The Earth Is Too Complex to Simulate and Too Hidden to Observe

[00:00:00]Can a tiny cloud droplet affect the growth of a hurricane?

[00:00:05]Or can a satellite looking only at the ocean surface tell us what's happening thousands of meters below?

[00:00:12]These sound like Earth science questions, but answering them forced mathematics to change.

[00:00:19]The Earth presents two fundamental challenges. It contains interacting processes and across an enormous range of scales.

[00:00:28]And most of its state cannot be observed directly.

[00:00:34]The first challenge led to multi-scale modeling.

[00:00:37]The second led to inverse problems.

[00:00:40]Together, they change how we use mathematics to understand complex systems.

[00:00:48]Let's begin with the hurricane. At first glance, a hurricane looks like one giant rotating storm. It can span hundreds or even thousands of kilometers.

[00:00:59]That's the large scale, but inside the storm many smaller processes are happening. There are rain bands, thunderstorms, and turbulent eddies.

[00:01:10]There are updrafts and downdrafts. And even on a smaller scale, there are cloud droplets, ice crystals, and microphysical processes.

[00:01:19]So, the hurricane is not just one object. It's a system of many interacting scales.

[00:01:26]The same thing happens in the ocean. At the largest scale, we see basin scale circulation, such as the Gulf Stream. At the meso-scale, we see eddies and meanders. At smaller scales, we see turbulence and mixing.

[00:01:41]And at the microscopic scales, we see dissipation and molecular processes.

[00:01:46]The key message is simple. Small scales influence large scale behavior.

[00:01:53]Tiny cloud processes can affect how a hurricane grows.

[00:01:57]Small ocean eddies can affect how heat is transported across the planet.

[00:02:02]This is what makes Earth science so mathematically difficult. We cannot just model the big picture and ignore the details.

[00:02:11]But we also cannot simulate every detail.

[00:02:15]That tension created the mathematics of multi-scale systems.

[00:02:20]Both large-scale and small-scale processes influence each other.

[00:02:25]The large-scale environment shapes the small-scale processes, but the small-scale processes also feed back and change the large-scale system.

[00:02:33]That's the central difficulty.

[00:02:36]Now, let's write a very simple mathematical form to describe this.

[00:02:40]Here, X is going to represent the variables we can resolve. For example, the large-scale wind, temperature, or ocean current.

[00:02:50]Y represents the small-scale variables we cannot fully resolve. For example, cloud droplets, turbulence, or small eddies. The true system depends on both X and Y. But in practice, we often cannot compute Y directly. So, we replace the effect of Y by an approximate term.

[00:03:12]This is one of the most important ideas in multi-scale modeling. The word closure means we're trying to close the large-scale equation by representing the missing small-scale physics. For example, instead of simulating every cloud droplet, a climate model uses a cloud parameterization. Instead of resolving every small ocean eddy, a coarse model would use an eddy closure.

[00:03:36]So, the mathematical problem becomes, how do we represent the effects of the scales we cannot see?

[00:03:42]That question is not just a detail. It created major parts of modern applied mathematics.

[00:03:49]Multi-scale analysis, homogenization, stochastic modeling, parameterization, reduced order modeling, and now machine learning closures.

[00:03:59]All of these ideas were pushed forward because Earth systems are too complex to resolve completely.

[00:04:07]We now have seen the first obstacle.

[00:04:10]The Earth contains too many interacting scales to simulate them all.

[00:04:14]But there's also a second obstacle.

[00:04:17]Much of the Earth is not merely unresolved. It's hidden.

[00:04:22]We cannot directly observe the full atmosphere or the deep ocean. So, mathematics must do something different.

[00:04:30]Infer what cannot be seen from what can be measured.

[00:04:35]This is the idea of an inverse problem.

[00:04:39]Let's use the ocean state as a concrete example.

[00:04:42]Suppose we want to know the full three-dimensional state of the ocean that includes temperature, salinity, and currents, not only at the surface, also deep below the surface.

[00:04:53]But what can we actually observe?

[00:04:56]Satellites observe the surface. They measure things like sea surface temperature, sea surface height, and ocean color.

[00:05:03]Surface drifters can tell us something about near surface currents. Buoys and moorings give us local measurements.

[00:05:10]Argo floats provide vertical profiles of temperature and salinity. But they are still sparse.

[00:05:17]So, we have a difficult situation. We want to know the full deep ocean, but we only observe limited, noisy, incomplete data.

[00:05:26]This is an inverse problem. We're trying to recover what is hidden from what is observed.

[00:05:32]Surface observations give us clues, but the deep ocean state is not directly visible.

[00:05:39]So, we can buy combine observations, physical models, and mathematical inference.

[00:05:45]This is the heart of data assimilation.

[00:05:48]It's not just filling in missing data.

[00:05:51]It is using physics and probability to infer a dynamically consistent ocean state.

[00:05:58]A forward problem is the usual direction. If we know the full ocean state, we can predict what the observations should look like. For example, if we know the temperature and currents everywhere, we can predict what a satellite might observe at the surface.

[00:06:15]The inverse problem goes backwards.

[00:06:17]Given observations, we try to infer the hidden state.

[00:06:21]In our ocean example, we use surface observations to infer what's happening below the surface.

[00:06:28]But, this is hard because the answer may not be unique.

[00:06:32]Different oceans deep ocean states may produce very similar surface observations.

[00:06:38]That means the problem is ill-posed.

[00:06:42]Here's a simple analogy to make it clear.

[00:06:45]Imagine looking at only the top of an iceberg. You see what is above the water, but you want to know the full shape below the water.

[00:06:53]Many different underwater shapes could produce the same visible top.

[00:06:58]The deep ocean problem is similar. The surface gives clues, but it does not uniquely determine everything below.

[00:07:05]That's why uncertainty is unavoidable.

[00:07:09]So, instead of finding a single perfect answer, we seek a probability distribution.

[00:07:16]This means we do not ask, "What is the ocean state?" We ask, "What states are likely given the observations and the model?"

[00:07:27]This is a major shift in thinking.

[00:07:29]Understanding becomes inference.

[00:07:32]Prediction becomes probabilistic.

[00:07:35]And data becomes useful when combined with models and uncertainty.

[00:07:41]This example is especially powerful because the ocean stores and transports enormous amounts of heat.

[00:07:49]Small errors in the deep ocean can affect long-term climate predictions.

[00:07:55]But the deep ocean is difficult to observe. We cannot place sensors everywhere. We cannot continuously measure every layer. So, mathematics becomes essential.

[00:08:05]A model provides physical constraints.

[00:08:08]Observations provide partial information.

[00:08:12]Probability describes uncertainty.

[00:08:15]And data assimilation combines them.

[00:08:19]This is why inverse problems are not just technical tools. They're a new way of knowing.

[00:08:25]They tell us how to reason when we cannot see the full system.

[00:08:30]Now, let's connect these two problems.

[00:08:34]In Earth science, we rarely face only one challenge at a time. We face both.

[00:08:39]The system is multi-scale and the observations are incomplete.

[00:08:44]For example, in the ocean, small-scale turbulence affects large-scale heat transport. And at the same time, we may only observe the surface.

[00:08:53]So, we need to infer hidden states and represent unresolved processes at the same time.

[00:09:00]That's extremely hard and this difficulty forced mathematics to evolve.

[00:09:05]We needed methods for multi-scale modeling. We needed methods for inverse problems. We needed stochastic models to represent uncertainty. And we needed data assimilation to combine models and observations.

[00:09:19]Today, we also use machine learning to learn closures, detect patterns, and improve reconstruction.

[00:09:26]But the key point is not that AI replaces physics. The key point is that modern mathematics now combines physics, data, and uncertainty.

[00:09:37]Earth science has helped create this way of thinking.

[00:09:41]So, what did Earth science teach mathematics?

[00:09:45]First, multi-scale systems taught us that small things matter.

[00:09:50]Tiny cloud droplets, turbulent eddies, and small-scale mixing can shape planetary-scale behavior.

[00:09:58]Second, inverse problems taught us that data is incomplete and we never really fully observe the system directly. So, we need to infer hidden states from partial observations.

[00:10:10]Third, understanding becomes inference.

[00:10:14]We no longer just simply solve equations. We combine models, data, and probability.

[00:10:20]Earth science did not just use mathematics, it transformed it. It changed mathematics from a tool for solving idealized equations into a framework for reasoning about complex, uncertain, partially observed systems.

[00:10:35]And that is one of the deepest lessons of modern applied mathematics.

[00:10:39]The Earth is too complex to simulate in full and too hidden to observe directly.

[00:10:45]Multi-scale methods help us represent the processes we cannot resolve.

[00:10:51]Inverse methods help us infer the states we cannot see.

[00:10:56]Earth science is not simply applying existing mathematics.

[00:11:01]It pushed mathematics to change.

[00:11:04]In the next lecture, we will dive into what exactly applied mathematics is.