This webpage gives a basic description of how meteorological models generate their predictions, with an emphasis on model errors and the influence of model resolution.  It is intended to provide a framework for discussions of weather prediction models and their errors.  Additional information on errors of particular importance to BLIPMAP predictions is given in the Model Notes webpage.

The Grid

      Numerical meteorological models produce predictions by solving equations at grid points created by dividing the atmosphere into cells and forecasting the average of each predicted variable over that cell.  Generally cells are regularly spaced in the horizontal but vary in the vertical, with vertical spacings becoming smaller near the surface since atmospheric conditions change more rapidly with height there.  Similarly, vertical cell spacings in general are much smaller than horizontal spacings since conditions change much more rapidly in the vertical than in the horizontal.  (See the Grid Cell Diagram) 

The Equations of Motion

      A meterological model produces forecasts by solving the "equations of motion" for a fluid, which are derived from fundamental physical laws: conservation of mass (both of air and moisture), conservation of momentum (Newton's laws), and conservation of thermal energy (thermodynamics). These equations predict the change in variables such as temperature or velocity resulting from the physical phenomena which affect them. These equations strongly interact with each other - for example, a change in thermal energy (affecting temperature) will change atmospheric density (affecting mass), in turn changing pressure differences (affecting velocity). These feedback interactions tend to counter a forced change, e.g. to oppose an atmospheric heating by introducing a cooling effect. If forcing were to remain constant then eventually the feedbacks would produce an "equilibrium" in which changes with time would become increasingly smaller - but the forcing of the atmosphere is never constant, as solar heating is constantly changing, so the atmosphere is in a state of constantly adjusting "quasi-equilibrium".

Solution Methodology

      What is being solved is a set of "Partial Differential Equations", where the PDEs predict the time-change of a variable based upon conditions in each central cell and its horizontal/vertical neighbors.  Those who have taken calculus will remember that differential equations only apply to "infinitesimally" small spacings so we are assuming that a "finite difference" over the finite grid spacing is a reasonable approximation to a differential.  The equations are solved in a "marching-forward-in-time" fashion, starting from the assumed 3D initial conditions (which are usually based on available observations at that time), and predicting how the variables change at each time step at each grid point to give forecasts at a given time.  PDEs are solved for the "prognostic variables" of temperature, humidity, the three components of velocity, condensed moisture (clouds), etc.  The model also calculates "diagnostic" variables which depend solely upon conditions at the present time (so they depend upon the prognostic variables but do not have to be solved in a "time-marching" fashion).  Since all the equations are inter-related, an error in any one will to some degree affect the others.  These PDEs are not empirical equations, since they are fundamentally derived from well-established physical laws of conservation of energy and momentum and mass.

Model Initialization

      Since the equations of motion predict changes there must be an initially prescribed 3D atmosphere for them to start from. This is obtained from atmospheric observations (baloon, satellite, and aircraft data) at a limited number of points which are then interpolated to complete the 3D grid. There will of course be errors in the interpolated initial state (it is not a solution of the equations of motion), so initially the equations predict relatively large changes as they evolve towards the equation of motion quasi-equilibrium. Forecasts during this "spinup" period are subject to large errors and should be taken with a grain of salt, since they reflect the initial "interpolated" guess. Spinup inaccuracies decrease with time - within the BL, the minimum-to-maximum spinup time is roughly 1-to-4 hours, the longer time being for larger grid spacings and larger initial errors (spinup times are longer above the BL).

Overall Prediction Accuracy

      Forecast accuracy of the many parameters predicted by a meteorological model can be generally ordered, from most accurate to least accurate, as:  (1) Winds,  (2) Thermal parameters,  (3) Moisture parameters,  (4) Cloud parameters,  (5) Rainfall. 

Model Errors

      The accuracy of the a model depends on many factors, which can be roughly grouped as:

• Time Step Errors: 
        Since the equations represent the change over an "infinitesimal" time but the actual time step is finite, errors will occur with larger time steps creating larger errors.  The actual time step used is, like much else in numerical modeling, a compromise between the ideal and the practical.
  
  
• Initial Condition Errors: 
        The model requires "Initial Conditions" to start from, needing "knowledge" of the full three-dimensional atmosphere i.e. all prognostic variables at every grid point!  This is obtained by combining available observations - notably, but not limited to, observed soundings - with previous model predictions to produce an initial "analysis".  This analysis must be done carefully while respecting the dynamic constraints of the equations, i.e. is not a matter of simple interpolation of observations, since a large error introduced between observing points would cause large and unrealistic changes immediately after startup as the equations tried to adjust to an unrealistic initial imbalance.  Correct initial conditions can be important to model predictions, but are less important at later times and closer to the surface.  Initial conditions are most important for predicting the progress of disturbances above the BL, such as frontal movement.
  
  
• Lateral Boundary Condition Errors: 
        A 20 km resolution model cannot cover the entire world, so such a "fine-mesh" model is "nested" inside a "coarse-mesh" model, having larger-spacing and larger domain, which provides "neighboring" cells along the lateral (outside) boundary of the fine-mesh model.  The coarse-grid model typically covers the entire globe (though sometimes there will be multiple "nestings" to get to the global model).  Errors in the lateral boundary conditions will, over time, reach further into the fine-mesh grid so lateral BC errors can be reduced by increasing the size of the fine-mesh domain - but of course that takes additional grid points and computational time and given finite constraints on each the choice of a lateral boundary is always a trade off. The model domain used by the RAP model can be viewed here.  By comparison, the NAM grid is much larger, reaching from the North Pole to the Equator!
  
  
• Surface Boundary Condition Errors: 
        The model must know the type of surface the atmosphere is interacting with since its roughness, vegetation type, soil type and water content, etc. all affect model predictions.  This is the one true boundary the atmosphere has and creates the "Boundary Layer" (BL) so of course the predicted BL is especially affected by surface boundary errors!  A major difficulty is that "average" conditions for all of surface quantities must be known for each of the surface model grid cells while in reality these quantities vary over much smaller scales so trying to, first, know what the actual surface consists of and, secondly, create a meaningful average for each grid cell are both subject to much error.  Additionally, certain variables such as vegetation and soil moisture vary with time.  Further, trying to predict soil moisture in detail would require calculations as intensive as that for the atmosphere itself, so greatly simplified equations are used instead.
  
  
• Inadequate Spatial Resolution Errors: 
        The use of differential equations assumes/requires that a feature is "resolved" by the grid resolution.  If the effect of a lake or mountain ridge or whatever upon the atmosphere is to be predicted, the model must adequately "know" about the existence of that lake or ridge or whatever.  Realistic resolution requires a minimum of four grid points inside such a feature.  Modelers keep trying, within the bounds of available computer power, to use finer and finer grid resolutions since with present grid spacings there is still much that is not being resolved, particularly if the forcing is controlled by topography.  There are also many atmospheric features such as convergence-created upward motions which are smaller than can be resolved with present model grids and so are not well predicted.  This error can be thought of as error that occurs because the PDE equations assume changes over an infinitesimally small distance whereas the model can only estimate changes over a finite distance - if in reality changes occurs over a smaller distance than the grid can effectively resolve, the finite-difference equations then cannnot accurately represent the actual differences existing in the atmosphere.
  
  
• Model "Noise" Errors: 
        One result of the limited resolution resulting from use of a finite-spacing grid is that model "noise" develops, particularly at the smallest resolveable scales.  Physically this is because energy in the atmosphere tends to be generated at relatively large scales and then break down into successively smaller eddies.  The "differential" PDEs can and do simulate this behavior but only when resolution of atmospheric eddies is adequate, which is not true when the eddy size becomes comparable to the grid spacing - so in the model energy breaks down until it reaches the smallest model scale (i.e. the smallest resolvable eddy size) and is then trapped there.  As a result, model predictions often vary at the smallest model scale in a saw-tooth manner, e.g. a forecast variable will be too large at one cell, then too small at the neighboring cell, and then again too large at the next cell, etc.  This is typically reduced by "numerical filtering", but too much filtering also throws away part of the true signal so a compromise is required (as is typical in numerical modeling!).  Often one can see evidence of this model noise not being fully controlled when a "bullseye" pattern appears in a BLIPMAP, with the value at one grid point being much larger/smaller than its surrounding neighbors.  Decreasing model grid spacing helps to reduce model noise, but it will always exist to some degree.
  
  
• Model Topography Errors: 
        Generally model conditions represent "average" conditions over the extent of its grid cell, but the effect of surface elevation on the model is somewhat different since the topography used by a model is typically smoothed to a coarser resolution than that of the model grid spacing and can differ significantly from the actual topography, particularly when the actual surface elevation changes abruptly.  The reason for this degradation is the model noise problem discussed above:  if the topography were to be fully resolved then much noise would be generated at the very smallest scale, aggravating the normal model noise build-up problem at that scale, so to avoid this the very smallest scales are filtered out of the topography.  Note that this means that 8 model points are now required to resolve a ridge, so resolution of surface elevation influences requires a finer grid spacing than for many other atmospheric influences.         Another terrain factor is that models often use an "envelope topography" to produce better velocity predictions - but that can result in worsened BL predictions, such as for BL top.  The idea behind an "envelope topography" is that in reality velocities on either side of a mountain ridge, the Sierras being a good example, are separated by a relatively high ridge; but if one uses elevations averaged over 20 km (or larger!) cells then the ridge largely disappears, so flows at a level which are not interacting in reality will be interacting in the model.  Therefore a weighting is employed which pushs model topography toward the higher elevations that actually exist over each grid cell rather than to a simple average.  However, other parameters such as surface temperature and the BL driven by it do depend upon the average elevation over a grid cell, so use of an envelope topography makes those predictions less accurate!  Sometimes a compromise solution is attempted - for example, the RAP model has two topographies, a "normal" (envelope) topography used for most calculations and a "minimum" topography used for surface temperature adjustments.
        Additional discussion of differences between the smoothed model topography and the real topography, with two illustrations, can be found on the Grid Orientation webpage. 
  
  
• Parameterization Errors: 
        "Parameterization" refers to model terms which cannot be obtained from fundamental principles so instead are computed from approximated equations.  For example, a model which has a 20 km resolution cannot resolve many small clouds, yet those clouds affect the atmosphere through effects such as release of heat aloft in condensation, reduction of solar radiation reaching the surface, etc.  Since these effects are important but can't be predicted explicitly by model equations/resolution, they must be "parameterized".  Parametrization is the "voodoo" part of numerical modeling since it tries to predict complex processes using necessarily over-simplified assumptions.  Many cloud forecast terms fall into this category.   [In one sense this might be described as an "inadequate resolution" error, but here the resolution increase required to obtain fundamnetally correct equations is so large that it cannot be achieved in the foreseeable future, if ever, so is a unique problem.  An example familiar to soaring pilots is the formation of small puffy cumulus clouds - these start out is small wisps of visible vapor and even the cell resolution needed to resolve such wisps is well beyond anything presently possible but to predict this truly correctly one would have to resolve down to the cloud droplet scale!  A non-cloud example which is theoretically more do-able but still impossible in practice is predicting the upward transfer of heat in the BL - in reality this occurs though eddies such as thermals and downdrafts which would need to be resolved, but that would take a grid spacing of less than 50 m so instead it is parameterized as a grid-average vertical transfer.]
  
  
• End User Errors: 
        I cannot resist adding this after noting that some users misuse the predictions by incorrectly applying the model-produced forecasts, apparently due to a lack of appropriate knowledge.

Accuracy and Model Resolution

      Many of the errors listed above are reduced when the spatial resolution of a model is increased, notably those errors resulting from surface averaging effects, from lack of resolved topography, and from the model's finite-difference equations lacking proper resolution of actual atmospheric differences.  Therefore meteorologists are constantly striving to obtain more powerful computers so that they can use ever smaller grid spacings (in both the horizontal and vertical) to reduce these errors and provide better predictions.  Of course, there will continue to be errors due to other factors, such as initial condition errors, which cannot be reduced by using a finer resolution - but it is best to at least eliminate the errors that one can control.  BL predictions are particularly affected by grid resolution errors, and often much less affected by other errors, so better model resolution is especially valuable for soaring predictions.  Some parameterization errors are reduced as resolution is improved, but other are not so parameterziation error will often continue to be a significant source of BL prediction error, particularly for cloud forecasts.
      I should point out that the computer power needed for improved model resolution increases as the fourth power of the resolution increase, e.g. halving the grid spacing requires the computer power to be 16 times larger!  So doubling a model's resolution is a big accomplishment!   [This is because in addition to a factor of 8 increase corresponding to the increased number of horizontal and vertical grid points, since the finer resolution should occur in all directions, there is an additional factor of two because the time step must now be halved to accommodate the smaller grid size.]