Estimate an initial state without treating either source as perfect.
By the end: Use uncertainty weights and identify assumptions that can fail.
Observations arrive at different places and times and measure different aspects of the atmosphere. A short prior forecast supplies a background state. Data assimilation combines the background with observations to make an analysis, which becomes the starting state for another forecast.
A satellite radiance is not directly a grid-cell temperature. An observation operator predicts what a sensor would measure from the model state. Quality control, bias correction, and representation of errors matter before differences are used to update the state.
An observation has a location, a time, a measurement type, and uncertainty. A model value also has a meaning, which may be an average over a grid volume rather than the temperature at one instrument. Before subtracting them, the analysis must compare quantities with compatible meanings. An observation operator supplies the model's predicted measurement. For a directly comparable scalar temperature it can simply return the temperature. For a satellite measurement it may require a more complex calculation. The word observation does not make the recorded number a direct reading of every model variable.
Consider an invented sensor that reports a weighted temperature of two layers. Its stipulated operator takes one quarter of the lower-layer temperature plus three quarters of the upper-layer temperature. If the model layers are 16°C and 8°C, it predicts a reading of 10°C. A reported reading of 11°C then differs by one degree. Subtracting the lower-layer value alone from that reading would instead give negative five degrees and suggest the opposite correction. Neither number is a new weather measurement; the example shows why the comparison must follow the sensor's defined meaning.
| Candidate model layers | Predicted sensor value | 11°C minus prediction |
|---|---|---|
| 16°C, 8°C | 10°C | +1°C |
| 20°C, 8°C | 11°C | 0°C |
| 16°C, 9⅓°C | 11°C | 0°C |
The observation minus the predicted measurement is called an innovation. A nonzero innovation does not identify which input is wrong. It can reflect an inaccurate background, a sensor error, or a mismatch between the real measurement and its modeled counterpart. Even in the two-layer example, one scalar innovation cannot uniquely determine two temperature corrections. Raising only the lower layer by four degrees or raising only the upper layer by four thirds of a degree gives the same one-degree predicted-reading increase. Extra uncertainty relationships determine how a multivariable analysis distributes that correction.
Location mismatch produces a similar issue without a complicated instrument. A thermometer in a sheltered courtyard can differ from an average over a much larger grid cell even if the instrument is perfectly calibrated. That discrepancy concerns representativeness, meaning whether a measurement describes the quantity the model stores. Treating instrument precision as the entire observation uncertainty can then give the observation too much weight. A smaller printed sensor error bar does not remove differences in height, spatial averaging, or observation time. Those differences need a defensible comparison model and error assumptions.
Time also belongs in that comparison. A temperature recorded earlier is not automatically an observation of the state now being estimated. A forecast model can connect states at different times and predict what an earlier sensor should have seen. That adds dynamical assumptions to the analysis rather than converting an old measurement into a simultaneous one. Our scalar lab avoids this issue by stipulating one comparable temperature and no time offset. It has no observation-window control, moving station, or satellite operator, so the two-layer example is explanatory arithmetic outside the available controls.
| Stipulated mismatch | Compared quantity | Consequence |
|---|---|---|
| Point sensor versus cell | Local versus volume mean | Precision alone omits representation error |
| Earlier versus current state | Different valid times | A model must connect the times |
| Layer-weighted sensor | Weighted layers versus one layer | Use the weighted prediction |
Quality control concerns the plausibility and interpretation of observations before accepting their influence. An extreme innovation might come from a failed instrument, but it might also reveal that the background missed a real feature. Automatically rejecting every large disagreement would keep the analysis close to the forecast precisely when correction matters most. Automatically trusting every disagreement would admit faults. The scalar example cannot resolve that choice because it stipulates usable unbiased inputs. Its uncertainty weight answers how to combine accepted estimates under a stated error model, not how to certify that every incoming observation is valid.
Our example estimates one temperature. Background is 18°C; observed temperature is 22°C. Independent unbiased errors have background variance B = 4°C² and observation variance R = 1°C². The observation weight B divided by B plus R is 0.8. The analysis is 18 plus 0.8 times the four-degree difference, or 21.2°C.
Variance is the square of standard deviation. These assumptions give analysis variance BR divided by B plus R, or 0.8°C². The displayed bars are plus or minus one standard deviation, not guaranteed bounds. The Gaussian interpretation makes this a posterior mean and variance; the linear minimum-variance weighting needs weaker distributional assumptions.
Let b denote the background temperature, o the comparable observation, and K a dimensionless observation weight. The analysis is b plus K times o minus b. Equivalently, it gives weight one minus K to b and weight K to o. To choose K, imagine many repetitions of this estimation problem with a fixed but unknown true temperature in each repetition. Error variance describes the average squared departure of an unbiased estimate from that truth. It is a property of the error model across repetitions, not the squared difference between today's two estimates.
Write B for background error variance and R for observation error variance. With independent unbiased errors, the analysis error variance is the background contribution, B times the square of one minus K, plus the observation contribution, R times K squared. Squaring the weights is essential because variance measures squared error. For example, halving a random error halves each departure but quarters its squared departure. Independence removes the cross-product contribution in the average. Without that assumption, the simple two-term expression omits a relationship that can change both the best weight and the reported analysis uncertainty.
| Calculation | Value | Units |
|---|---|---|
| Innovation | 17 − 12 = 5 | °C |
| Observation weight | 9/13 ≈ 0.6923 | Dimensionless |
| Analysis increment | 45/13 ≈ 3.4615 | °C |
| Analysis variance | 36/13 ≈ 2.7692 | °C² |
Use a separate numerical example with b equal 12°C, o equal 17°C, B equal 9°C², and R equal 4°C². The minimum-variance weight is 9/13, approximately 0.6923. The innovation is five degrees, so the analysis increment is 45/13 degrees, approximately 3.4615. Adding that increment to twelve gives 15.4615°C. This estimate lies nearer the observation because its stipulated error variance is smaller. It is still not a measurement of truth. Both the location of the estimate and its expected accuracy depend on the supplied uncertainty assumptions.
b = 12°C is background, o = 17°C is observation, B = 9°C² and R = 4°C² are independent unbiased-error variances. K is dimensionless.
The optimum can be found without calculus by completing a square. Expand the variance into B, minus twice B times K, plus the product of B plus R with K squared. Next subtract B divided by B plus R from K and square that difference. Multiplying this square by B plus R and adding BR divided by B plus R recovers the variance. The squared part cannot be negative. It is zero when K equals B divided by B plus R. This argument establishes the minimum among linear unbiased combinations under the stipulated error model, without requiring Gaussian error distributions.
V(K) is expected squared analysis error in °C² under independent unbiased errors. K is the observation weight.
The positive coefficient B + R makes the square nonnegative; this establishes the linear minimum-variance result without a Gaussian assumption.
For the separate example, the smallest variance is 36/13°C², approximately 2.7692°C². Its standard deviation is the square root, approximately 1.6641°C. Equal weighting would instead give variance 3.25°C², and using only the background would give nine. These comparisons concern expected squared error over the assumed repetitions. They do not guarantee that the weighted analysis beats each input on every individual occasion. A noisy observation can happen to land exactly on truth, and a statistically better combination can then move away from it in that particular case.
σₐ is analysis error standard deviation, not a guaranteed error bound. Variance and standard deviation have different units.
Analysis error variance, °C²
Observation weight K, dimensionless
If the errors are Gaussian as well as independent and unbiased, the same scalar update can be read as a posterior mean and variance conditional on the observation. That stronger interpretation assigns a distribution after observing data. The linear minimum-variance result itself needed only the means, variances, and zero error covariance used above. Neither interpretation makes the displayed standard deviation a hard temperature bound. Under a Gaussian interpretation, values beyond one standard deviation remain possible. A narrower bar reports concentration of the assumed error distribution, not certainty that an estimate is close to reality. For that posterior interpretation, treat the background as a Gaussian prior for the true temperature and the observation error as independent Gaussian noise.
B and R are error variances in °C². K is dimensionless. Background b, observation o, and analysis a are temperatures in °C.
Background stays at 18°C with standard deviation 2°C; observation stays at 22°C. Changes replace only the assumed observation uncertainty, not the reading. Errors are stipulated independent and unbiased.
Observation weight: 0.800. Analysis: 21.200°C. Analysis variance: 0.800°C². Bars show ±1 standard deviation, not hard limits.
| Estimate | Temperature | Error variance |
|---|---|---|
| Background | 18°C | 4°C² |
| Observation | 22°C | 1°C² |
| Analysis | 21.2°C | 0.8°C² |
In the actual lab, background remains 18°C with standard deviation 2°C and observation remains 22°C. The observation-uncertainty slider ranges from 1°C to 4°C in half-degree steps. Precise sensor, Equal uncertainty, and Noisy sensor presets select standard deviations one, two, and four. Changing them replaces only the assumed observation uncertainty. It does not collect a new reading or advance a forecast in time. The three bars compare estimates on one common temperature axis and show plus or minus one standard deviation, not observed minima and maxima.
At observation standard deviation 1.5°C, the variance is 2.25°C² rather than 1.5°C². The weight is four divided by 6.25, or 0.64. The unchanged four-degree innovation therefore gives a 2.56-degree increment and analysis 20.56°C. Analysis variance is 1.44°C², so its standard deviation is 1.2°C. This intermediate slider setting separates the three operations that are easy to confuse: squaring a standard deviation to obtain variance, using variances to calculate a weight, and taking a square root to draw an uncertainty bar in temperature units.
Analysis temperature, °C
Observation standard deviation, °C
Now select an observation standard deviation of 2.5°C while leaving the reading unchanged. Its variance becomes 6.25°C², weight becomes about 0.3902, and analysis becomes about 19.5610°C. The analysis moves toward the background because the same disagreement is now attributed to a less precise observation. It does not mean the temperature physically cooled during the interaction. The diagram compares alternative assumptions about one estimation problem. Its positions are estimates under those assumptions; reading the slider as elapsed time would invent a process that the model does not compute.
The analysis variance also changes with the assumed observation variance. For standard deviations 1.5, 2.5, and 3.5°C it is approximately 1.44, 2.4390, and 3.0154°C². All remain below the background variance four under this independent unbiased-error model. At increasing observation uncertainty, the observation contributes less information, so the analysis approaches the background rather than becoming worse than it in expected squared error. This comparison presumes that the supplied error variances are valid. Merely entering a small variance into the interface cannot make an unreliable instrument informative in the physical world.
The plotted uncertainty bars can overlap even when the estimates have different means. Overlap is not a test that two estimates are identical, and separation is not proof that one has failed. Their meaning depends on the distributions and error relationships behind them. In the scalar model, the innovation's error variance before seeing the reading is B plus R, since independent background and observation errors subtract. At standard deviation 1.5°C, that variance is 6.25°C² and its standard deviation is 2.5°C. The four-degree innovation is then 1.6 such standard deviations.
| Observation SD, °C | Weight | Analysis variance, °C² |
|---|---|---|
| 1.5 | 0.6400 | 1.4400 |
| 2.5 | 0.3902 | 2.4390 |
| 3.5 | 0.2462 | 3.0154 |
That standardized innovation provides a scale for disagreement rather than a verdict. One unusual difference can occur under a valid model, while a run of systematically positive differences can suggest bias or a comparison mismatch. Testing those possibilities needs repeated cases and information beyond this one-temperature illustration. The lab supplies no innovation archive, rejection rule, or quality-control threshold. Use its readouts to trace the arithmetic of an uncertainty assumption. Do not read the changing bar width as empirical evidence that a sensor's real error distribution has been measured or that a forecast system has passed validation.
If two observations share a bias or are correlated, treating them as independent double-counts information. An apparently precise analysis can be wrong. Weather systems use spatial and cross-variable error relationships, not separate averages at every location.
After an analysis, the model evolves again and errors grow. Assimilation therefore repeats. Reanalysis applies a consistent analysis system to historical observations; it is a model-assisted reconstruction, not a set of direct measurements everywhere.
Independent observations can reduce uncertainty when they add genuinely separate information about the same quantity. Suppose two invented sensors each have error variance four and their average is used. With independent zero-mean errors, the average has variance two because each one-half weight contributes one quarter times four. Covariance is the expected product of the two error deviations from their means. If their errors instead have covariance three, the average's variance is 3.5. The cross-product adds 1.5 to the independent calculation. Both sensor variances stay four in these examples; what changes is their tendency to be wrong together, which individual error bars do not display.
Covariance is the mean product of two zero-mean errors. Positive covariance means departures tend to have the same sign; negative covariance means opposite departures tend to occur. It has squared-temperature units here. For two sensors with variance four each, covariance three corresponds to correlation 0.75, since correlation divides covariance by the product of their standard deviations. Averaging can still help in that case, but much less than the independent calculation predicts. Repeating the same sensor report twice is the limiting case of no new information, not two independent confirmations of its value.
| Error correlation | Error covariance, °C² | Average variance, °C² |
|---|---|---|
| 0 | 0 | 2 |
| 0.75 | 3 | 3.5 |
| 1 | 4 | 4 |
A background can also share error with an observation, especially when previous analyses already used related measurements. In a scalar algebraic extension with covariance S between the two errors, the analysis variance gains twice K times one minus K times S. The minimizing weight becomes B minus S divided by B plus R minus twice S when that denominator is positive. For B nine, R four, and S three, the weight is 6/7 and the minimum variance is 27/7. This extension is a paper calculation; the interface implements only S equal zero.
S is the error covariance in °C². This extends the linear unbiased-error model, without assuming independence or requiring Gaussian errors.
Require B + R − 2S > 0. The stipulated variances and covariance must describe admissible errors; here S² = 9 is at most BR = 36.
Here B = 9°C², R = 4°C² and S = 3°C²; K is dimensionless.
For these values the subtraction is 9 − 36/7 = 27/7. Covariance changes the variance formula, not the physical units.
Cycling introduces another reason not to reuse uncertainty without thought. Start a separate scalar example with background variance nine and assimilate one independent observation of variance four. The resulting analysis variance is 36/13. If the next forecast simply preserves that temperature but adds independent model-error variance two, the next background variance becomes 62/13, about 4.7692. Assimilating a new independent observation of variance four then gives analysis variance 124/57, about 2.1754. The intervening forecast has lost some information. The stated identity forecast and added independent error are invented assumptions, not measured atmospheric behavior.
Assimilating the same original observation again as if it were independent would instead double-count information. In the preceding static problem with no forecast-error addition, the naive variance would shrink to 36/22, about 1.6364, because the method falsely counts two independent sensor constraints. A correctly tracked analysis knows that the report already influenced its background. This is distinct from receiving a new independent measurement with the same numerical reading. The values may match, but their error dependence differs. Information content depends on how reports were produced and previously used, not simply on the number of matching numbers.
| Stage | Variance, °C² | Assumption |
|---|---|---|
| Initial background | 9 | Unbiased scalar error |
| First analysis | 36/13 ≈ 2.7692 | New independent observation R = 4 |
| Next background | 62/13 ≈ 4.7692 | Identity forecast adds independent variance 2 |
| Next analysis | 124/57 ≈ 2.1754 | Another independent observation R = 4 |
Reanalysis applies analysis methods to historical observations and forecasts, which can provide estimates where direct measurements were sparse. It remains a model-assisted reconstruction with uncertainty, rather than a complete historical thermometer record. A consistent method aids comparison, but changes in observing coverage and imperfect error assumptions can still affect interpretation. The scalar examples explain why the background, observation model, and dependence assumptions belong in that interpretation. Their arithmetic supports restricted estimates of expected error; it cannot certify the accuracy of a spatial historical reconstruction or of the next operational forecast initialized from an analysis.
Keep background 18°C and observation 22°C, but use B = 1 and R = 4. Find the analysis.
Less trustworthy observations get less weight.
| B | R | Observation weight | Analysis |
|---|---|---|---|
| 4 | 1 | 0.8 | 21.2°C |
| 1 | 4 | 0.2 | 18.8°C |
The observation weight is 1/5 = 0.2. The analysis is 18 + 0.2 × 4 = 18.8°C. Its variance is 0.8°C² under the same independent unbiased-error assumptions.
Two sensors agree exactly, but both use the same faulty calibration. Does averaging them establish the true temperature?
Agreement does not remove common error.
| Quantity | Temperature |
|---|---|
| True temperature in the example | 20°C |
| Sensor A | 23°C |
| Sensor B | 23°C |
| Average | 23°C |
No. Independent random noise can be reduced by averaging, but a shared bias survives. The reported spread can be small while both readings and their average are wrong. Investigate calibration and independent evidence.
Selected reference passages support the definitions and modeling distinctions below. The examples, explanations, and implementations are original. Terminology and assumptions are stated within each lesson; source review is not an independent audit of this course. Optional links are not runtime dependencies.
Background, analysis, observations, cycling, and uncertainty. The scalar weighted-average example is a restricted teaching case, not the operational assimilation system.
Initial-condition and model perturbations. Operational configurations change; the twelve invented rain members are not ECMWF output.