Sea Level Rise Models – V

To take a look at the payoff surface, we need to do more than the naive calibrations I’ve used so far. Those were adequate for choosing constant terms that aligned the model trajectory with the data, given a priori values of a and tau. But that approach could give flawed estimates and confidence bounds when used to estimate the full system.

Elaborating on my comment on estimation at the end of Part II, consider a simplified description of our model, in discrete time:

(1) sea_level(t) = f(sea_level(t-1), temperature, parameters) + driving_noise(t)

(2) measured_sea_level(t) = sea_level(t) + measurement_noise(t)

The driving noise reflects disturbances to the system state: in this case, random perturbations to sea level. Measurement noise is simply errors in assessing the true state of global sea level, which could arise from insufficient coverage or accuracy of instruments. In the simple case, where driving and measurement noise are both zero, measured and actual sea level are the same, so we have the following system:

(3) sea_level(t) = f(sea_level(t-1), temperature, parameters)

In this case, which is essentially what we’ve assumed so far, we can simply initialize the model, feed it temperature, and simulate forward in time. We can estimate the parameters by adjusting them to get a good fit. However, if there’s driving noise, as in (1), we could be making a big mistake, because the noise may move the real-world state of sea level far from the model trajectory, in which case we’d be using the wrong value of sea_level(t-1) on the right hand side of (1). In effect, the model would blunder ahead, ignoring most of the data.

In this situation, it’s better to use ordinary least squares (OLS), which we can implement by replacing modeled sea level in (1) with measured sea level:

(4) sea_level(t) = f(measured_sea_level(t-1), temperature, parameters)

In (4), we’re ignoring the model rather than the data. But that could be a bad move too, because if measurement noise is nonzero, the sea level data could be quite different from true sea level at any point in time.

The point of the Kalman Filter is to combine the model and data estimates of the true state of the system. To do that, we simulate the model forward in time. Each time we encounter a data point, we update the model state, taking account of the relative magnitude of the noise streams. If we think that measurement error is small and driving noise is large, the best bet is to move the model dramatically towards the data. On the other hand, if measurements are very noisy and driving noise is small, better to stick with the model trajectory, and move only a little bit towards the data. You can test this in the model by varying the driving noise and measurement error parameters in SyntheSim, and watching how the model trajectory varies.

The discussion above is adapted from David Peterson’s thesis, which has a more complete mathematical treatment. The approach is laid out in Fred Schweppe’s book, Uncertain Dynamic Systems, which is unfortunately out of print and pricey. As a substitute, I like Stengel’s Optimal Control and Estimation.

An example of Kalman Filtering in everyday devices is GPS. A GPS unit is designed to estimate the state of a system (its location in space) using noisy measurements (satellite signals). As I understand it, GPS units maintain a simple model of the dynamics of motion: my expected position in the future equals my current perceived position, plus perceived velocity times time elapsed. It then corrects its predictions as measurements allow. With a good view of four satellites, it can move quickly toward the data. In a heavily-treed valley, it’s better to update the predicted state slowly, rather than giving jumpy predictions. I don’t know whether handheld GPS units implement it, but it’s possible to estimate the noise variances from the data and model, and adapt the filter corrections on the fly as conditions change.

Continue reading “Sea Level Rise Models – V”

Sea Level Rise Models – IV

So far, I’ve established that the qualitative results of Rahmstorf (R) and Grinsted (G) can be reproduced. Exact replication has been elusive, but the list of loose ends (unresolved differences in data and so forth) is long enough that I’m not concerned that R and G made fatal errors. However, I haven’t made much progress against the other items on my original list of questions:

  • Is the Grinsted et al. argument from first principles, that the current sea level response is dominated by short time constants, reasonable?
  • Is Rahmstorf right to assert that Grinsted et al.’s determination of the sea level rise time constant is shaky?
  • What happens if you impose the long-horizon paleo constraint to equilibrium sea level rise in Rahmstorf’s RC figure on the Grinsted et al. model?

At this point I’ll reveal my working hypotheses (untested so far):

  • I agree with G that there are good reasons to think that the sea level response occurs over multiple time scales, and therefore that one could make a good argument for a substantial short-time-constant component in the current transient.
  • I agree with R that the estimation of long time constants from comparatively short data series is almost certainly shaky.
  • I suspect that R’s paleo constraint could be imposed without a significant degradation of the model fit (an apparent contradiction of G’s results).
  • In the end, I doubt the data will resolve the argument, and we’ll be left with the conclusion that R and G agree on: that the IPCC WGI sea level rise projection is an underestimate.

Continue reading “Sea Level Rise Models – IV”

Sea Level Rise Models – III

Starting from the Rahmstorf (R) parameterization (tested, but not exhaustively), let’s turn to Grinsted et al (G).

First, I’ve made a few changes to the model and supporting spreadsheet. The previous version ran with a small time step, because some of the tide data was monthly (or less). That wasted clock cycles and complicated computation of residual autocorrelations and the like. In this version, I binned the data into an annual window and shifted the time axes so that the model will use the appropriate end-of-year points (when Vensim has data with a finer time step than the model, it grabs the data point nearest each time step for comparison with model variables). I also retuned the mean adjustments to the sea level series. I didn’t change the temperature series, but made it easier to use pure-Moberg (as G did). Those changes necessitate a slight change to the R calibration, so I changed the default parameters to reflect that.

Now it should be possible to plug in G parameters, from Table 1 in the paper. First, using Moberg: a = 1290 (note that G uses meters while I’m using mm), tau = 208, b = 770 (corresponding with T0=-0.59), initial sea level = -2. The final time for the simulation is set to 1979, and only Moberg temperature data are used. The setup for this is in change files, GrinstedMoberg.cin and MobergOnly.cin.

Moberg, Grinsted parameters

Continue reading “Sea Level Rise Models – III”

Sea Level Rise Models – II

Picking up where I left off, with model and data assembled, the next step is to calibrate, to see whether the Rahmstorf (R) and Grinsted (G) results can be replicated. I’ll do that the easy way, and the right way.

An easy first step is to try the R approach, assuming that the time constant tau is long and that the rate of sea level rise is proportional to temperature (or the delta against some preindustrial equilibrium).

Rahmstorf estimated the temperature-sea level rise relationship by regressing a smoothed rate of sea level rise against temperature, and found a slope of 3.4 mm/yr/C.

Rahmstorf figure 2

Continue reading “Sea Level Rise Models – II”

Sea Level Rise Models – I

A recent post by Stefan Rahmstorf at RealClimate discusses a new paper on sea level projections by Grinsted, Moore and Jevrejeva. This paper comes at an interesting time, because we’ve just been discussing sea level projections in the context of our ongoing science review of the C-ROADS model. In C-ROADS, we used Rahmstorf’s earlier semi-empirical model, which yields higher sea level rise than AR4 WG1 (the latter leaves out ice sheet dynamics). To get a better handle on the two papers, I compared a replication of the Rahmstorf model (from John Sterman, implemented in C-ROADS) with an extension to capture Grinsted et al. This post (in a few parts) serves as both an assessment of the models and a bit of a tutorial on data analysis with Vensim.

My primary goal here is to develop an opinion on four questions:

  • Can the conclusions be rejected, given the data?
  • Is the Grinsted et al. argument from first principles, that the current sea level response is dominated by short time constants, reasonable?
  • Is Rahmstorf right to assert that Grinsted et al.’s determination of the sea level rise time constant is shaky?
  • What happens if you impose the long-horizon paleo constraint to equilibrium sea level rise in Rahmstorf’s RC figure on the Grinsted et al. model?

Paleo constraints on equilibrium sea level

Continue reading “Sea Level Rise Models – I”

Exhibit A – the Social Cost of Carbon

I recently discovered a cool set of tools from MIT’s Simile project. My favorites are Timeline and Exhibit, which provide a fairly easy way to create web sites where visitors can interact with data. As a test, I built an Exhibit containing Richard Tol’s survey of assessments of the social cost of carbon (SCC):

Social Cost of Carbon Exhibit

Continue reading “Exhibit A – the Social Cost of Carbon”