MPE StudioMath of Planet Earth
Opening / Prologue

The Planet We Never See Directly

Time9 min

LevelBegin

Ideashidden states · reconstruction · observations

The question

How can we reason about a planet whose most important states and mechanisms are never observed in full?

Thumbnail for More Data ≠ More Understanding

Watch first · short video

More Data ≠ More Understanding

Begin with the difference between collecting information and understanding the hidden system that produced it.

Watch video ↗Applied Mathematics in Geosciences · Episode 1
Then explore it yourself ↓
Story6 min reading

The map that was never measured

A weather map arrives as a smooth sheet of color. Temperature, pressure, wind, and rain seem to exist at every point, all at the same instant. It is natural to read the image as a photograph of the atmosphere. Now erase every location that was not directly sampled. What remains is a scatter of surface stations, balloon profiles, aircraft tracks, radar volumes, and satellite swaths, each with its own timing, footprint, and error.

The continuous field did not come from a continuous instrument. It was assembled by asking what atmospheric state is compatible with those measurements and with a model of how nearby places, variables, and times are related. The same is true below the ocean surface, where profiles are separated by immense unsampled volumes, and below the ground, where no instrument can directly see the structures inferred from waves.

A reconstruction can be extraordinarily useful without being a photograph. It combines evidence with an observation process, a dynamical or statistical model, and assumptions about error. Dense observations can anchor much of the result. Sparse regions depend more strongly on relationships supplied by the model. If those relationships are wrong, an apparently seamless map can carry a seamless error.

The complete picture was never observed. It was reconstructed. That sentence is not a reason to distrust every map. It is a reason to ask a better set of questions: where did the information come from, which directions were invisible to the observing system, how did assumptions fill the gaps, and how uncertain is the answer?

IdeaOne durable insight

The mathematical idea

  • Measurements constrain only the parts of a state that the observing system can see.
  • A model can propagate information into unobserved places, but it can also propagate structural error.
  • Uncertainty belongs in the reconstruction itself, not in a footnote added afterward.
ExperimentSynthetic model · 7 min

Try this:

Remove the inference

Distinguish a measured value from a field reconstructed with models and assumptions.

Seed 17
Hidden field, point observations, reconstruction, and uncertaintylocationstatesolid truth · dashed reconstruction · points observations

What changedDense observations anchor more of the reconstruction, while uncertainty remains between them.

Text description: Dense observations anchor more of the reconstruction, while uncertainty remains between them. The visualization uses labels, line styles, symbols, and position in addition to color.

See the mathematics

The interactive uses a synthetic hidden field, samples it at a finite number of locations, and reconstructs the gaps with a transparent weighted rule. The equations below are the equations used by this demonstration.

01 · Hidden field

xi = 10i/89,   i = 0, …, 89
ui = 1.05 sin(0.9xi) + 0.50 sin(2.1xi) + 0.85 exp[−(xi − 6.8)2/0.35]

The demonstration divides the domain 0 ≤ x ≤ 10 into 90 grid points. ui is the complete synthetic state at grid point i. It is the solid black truth curve in the plot. In a real application, this complete field would be unknown.

02 · Measurements

yj = usj + εj,   εj ∼ Uniform(−σ, σ),   j = 1, …, N

The red points are the measurements yj. Observation density sets N; Observation noisesets σ. New realization redraws the errors εj.

03 · Reconstruction

wij = exp[−(i − sj)2/(2ℓ2)]

Wi = Σj wij,   yi = Σj wijyj / Wi

ûi ={(0.60 + 0.40c) yi,   Wi > 0.080.20c sin(9i/89),   Wi ≤ 0.08

The Gaussian weights give nearby measurements more influence. Smoothness sets the kernel width ℓ = 2 + 5s, wheres is the slider value. Model relationship sets c: it controls how strongly the reconstruction retains the measured pattern and supplies the fallback pattern where local observational support Wi is too weak. The result ûiis the dashed blue curve.

04 · Displayed uncertainty band

B = 0.15 + σ + N−1/2 + (1 − c),   displayed range = ûi ± B

The pale blue band widens when measurements are noisier or sparser, or when the model relationship is weaker. It is a deliberately simple teaching indicator, not a calibrated posterior confidence interval.

xi, i
Horizontal location and its grid-point index
ui
Hidden state · solid black curve
sj, yj
Sensor indices and red observations
N
Observation density slider
σ
Observation noise slider
s, ℓ
Smoothness slider and resulting kernel width
c
Model relationship slider
ûi
Reconstruction · dashed blue curve
Earth

Where it appears on Earth

Weather analyses and ocean reanalyses

Operational analyses combine many observation types with a forecast model. Ocean heat, soil moisture, and atmospheric winds are therefore estimated fields, not products of a single pure observation.

Follow the Earth connection →
What this model leaves out

This demonstration uses a one-dimensional smooth field, simple independent errors, and a transparent reconstruction rule. Real observing networks change in space and time; satellite retrievals have footprints and biases; data-assimilation systems are high-dimensional; and uncertainty may be correlated, non-Gaussian, and model dependent.

Sources and further reading