoint of yet
- in progress: a feature currently under development
- in pipeline: a feature which has been submitted and awaiting integration into the CVS repository
Primary code developers
Monika Kopacz, Kumaresh Singh, Changsub Shim, Daven Henze
Adjoint model lead scientist
A user's guide v35 is available. User's Guide v35
Previous version v32-v33 available at User's Guide v32
Meemong Lee has created a detailed flowchart of the inverse model code structure. http://adjoint.colorado.edu/~daven/gcadj_std/flowchart.pdf
Some IDL and MATLAB routines for plotting benchmark results. http://adjoint.colorado.edu/~daven/gcadj_std/tools.tar.gz
Background papers and presentations
Several articles and presentations (including a GC adjoint modeling clinic overview from IGC5) providing background information about adjoints. http://adjoint.colorado.edu/~daven/gcadj_std/adj_articles.tar.gz
Distribution and Use
Code for the adjoint is distributed through GITLAB, a web interface connected to a GIT server located at adjoint.colorado.edu. You can access GITLAB at http://adjoint.colorado.edu:8080 after your account is created. Here is our Quick Start Guide.
Even if your office mate has a copy of the code, the best way to obtain the model is to get an account for yourself and download a version from the repository. So please do not copy code directly from others or pass the code along to third parties. This vastly helps with tracking developments and keeping up with model updates.
Use of the adjoint model code follows standard practice for GEOS-Chem. It is expected that any developments that come of individual applications based on this community model will eventually be given back to the community by incorporation of new developments into the standard adjoint code. New development should be submitted to Daven Henze for inclusion in the standard adjoint model code.
Using GIT gives the users the ability to change the code and commit their changes without affecting the main repository hosted at adjoint.colorado.edu. Users can work with their tuned versions of the code and even create their own tags because GIT acts as a local repository. When ready to submit your update to the comunity just send Daven Henze your patch and we'll take care of the rest.
Quick guide to GIT
Useful GIT commands:
Status of project vs the current repository:
Check difference of files (differences have colors for easy reading)
git diff --word-diff=color <wildcard> [<wildcard>] <path>/foo_mod.f
Checkout specific version
git checkout <wildcard>
Replacing a file with the newest version from the repository:
git checkout origin/master -- <path>/foo_mod.f
Merging changes in a file: (Reference)
git merge -m <wildcard>
git commit -a
Tagging a version
git tag -a TAGNAME
Deleting a tag
git tag -d TAGNAME git push origin :refs/tags/TAGNAME
Saving changes to repository
Saving tags to repository
git push --tags
List the history of a file:
git log -- <path>/foo_mod.f
Add a file to the repository
git add <file_name>
Delete a file from the repository
git rm <file_name>
Determine current version
git show HEAD [ | grep commit]
Download remote changes, rewind your local branch, then replays all your changes over the top of your current branch one by one, until you’re all up to date.
git pull --rebase
There are several wildcards that you can use on git for example:
"origin/master" - Latest version on the repository
"HEAD" - Latest version as of your last download
"v33i" - Specific TAG, find all tag names on GitLab
"32d5c926e" - Specific COMMIT, find all commit numbers on GitLab
Backward compatibility (CVS)
For people using old version of the code we still have active our CVS repository, but note that the latest version on CVS is v33i-patch2. Here you can find our Quick Guide to CVS
Crediting GEOS-Chem adjoint developers
We aim to make distribution of adjoint model code as immediate as possible. A consequence is that many features may not yet be publicly documented. Therefore, giving code developers due credit is of utmost importance.
Authors of new additions to the standard code should be offered co-authorship on the first round of presentations and publications to come of their development. Features currently falling in this category and their developers are:
- (v35) Deposition based cost function. Fabien Paulot, Harvard; Daven Henze and Yanko Davila, CU Boulder.
- (v34) Implementation of the sensitivity to reaction rate constants. Developers: Hyungmin Lee, CU Boulder; Thomas ; Fabien Paulot (Harvard); Daven Henze, CU Boulder and Yanko Davila, CU Boulder.
- (v34) ISOROPIA II adjoint. Developer: Shannon Capps, EPA.
- (v34) Off-diagonal covariance error matrices implementation. Developers: Nicolas Bousserez, CU Boulder; Kumaresh; Yanko Davila, CU Boulder.
- (v32) MOPITT v5 CO observation operator. Developer: Zhe Jiang, University of Toronto.
- (v32) Dust adjoint. Developer: Xiaoguang (Richard) Xu, University of Nebraska Lincoln.
- (v32) Black carbon offline aerosol adjoint. Developer: Yuhao Mao (UCLA).
- (v32) Nested full chemistry adjoint. Developers: Zhe Jiang, University of Toronto; Daven Henze, CU Boulder.
- (v32) Stratospheric production / loss rate sensitivities. Developer: Hyung-Min Lee, CU Boulder.
- (v32) CH4 adjoint. Developer: Kevin Wecht, Harvard University
- (v29) LIDORT. Developer: Daven Henze, University of Colorado Boulder. Collaborator: Rob Spurr.
- (v28) CO2 adjoint. Developer: Daven Henze, University of Colorado Boulder. Collaborators: Ray Nassar, Kevin Bowman, Dylan Jones.
Overall, if you have any questions about authorship, even for a conference presentation, please contact Daven Henze.
Current GEOS-Chem Adjoint Research Projects (please add yours!)
|User Group||Description||Contact Person|
|CU Boulder||Aerosol precursors, CO2, O3; general adjoint code maintenance||Daven Henze|
|CU Boulder||Inverse modeling/optimization; CO2 fluxes inversion (CMS project); general adjoint code maintenance||Nicolas Bousserez|
|CU Boulder||Sensitivity of nitrate deposition over Antarctica including stratospheric tracers||Hyung-Min Lee|
|Harvard||Methane||Kevin Wecht, wecht [at] fas.harvard.edu|
|Harvard||Smoke Emissions in SE Asia||Patrick Kim, kim68 [at] fas.harvard.edu|
|Harvard||Methane from GOSAT||Alex Turner, aturner [at] fas.harvard.edu|
|Purdue University||Methane (SICAMACHY, AIRS and IASI)||Jinyun Tang|
|MIT||Aircraft emissions||Jamin Koo|
|MIT||Air quality, aircraft emissions and sensitivities||Bogdan V. Constantin|
|Princeton||BC sensitivities, general adjoint code development||Monika Kopacz, mkopacz [at] princeton.edu|
|Dalhousie University||Lightning NOx emissions and impact on tropical ozone using the adjoint||Nicolas Bousserez (now at CU-Boulder)|
|Dalhousie University||Surface NOx emissions inversion using SCIAMACHY/OMI NO2 measurements||Akhila Padmanabhan akhila [at] dal.ca; Nicolas Bousserez  (now at CU-Boulder)|
|JPL||Microwave Limb Sounder (MLS) Ozone assimilation||Meemong Lee|
|JPL||TES ozone assimilation/attribution of ozone radiative forcing||Kevin Bowman|
|University of Edinburgh||Quantifying the impact of boreal forest fires on tropospheric oxidants over the Atlantic||Mark Parrington|
|US EPA||Integration with economic models for future emission inventory scenario development||Farhan Akhtar|
|Peking University||Satellite constraints on VOC emissions||May Fu|
|Purdue University||Feedback between terrestrial ecosystem processes and atmospheric co2 signals||Qing Zhu|
|Purdue University||Feedback between aquatic ecosystem processes and atmospheric CH4 signals||Zeli Tan|
|University of Toronto||Sensitivity of ozone and reactive nitrogen to precursor emissions||Thomas Walker|
|University of Toronto||Adjoint analysis for carbon monoxide||Zhe Jiang|
|University of Toronto||Sensitivity of ozone and CO to precursor emissions||Cynthia Whaley|
|Georgia Tech / US EPA||ISORROPIA adjoint development; NH3 assimilation; cloud droplet sensitivities||Shannon Capps|
|Peking University||Source attributions of tropospheric ozone over North China||Jintai Lin|
|University of Wollongong||Sensitivity of ozone and adjoint analysis of CO over Australasia.||Rebecca Buchholz|
|University of Toronto||CO2 assimilation & transport model bias estimation||Martin Keller|
|University of Wisconsin||CO2 assimilation and forecast & Temperature profile retrieval||Wenguang Bai|
|Dalhousie University||Sensitivity of global PM2.5-induced mortality to emissions||Colin Lee|
|University of Leicester (UK)||Top-down estimates of Amazon isoprene emissions||Michael Barkley|
|Anyang University||Aerosol emission modeling in East Asia||Youn Seo Koo|
|Nanjing University||Inverse modeling of terrestrial ecosystem carbon flux||Hengmao Wang|
|Tsinghua University||Nested-gird simulations with the adjoint model||Nan Yang|
|University of Minnesota||Inverse modeling of VOC sources based on TES and IASI measurements||Kelley Wells, Dylan Millet|
|University of Minnesota & CU Boulder||Inverse modeling of N2O sources||Dylan Millet, Daven Henze|
|Tsinghua University||Inverse modeling of anthropogenic emissions over East Asia||Qiang Zhang|
|UCLA||Constrain black carbon emission||Ling Qi|
- In press or sumbitted
- Jiang, Z., D. B. A. Jones, H. M. Worden, M. N. Deeter, D. K. Henze, J. Worden, and K. W. Bowman, Quantifying the impact of model biases in convective transport on inferred CO source estimates using multi-spectral CO retrievals from MOPITT, JGR, in press.
- Kharol, S., R. V. Martin, S. Philip, S. Vogel, D. K. Henze, D. Chen, Y. Wang, Q. Zhang, C. L. Heald, Persistent Sensitivity of Asian Aerosol to Emissions of Nitrogen Oxides, GRL, in press.
- Koo, J., Q. Wang, D. K. Henze, I. A. Waitz, S.R.H. Barrett, Spatial sensitivities of human health risk to intercontinental and high-altitude pollution, Atmos. Environ, in press.
- L. Zhu, D. K. Henze, K. E. Cady-Pereira, M. W. Shephard, M. Luo, R. W. Pinder, J. O. Bash, G. Jeong, Constraining U.S. ammonia emissions using TES remote sensing observations and the GEOS-Chem adjoint model, JGR, in press.
- Paulot, F., D. J. Jacob and D. K. Henze, Sources and processes contributing to nitrogen deposition in biodiversity hotspots worldwide, submitted.
- Singh, K., A. Sandu, Variational Chemical Data Assimilation with Approximate Adjoints, submitted.
- Xu, X., J. Wang, D. K. Henze, W. Qu, M. Kopacz, Constraints on Aerosol Sources Using GEOS-Chem Adjoint and MODIS Radiances, and Evaluation with Multi-sensor (OMI, MISR) data, submitted.
- Bowman, K. W., and D. K. Henze, Attribution of direct ozone radiative forcing to spatially-resolved emissions, Geophys. Res. Lett., 39, L22704, doi:10.1029/2012GL053274. GRL Editor's highlight, Science Daily, KGNU radio interview.
- Henze, D. K., D. T. Shindell, F. Akhtar, R. J. D. Spurr, R. W. Pinder, D. Loughlin, M. Kopacz, K. Singh, and C. Shim, Spatially refined aerosol direct radiative forcing efficiencies, Environ. Sci. Technol., 46, 9511 - 9518, dx.doi.org/10.1021/es301993s.
- Karydis, V. A., S. L. Capps, R. H. Moore, A. Russell, D. K. Henze, and A. Nenes, Using a global aerosol model adjoint to unravel the footprint of spatially-distributed emissions on cloud droplet number and cloud albedo, Geophys. Res. Lett., 39, L24804, doi:10.1029/2012GL053346.
- Parrington, M., P. I. Palmer, D. K. Henze, D. W. Tarasick, E. J. Hyer, R. C. Owen, C. Clerbaux, K. W. Bowman, M. N. Deeter, E. M. Barratt, P.-F. Coheur, D. Hurtmans, M. George, and J. R. Worden, The influence of boreal biomass burning emissions on the distribution of tropospheric ozone over North America and the North Atlantic during 2010, Atmos. Chem. Phys., 12, 2077-2098, online.
- Paulot, F., D. K. Henze, and P. O. Wennberg, Impact of the isoprene photochemical cascade on tropical ozone, Atmos. Chem. Phys., 12, 1307-1325, online.
- Singh, K. and A. Sandu, 2012: Variational chemical data assimilation with approximate adjoints. Computers and Geosciences, 40, 10-18.
- Turner, A., D. K. Henze, R. V. Martin, and A. Hakami, The spatial extent of source influences on modeled column concentrations of short-lived species, Geophys. Res. Lett., 39, L12806, doi:10.1029/2012GL051832.
- Walker, T., D. B. A. Jones, M. Parrington, D. K. Henze, L. T. Murray, J. W. Bottenheim, K. Anlauf, J. R. Worden, K. W. Bowman, C. Shim, K. Singh, M. Kopacz, D. W. Tarasick, J. Davies, P. von der Gathen, and C. C. Carouge, Impacts of midlatitude precursor emissions and local photochemistry on ozone abundances in the Arctic, J. Geophys. Res.,117, D01305 doi:10.1029/2011JD016370.
- Wang, J., X. Xu, D. K. Henze, Q. Ji, S.-C. Tsay, J. Huang, Top-Down Estimate of Dust Emissions through Integration of MODIS and MISR Aerosol Retrievals with the GEOS-Chem adjoint model, Geophys. Res. Lett., 39, L08802.
- Wecht, K. J., D. J. Jacob, S. C. Wofsy, E. A. Kort, J. R. Worden, S. S. Kulawik, D. K. Henze, M. Kopacz, and V. H. Payne, Validation of TES methane with HIPPO aircraft observations: implications for inverse modeling of methane sources, Atmos. Chem. Phys., 12, 1823-1832, online.
- Jiang, Z., D. B. A. Jones, M. Kopacz, J. Liu, D. K. Henze, and C. Heald, Quantifying the impact of model errors on top-down estimates of carbon monoxide emissions using satellite observations, J. Geophys. Res., 116, D15306, doi:10.1029/2010JD015282.
- Jiang, Z., D. B. A. Jones, M. Kopacz, J. Liu, D. K. Henze, and C. Heald (2011), Quantifying the impact of model errors on top-down estimates of carbon monoxide emissions using satellite observations, J. Geophys. Res., 116, D15306, doi:10.1029/2010JD015282.
- Kopacz, M., D. L. Mauzerall, J. Wang, E. M. Leibensperger, D. K. Henze, and K. Singh, Origin and radiative forcing of black carbon transported to the Himalayas and Tibetan Plateau, Atmos. Chem. Phys., 11, 2837-2852 online.
- Kopacz, M., D. J. Jacob, J. A. Fisher, J. A. Logan, L. Zhang, I. A Megretskaia, R. M. Yantosca, K. Singh, D. K. Henze, J. P. Burrows, M. Buchwitz, I. Khlystova, W. W. McMillan, J. C. Gille, D. P. Edwards, A. Eldering, V. Thouret, and P. Nedelec (2010), Global estimates of CO sources with high resolution by adjoint inversion of multiple satellite datasets (MOPITT, AIRS, SCIAMACHY, TES), Atoms. Chem. Phys., 10, 855-876, online.
- Kopacz, M., D.J. Jacob, J.A. Fisher, J. A. Logan, L. Zhang, I. A. Megretskaia, R. M. Yantosca, K. Singh, D. K. Henze, J. P. Burrows, M. Buchwitz, I. Khlystova, W. W. McMillan, J. C. Gille, D. P. Edwards, A. Eldering, V. Thouret, and P. Nedelec (2010): Global estimates of CO sources with high resolution by adjoint inversion of multiple satellite datasets (MOPITT, AIRS, SCIAMACHY, TES), Atmos. Chem. Phys., 10, 855-876. http://www.atmos-chem-phys.net/10/855/2010/acp-10-855-2010.
- Parrington, M., P. I. Palmer, D. K. Henze, D. W. Tarasick, E. J. Hyer, R. C. Owen, C. Clerbaux, K. W. Bowman, M. N. Deeter, E. M. Barratt, P.-F. Coheur, D. Hurtmans, M. George, and J. R. Worden (2012), The influence of boreal biomass burning emissions on the distribution of tropospheric ozone over North America and the North Atlantic during 2010, Atmos. Chem. Phys., 12, 2077-2098
- Singh, K., Jardak, M., Sandu, A., Bowman, K., Lee, M., and Jones, D. (2010): Construction of non-diagonal background error covariance matrices for global chemical data assimilation, Geosci. Model Dev. Discuss., 3, 1783-1827, doi:10.5194/gmdd-3-1783-2010. http://www.geosci-model-dev-discuss.net/3/1783/2010/gmdd-3-1783-2010.html
- Eller, P., K. Singh, A. Sandu, K. Bowman, D. K. Henze, and M. Lee (2009), Implementation and evaluation of an array of chemical solvers in a global chemical transport model, Geosci. Mod. Devel., 2, 185-207. online
- Henze, D. K., J. H. Seinfeld and D. T. Shindell, (2009), Inverse modeling and mapping U.S. air quality influences of inorganic PM2.5 precursor emissions with the adjoint of GEOS-Chem, Atoms. Chem. Phys., 9, 5877-5903, online.
- Kopacz, M., D. J. Jacob, D. K. Henze, C. L. Heald, D. G. Streets, and Q. Zhang (2009), A comparison of analytical and adjoint Bayesian inversion methods for constraining Asian sources of CO using satellite (MOPITT) measurements of CO columns, J. Geophys. Res., 114, D04305, doi:10.1029/2007JD009264.
- Pye, H. O. T., H. Liao, S. Wu, L. J. Mickely, D. J. Jacob, D. K. Henze, and J. H. Seinfeld (2009), Effect of changes in climate and emissions on future sulfate-nitrate-ammonium aerosol levels in the United States, J. Geophys. Res., 114, D01205, doi:10.1029/2008JD010701.
- Zhang, L., D. J. Jacob, M. Kopacz, D. K. Henze, K. Singh, and D. A. Jaffe (2009), Intercontinental source attribution of ozone pollution at western U.S. sites using an adjoint method, Geophys. Res. Lett., 36, L11810, doi:10.1029/2009GL037950.
- Henze, D. K., A. Hakami and J. H. Seinfeld (2007), Development of the adjoint of GEOS-Chem, Atmos. Chem. Phys., 7, 2413-2433.
- Adjoint inversion of CO sources using combined MOPITT, SCIAMACHY and AIRS CO columns, presented by Monika Kopacz at the COSPAR Scientific Assembly, Montreal, July 18, 2008. http://acmg.seas.harvard.edu/presentations/powerpoints/mak2008/COSPAR_MKopacz_July2008.ppt
- Singh, K., P. Eller, A. Sandu, D. K. Henze, K. Bowman, M. Kopacz, and M. Lee (2009), Towards the construction of a standard geos-chem adjoint model, ACM High Performance Computing Conference.
- Kopacz, M., Mauzerall, D.L., Leibensperger, E.M., Wang, J., Henze, D.K., Singh, K., Shim, C. Identifying the origin and estimating the radiative forcing of BC in the Himalayas: an analysis using the global GEOS-Chem adjoint model, European Geophysical Union meeting, Vienna, May 4, 2010.
- Kopacz, M., Jacob, D.J., Fisher, J.A., Logan, J.A., Zhang, L., Megretskaia, I.A., Yantosca, R.M., Singh, K., Henze, D.K., Burrows, J.P., Buchwitz, M., Khlystova, I., McMillan, W.W., Gille, J.C., Edwards, D.P., Eldering, A., Thouret, V., Nedelec, P. Global estimates of CO sources with high resolution by adjoint inversion of multiple satellite datasets (MOPITT, AIRS, SCIAMACHY, TES), European Geophysical Union meeting, Vienna, May 7, 2010.
- Tang, J., Zhuang, Q. and Xiong, X. (2010), 4D-Var inversion of atmospheric methane fluxes by assimilating SCIAMACHY and AIRS satellite retrievals, AGU, Dec. 18, 2010, http:/web.ics.purdue.edu/~tang16/agu2010_tang.ppt
- Bousserez, N., R. V. Martin, K. W. Bowman, D.K. Henze, M. Kopacz, K. Singh, C. Shim, C. Wespes, Improving the lightning NOx source using satellite observations: a 4D-var analysis approach, AGU, Dec., 2010, http://myweb.dal.ca/nc689777/AGU_liNOx_poster_final.pdf