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[[File:GoldenMedows.jpg|thumb|right|Field of [[cumulus cloud]]s, which are parameterized since they are too small to be explicitly included within numerical weather prediction]]
{{Main|Parametrization (climate)}}
Some meteorological processes are too small-scale or too complex to be explicitly included in numerical weather prediction models. ''[[Parameterization]]'' is a procedure for representing these processes by relating them to variables on the scales that the model resolves. For example, the gridboxes in weather and climate models have sides that are between {{convert|5|km|mi|0|sp=us}} and {{convert|300|km|mi|-2|sp=us}} in length. A typical [[cumulus cloud]] has a scale of less than {{convert|1|km|mi|1|sp=us}}, and would require a grid even finer than this to be represented physically by the equations of fluid motion. Therefore, the processes that such [[cloud]]s represent are parameterized, by processes of various sophistication. In the earliest models, if a column of air within a model gridbox was conditionally unstable (essentially, the bottom was warmer and moister than the top) and the water vapor content at any point within the column became saturated then it would be overturned (the warm, moist air would begin rising), and the air in that vertical column mixed. More sophisticated schemes recognize that only some portions of the box might [[convection|convect]] and that [[Entrainment (meteorology)|entrainment]] and other processes occur. Weather models that have gridboxes with sizes between {{convert|5|and|25|km|mi|0|sp=us}} can explicitly represent convective clouds, although they need to parameterize [[cloud microphysics]] which occur at a smaller scale.<ref>{{cite journal|url=http://ams.confex.com/ams/pdfpapers/126017.pdf|title=3.7 Improving Precipitation Forecasts by the Operational Nonhydrostatic Mesoscale Model with the Kain-Fritsch Convective Parameterization and Cloud Microphysics|author1=Narita, Masami |author2=Shiro Ohmori |name-list-style=amp |date=2007-08-06|access-date=2011-02-15|journal=12th Conference on Mesoscale Processes}}</ref> The formation of large-scale ([[stratus cloud|stratus]]-type) clouds is more physically based; they form when the [[relative humidity]] reaches some prescribed value.
The amount of solar radiation reaching the ground, as well as the formation of cloud droplets occur on the molecular scale, and so they must be parameterized before they can be included in the model. [[Drag (physics)|Atmospheric drag]] produced by mountains must also be parameterized, as the limitations in the resolution of [[elevation]] contours produce significant underestimates of the drag.<ref>{{cite book|url=https://books.google.com/books?id=lMXSpRwKNO8C&pg=PA56|title=Parameterization schemes: keys to understanding numerical weather prediction models|author=Stensrud, David J.|page=6|year=2007|publisher=Cambridge University Press|isbn=978-0-521-86540-1}}</ref> This method of parameterization is also done for the surface flux of energy between the ocean and the atmosphere, in order to determine realistic sea surface temperatures and type of sea ice found near the ocean's surface.<ref>{{cite book|page=188|title=A climate modelling primer|author1=McGuffie, K. |author2=A. Henderson-Sellers |name-list-style=amp |publisher=John Wiley and Sons|year=2005|isbn=978-0-470-85751-9}}</ref> Sun angle as well as the impact of multiple cloud layers is taken into account.<ref>{{cite book|url=https://books.google.com/books?id=vdg5BgBmMkQC&pg=PA226|author1=Melʹnikova, Irina N. |author2=Alexander V. Vasilyev |name-list-style=amp |pages=226–228|title=Short-wave solar radiation in the earth's atmosphere: calculation, oberservation, interpretation|year=2005|publisher=Springer|isbn=978-3-540-21452-6}}</ref> Soil type, vegetation type, and soil moisture all determine how much radiation goes into warming and how much moisture is drawn up into the adjacent atmosphere, and thus it is important to parameterize their contribution to these processes.<ref>{{cite book|url=https://books.google.com/books?id=lMXSpRwKNO8C&pg=PA56|title=Parameterization schemes: keys to understanding numerical weather prediction models|author=Stensrud, David J.|pages=12–14|year=2007|publisher=Cambridge University Press|isbn=978-0-521-86540-1}}</ref> Within air quality models, parameterizations take into account atmospheric emissions from multiple relatively tiny sources (e.g. roads, fields, factories) within specific grid boxes.<ref>{{cite book|url=https://books.google.com/books?id=wh-Xf0WZQlMC&pg=PA11|pages=11–12|title=Meteorological and Air Quality Models for Urban Areas|author=Baklanov, Alexander, Sue Grimmond, Alexander Mahura|access-date=2011-02-24|year=2009|publisher=Springer|isbn=978-3-642-00297-7}}</ref>
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The horizontal [[Domain of a function|___domain of a model]] is either ''global'', covering the entire Earth, or ''regional'', covering only part of the Earth. Regional models (also known as ''limited-area'' models, or LAMs) allow for the use of finer grid spacing than global models because the available computational resources are focused on a specific area instead of being spread over the globe. This allows regional models to resolve explicitly smaller-scale meteorological phenomena that cannot be represented on the coarser grid of a global model. Regional models use a global model to specify conditions at the edge of their ___domain ([[boundary condition]]s) in order to allow systems from outside the regional model ___domain to move into its area. Uncertainty and errors within regional models are introduced by the global model used for the boundary conditions of the edge of the regional model, as well as errors attributable to the regional model itself.<ref>{{cite book|url=https://books.google.com/books?id=6RQ3dnjE8lgC&pg=PA261|title=Numerical Weather and Climate Prediction|author=Warner, Thomas Tomkins |publisher=[[Cambridge University Press]]|year=2010|isbn=978-0-521-51389-0|page=259}}</ref>
The vertical coordinate is handled in various ways. Lewis Fry Richardson's 1922 model used geometric height (<math>z</math>) as the vertical coordinate. Later models substituted the geometric <math>z</math> coordinate with a pressure coordinate system, in which the [[geopotential height]]s of constant-pressure surfaces become [[dependent variable]]s, greatly simplifying the primitive equations.<ref name="Lynch Ch2">{{cite book|last=Lynch|first=Peter|title=The Emergence of Numerical Weather Prediction|url=https://archive.org/details/emergencenumeric00lync|url-access=limited|year=2006|publisher=[[Cambridge University Press]]|isbn=978-0-521-85729-1|pages=[https://archive.org/details/emergencenumeric00lync/page/n55 45]–46|chapter=The Fundamental Equations}}</ref> This correlation between coordinate systems can be made since pressure decreases with height through the [[Earth's atmosphere]].<ref>{{cite book|author=Ahrens, C. Donald|page=10|isbn=978-0-495-11558-8|year=2008|publisher=Cengage Learning|title=Essentials of meteorology: an invitation to the atmosphere|url=https://books.google.com/books?id=2Yn29IFukbgC&pg=PA244}}</ref> The first model used for operational forecasts, the single-layer barotropic model, used a single pressure coordinate at the 500-millibar (about {{convert|5500|m|ft|abbr=on}}) level,<ref name="Charney 1950">{{cite journal|last1=Charney|first1=Jule|last2=Fjørtoft|first2=Ragnar|last3=von Neumann|first3=John|title=Numerical Integration of the Barotropic Vorticity Equation|journal=Tellus|date=November 1950|volume=2|issue=4|bibcode=1950TellA...2..237C |doi=10.3402/tellusa.v2i4.8607|author-link1=Jule Charney|author-link2=Ragnar Fjørtoft|author-link3=John von Neumann|pages=237|doi-access=free}}</ref> and thus was essentially two-dimensional. High-resolution models—also called ''mesoscale models''—such as the [[Weather Research and Forecasting model]] tend to use normalized pressure coordinates referred to as [[sigma coordinates]].<ref>{{cite web|last=Janjic |first=Zavisa |title=Scientific Documentation for the NMM Solver |url=http://nldr.library.ucar.edu/collections/technotes/asset-000-000-000-845.pdf |publisher=[[National Center for Atmospheric Research]] |access-date=2011-01-03 |author2=Gall, Robert |author3=Pyle, Matthew E. |pages=12–13 |date=February 2010 |url-status=dead |archive-url=https://web.archive.org/web/20110823082059/http://nldr.library.ucar.edu/collections/technotes/asset-000-000-000-845.pdf |archive-date=2011-08-23 }}</ref> This coordinate system receives its name from the [[independent variable]] <math>\sigma</math> used to [[nondimensionalization|scale]] atmospheric pressures with respect to the pressure at the surface, and in some cases also with the pressure at the top of the ___domain.<ref>{{cite book|last=Pielke|first=Roger A.|title=Mesoscale Meteorological Modeling|url=https://archive.org/details/mesoscalemeteoro00srro|url-access=limited|year=2002|publisher=[[Academic Press]]|isbn=978-0-12-554766-6|pages=[https://archive.org/details/mesoscalemeteoro00srro/page/n147 131]–132}}</ref>
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[[Edward Epstein (meteorologist)|Edward Epstein]] recognized in 1969 that the atmosphere could not be completely described with a single forecast run due to inherent uncertainty, and proposed using an [[Ensemble (fluid mechanics)|ensemble]] of [[stochastic process|stochastic]] [[Monte Carlo method|Monte Carlo simulations]] to produce [[arithmetic mean|means]] and [[variance]]s for the state of the atmosphere.<ref>{{cite journal|last=Epstein|first=E.S.|title=Stochastic dynamic prediction|journal=[[Tellus A]]|date=December 1969|volume=21|issue=6|pages=739–759|doi=10.1111/j.2153-3490.1969.tb00483.x|bibcode=1969Tell...21..739E}}</ref> Although this early example of an ensemble showed skill, in 1974 [[Cecil Leith]] showed that they produced adequate forecasts only when the ensemble [[probability distribution]] was a representative sample of the probability distribution in the atmosphere.<ref>{{cite journal|last=Leith|first=C.E.|title=Theoretical Skill of Monte Carlo Forecasts|journal=[[Monthly Weather Review]]|date=June 1974|volume=102|issue=6|pages=409–418|doi=10.1175/1520-0493(1974)102<0409:TSOMCF>2.0.CO;2|bibcode=1974MWRv..102..409L|doi-access=free}}</ref>
Since the 1990s, ''ensemble forecasts'' have been used operationally (as routine forecasts) to account for the stochastic nature of weather processes – that is, to resolve their inherent uncertainty. This method involves analyzing multiple forecasts created with an individual forecast model by using different physical [[parametrization (climate)|parametrizations]] or varying initial conditions.<ref name="HPCens">{{cite web|url=http://www.wpc.ncep.noaa.gov/ensembletraining|author=Manousos, Peter|publisher=[[Hydrometeorological Prediction Center]]|date=2006-07-19|access-date=2010-12-31|title=Ensemble Prediction Systems}}</ref> Starting in 1992 with [[Ensemble forecasting|ensemble forecasts]] prepared by the [[European Centre for Medium-Range Weather Forecasts]] (ECMWF) and the [[National Centers for Environmental Prediction]], model ensemble forecasts have been used to help define the forecast uncertainty and to extend the window in which numerical weather forecasting is viable farther into the future than otherwise possible.<ref name="Toth"/><ref name="ECens"/><ref name="RMS"/> The ECMWF model, the Ensemble Prediction System,<ref name="ECens">{{cite web | url=http://ecmwf.int/products/forecasts/guide/The_Ensemble_Prediction_System_EPS_1.html <!--Added by H3llBot--> | title=The Ensemble Prediction System (EPS) | publisher=[[ECMWF]] | access-date=2011-01-05 | archive-url=https://web.archive.org/web/20101030055238/http://ecmwf.int/products/forecasts/guide/The_Ensemble_Prediction_System_EPS_1.html <!--Added by H3llBot--> | archive-date=2010-10-30}}</ref> uses [[Singular value decomposition|singular vectors]] to simulate the initial [[probability density function|probability density]], while the NCEP ensemble, the Global Ensemble Forecasting System, uses a technique known as [[Bred vector|vector breeding]].<ref name="Toth">{{cite journal|last=Toth|first=Zoltan|author2=Kalnay, Eugenia |title=Ensemble Forecasting at NCEP and the Breeding Method |journal=[[Monthly Weather Review]]|date=December 1997|volume=125|issue=12|pages=3297–3319|doi=10.1175/1520-0493(1997)125<3297:EFANAT>2.0.CO;2|bibcode=1997MWRv..125.3297T|citeseerx=10.1.1.324.3941}}</ref><ref name="RMS">{{cite journal|title=The ECMWF Ensemble Prediction System: Methodology and validation|journal=Quarterly Journal of the Royal Meteorological Society|date=January 1996|volume=122|issue=529|pages=73–119|doi=10.1002/qj.49712252905|bibcode=1996QJRMS.122...73M|author1=Molteni, F. |author2=Buizza, R. |author3=Palmer, T.N. |author3-link=Tim Palmer (physicist) |author4=Petroliagis, T. }}</ref> The UK [[Met Office]] runs global and regional ensemble forecasts where perturbations to initial conditions are
In a single model-based approach, the ensemble forecast is usually evaluated in terms of an average of the individual forecasts concerning one forecast variable, as well as the degree of agreement between various forecasts within the ensemble system, as represented by their overall spread. Ensemble spread is diagnosed through tools such as [[spaghetti plot|spaghetti diagrams]], which show the dispersion of one quantity on prognostic charts for specific time steps in the future. Another tool where ensemble spread is used is a [[meteogram]], which shows the dispersion in the forecast of one quantity for one specific ___location. It is common for the ensemble spread to be too small to include the weather that actually occurs, which can lead to forecasters misdiagnosing model uncertainty;<ref name="ensbook"/> this problem becomes particularly severe for forecasts of the weather about ten days in advance.<ref>{{cite journal|last1=Palmer |first1=T.N. |author1-link=Tim Palmer (physicist) |first2=G.J. |last2=Shutts |first3=R. |last3=Hagedorn |first4=F.J. |last4=Doblas-Reyes |first5=T. |last5=Jung |first6=M. |last6=Leutbecher|title=Representing Model Uncertainty in Weather and Climate Prediction|journal=[[Annual Review of Earth and Planetary Sciences]]|date=May 2005|volume=33|pages=163–193|doi=10.1146/annurev.earth.33.092203.122552|bibcode=2005AREPS..33..163P}}</ref> When ensemble spread is small and the forecast solutions are consistent within multiple model runs, forecasters perceive more confidence in the ensemble mean, and the forecast in general.<ref name="ensbook">{{cite book|url=https://books.google.com/books?id=6RQ3dnjE8lgC&pg=PA261|title=Numerical Weather and Climate Prediction|author=Warner, Thomas Tomkins |publisher=[[Cambridge University Press]]|year=2010|isbn=978-0-521-51389-0|pages=266–275}}</ref> Despite this perception, a ''spread-skill relationship'' is often weak or not found, as spread-error [[Correlation and dependence#Correlation and linearity|correlations]] are normally less than 0.6, and only under special circumstances range between 0.6–0.7.<ref>{{cite web|url=http://www.atmos.washington.edu/~ens/pdf/WEM_WKSHP_2004.epgrimit.pdf|title=Redefining the Ensemble Spread-Skill Relationship from a Probabilistic Perspective|author1=Grimit, Eric P.|author2=Mass, Clifford F.|publisher=[[University of Washington]]|date=October 2004|access-date=2010-01-02|archive-url=https://web.archive.org/web/20081012094121/http://www.atmos.washington.edu/~ens/pdf/WEM_WKSHP_2004.epgrimit.pdf|archive-date=2008-10-12|url-status=dead}}</ref> {{cnspan|The relationship between ensemble spread and [[forecast skill]] varies substantially depending on such factors as the forecast model and the region for which the forecast is made.|date=December 2021}}
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