Editing Numerical weather prediction (section)

==Ensembles==
{{main|Ensemble forecasting}}
[[File:WRF rita spread2.jpg|thumb|280px|''Top'': [[Weather Research and Forecasting model]] (WRF) simulation of [[Hurricane Rita]] (2005) tracks. ''Bottom'': The spread of NHC multi-model ensemble forecast.|alt=Two images are shown. The top image provides three potential tracks that could have been taken by Hurricane Rita. Contours over the coast of Texas correspond to the sea-level air pressure predicted as the storm passed. The bottom image shows an ensemble of track forecasts produced by different weather models for the same hurricane.]]
In 1963, [[Edward Lorenz]] discovered the [[chaos theory|chaotic nature]] of the [[fluid dynamics]] equations involved in weather forecasting.<ref name="Cox">{{cite book|title=Storm Watchers|pages=[https://archive.org/details/stormwatcherstur00cox_df1/page/222 222–224]|year=2002|author=Cox, John D.|publisher=John Wiley & Sons, Inc.|isbn=978-0-471-38108-2|url=https://archive.org/details/stormwatcherstur00cox_df1/page/222}}</ref> Extremely small errors in temperature, winds, or other initial inputs given to numerical models will amplify and double every five days,<ref name="Cox" /> making it impossible for long-range forecasts—those made more than two weeks in advance—to predict the state of the atmosphere with any degree of [[forecast skill]]. Furthermore, existing observation networks have poor coverage in some regions (for example, over large bodies of water such as the Pacific Ocean), which introduces uncertainty into the true initial state of the atmosphere. While a set of equations, known as the [[Liouville's theorem (Hamiltonian)|Liouville equations]], exists to determine the initial uncertainty in the model initialization, the equations are too complex to run in real-time, even with the use of supercomputers.<ref name="HPCens"/> These uncertainties limit forecast model accuracy to about five or six days into the future.<ref name="Klaus">{{cite web|last=Weickmann|first=Klaus|author2=Jeff Whitaker |author3=Andres Roubicek |author4= Catherine Smith  |date=2001-12-01 | url=http://www.esrl.noaa.gov/psd/spotlight/12012001/ | title = The Use of Ensemble Forecasts to Produce Improved Medium Range (3–15&nbsp;days) Weather Forecasts. | publisher=[[Climate Diagnostics Center]] | access-date=2007-02-16|archive-url=https://web.archive.org/web/20100528082602/http://www.esrl.noaa.gov/psd/spotlight/12012001/|archive-date=2010-05-28}}</ref><ref>{{cite journal|last=Chakraborty|first=Arindam|title=The Skill of ECMWF Medium-Range Forecasts during the Year of Tropical Convection 2008|journal=Monthly Weather Review|date=October 2010|volume=138|issue=10|pages=3787–3805|doi=10.1175/2010MWR3217.1|bibcode=2010MWRv..138.3787C|doi-access=free}}</ref>

[[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 &ndash; 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|s2cid=14668576 }}</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 used by 24 ensemble members in the Met Office Global and Regional Ensemble Prediction System (MOGREPS) to produce 24 different forecasts.<ref name="The Met Office ensemble system- MOGREPS">{{cite web | url=http://www.metoffice.gov.uk/research/areas/data-assimilation-and-ensembles/ensemble-forecasting/MOGREPS | title=MOGREPS | publisher=[[Met Office]] | access-date=2012-11-01 | url-status=dead | archive-url=https://web.archive.org/web/20121022215636/http://www.metoffice.gov.uk/research/areas/data-assimilation-and-ensembles/ensemble-forecasting/MOGREPS | archive-date=2012-10-22 }}</ref>

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&ndash;0.7.<ref name="grimit">{{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/20110926204842/http://www.atmos.washington.edu/~ens/pdf/WEM_WKSHP_2004.epgrimit.pdf|archive-date=2011-09-26|url-status=dead}}</ref> 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.<ref name="grimit" />

In the same way that many forecasts from a single model can be used to form an ensemble, multiple models may also be combined to produce an ensemble forecast. This approach is called ''multi-model ensemble forecasting'', and it has been shown to improve forecasts when compared to a single model-based approach.<ref>{{cite journal|url=http://www.emc.ncep.noaa.gov/mmb/SREF/2222289_WAF_Feb-2010.official.PDF|title=Fog Prediction From a Multimodel Mesoscale Ensemble Prediction System|author1=Zhou, Binbin |author2=Du, Jun |volume=25|issue=1|date=February 2010|access-date=2011-01-02|journal=[[Weather and Forecasting]]|page=303|doi=10.1175/2009WAF2222289.1|bibcode=2010WtFor..25..303Z|s2cid=4947206 }}</ref> Models within a multi-model ensemble can be adjusted for their various biases, which is a process known as ''superensemble forecasting''. This type of forecast significantly reduces errors in model output.<ref>{{cite journal|url=http://www.nat-hazards-earth-syst-sci.net/10/265/2010/nhess-10-265-2010.pdf|title=Multimodel SuperEnsemble technique for quantitative precipitation forecasts in Piemonte region|author1=Cane, D. |author2=Milelli, M. |date=2010-02-12|access-date=2011-01-02|journal=Natural Hazards and Earth System Sciences|doi=10.5194/nhess-10-265-2010|bibcode=2010NHESS..10..265C|volume=10|page=265|issue=2|doi-access=free}}</ref>