Suppose you have a mathematical or computational model M that maps input parameters θ to outputs y, i.e., y=M(θ) where θ and y are n- and k-dimensional real vectors. The parameters θ may be initial or boundary conditions of a partial differential equation, diffusion coefficients in elliptic equations, or growth rates in ecological models. The outputs y can be compared to data z, obtained by observing the physical process under study. For example, if M is an atmospheric model, z may represent temperature measurements at k different locations. It is common to assume that 1z=M(θ)+ε, where ε is a random variable with known probability density function pε(⋅) that describes errors or mismatches between model and data. The above equation defines a k-dimensional “likelihood”, l(z|θ)=pε(z-M(θ)|θ), that describes the probability of the data for a given set of parameters.

In addition to Eq. (), one may have prior information about the model parameters, e.g., one may know that some parameters are nonnegative. Such prior information can be represented by a prior distribution p0(θ). By Bayes' rule, the prior and likelihood define the posterior distribution 2p(θ|z)∝p0(θ)l(z|θ). The posterior distribution combines information from model and data and defines parameters θ that lead to model outputs that are “compatible” with the data. Here compatible means that model outputs are likely to be within the assumed errors ε.

Data assimilation problems of this kind appear in science and engineering, e.g., in numerical weather prediction, oceanography and geomagnetism , as well as in global seismic inversion , reservoir modeling or subsurface flow , target tracking , and robotics . The term “data assimilation” is common in geophysics, but in various applications and disciplines, different names are used, including parameter estimation, Bayesian inverse problems, history matching and particle filtering.

Computational methods for data assimilation can be divided into three groups. The first group is based on the Kalman filter and includes the ensemble Kalman filter . Kalman filters are particularly useful when data are assimilated sequentially, as is the case in numerical weather prediction. The second group consists of optimization algorithms, called “variational methods” in this context . The third group are Monte Carlo sampling methods, including particle filters or direct sampling , and MCMC (; ). We will use variational methods, MCMC and direct sampling for numerical solution of feature-based data assimilation problems and we briefly review these techniques here.

In variational data assimilation one finds the parameter set θ∗ that maximizes the posterior probability, which is also called the posterior mode. One can find the posterior mode by minimizing the negative logarithm of the posterior distribution 3F(θ)=-log⁡p0(θ)l(z|θ). The optimization is done numerically and one can use, for example, Gauss–Newton algorithms. In some of the numerical examples below, we need to optimize functions F(θ) that are computationally expensive to evaluate and noisy, i.e., F(θ) is a random variable with unknown distribution. The source of noise in the function F(θ) is caused by numerically approximating the feature. Suppose, e.g., that the feature is an expected value and in the numerical implementation this expected value is approximated by Monte Carlo. The Monte Carlo approximation, however, depends on the number of samples used and if this number is small (finite), the approximation is noisy, i.e., the Monte Carlo average for the same set of parameters θ, but with two different seeds in the random number generator, can lead to two different values for F(θ). In such cases, one can use a derivative free optimization method such as global Bayesian optimization (GBO); see, e.g., . The basic idea is to model the function F(θ) by a Gaussian process and then update the GP model based on a small number of function evaluations. The points where the function is evaluated are chosen based on an expected improvement (EI) criterion, which takes into account where the function is unknown or known. The GP model for the function F(θ) is then updated based on the function evaluations at the points suggested by EI. One can iterate this procedure and when the iteration is finished, e.g., because a maximum number of function evaluations is reached, one can use the optimizer of the mean of the GP model to approximate the optimizer of the (random) function F(θ).

In MCMC, a Markov chain is generated by drawing a new sample θ′ given a previous sample θj-1, using a proposal distribution q(θ′|θj-1). The proposed sample θ′ is accepted as θj or rejected based on the values of the posterior distribution of the new and previous samples; see, e.g., and . Averages over the samples converge to expected values with respect to the posterior distribution as the number of samples goes to infinity. However, since θj depends on θj-1, the samples are not independent and one may wonder how many effectively uncorrelated samples one has obtained. This number can be estimated by dividing the number of samples by the integrated auto-correlation time (IACT) . Thus, one wants to pick a proposal distribution that reduces IACT. The various MCMC algorithms in the literature differ in how the proposal distribution is constructed. In the numerical examples below, we use the MATLAB implementation of the affine invariant ensemble sampler , as described by , and we use the numerical methods described in to compute IACT.

In direct sampling (sometimes called importance sampling) one generates independent samples using a proposal density q and attaches to each sample a weight: θj∼q(θ),wj∝p0(θj)l(z|θj)q(θj). Weighted averages of the samples converge to expectations with respect to the posterior distribution as the number of samples goes to infinity. While the samples are independent, they are not all equally weighted and one may wonder how many “effectively unweighted” samples one has. For an ensemble of size Ne, the effective number of samples can be estimated as 4Neff=Ne/ρ,ρ=E(w2)E(w)2. For a practical algorithm, we thus choose a proposal distribution q such that ρ is near 1. There are several strategies for constructing such proposal distributions and in the numerical illustrations below we use “implicit sampling” and construct the proposal distribution to be a Gaussian whose mean is the posterior mode θ∗ and whose covariance is the Hessian of F in Eq. (), evaluated at θ∗ (see also ).

Evaluating the feature-based posterior distribution is difficult because evaluating the feature-based likelihood is cumbersome. Even under simplifying assumptions of additive Gaussian noise in Eq. (), the likelihood lF(f|θ), defined by Eq. (), is generally not known. The reason is that the feature function F makes the distribution of FM(θ)+ε non-Gaussian even if ε is Gaussian. Numerical methods for data assimilation typically require that the posterior distribution be known up to a multiplicative constant. This is generally not the case when a feature-based likelihood is used (the feature-based likelihood is Gaussian only if F is linear and if ε is Gaussian). Thus, variational methods, MCMC or direct sampling are not directly applicable to solve feature-based data assimilation problems defined by Eq. (). More advanced techniques, such as approximate Bayesian computation (ABC) , however, can be used because these are designed for problems with unknown likelihood (see ).

Difficulties with evaluating the feature-based likelihood arise because we assume that Eq. () is accurate and we require that the feature-based likelihood follows directly from it. However, in many situations the assumptions about the noise ε in Eq. () are “ad hoc”, or for mathematical and computational convenience. There is often no physical reason why the noise should be additive or Gaussian, yet these assumptions have become standard in many data assimilation applications. This leads to the question: why not “invent” a suitable and convenient noise model for the feature?

We explore this idea and consider an additive Gaussian noise model for the feature. This amounts to replacing Eq. () by 6f=MF(θ)+η, where MF=F∘M, is the composition of the model M and feature extraction F, and η is a random variable that represents uncertainty in the feature and which we need to define (see below). For a given η, the feature-based likelihood defined by Eq. (), lF(f|θ)=pηf-MF(θ)|θ, is now straightforward to evaluate (up to a multiplicative constant) because the distribution of η is known or chosen. The feature-based likelihood based on Eq. () results in the feature-based posterior distribution 7pF(θ|f)∝p0(θ)lF(f|θ), where p0(θ) is the prior distribution, which is not affected by defining or using features. The usual numerical tools, e.g., MCMC, direct sampling, or variational methods, are applicable to the feature-based posterior distribution (Eq. ).

Our simplified approach requires that one defines the distribution of the errors η, similar to how one must specify the distribution of ε in Eq. (). We suggest using a Gaussian distribution with mean zero for η. The covariance that defines the Gaussian can be obtained by borrowing ideas from the ensemble Kalman filter (EnKF). In a “perturbed observation” implementation of the EnKF, the analysis ensemble is formed by using artificial perturbations of the data . We suggest using a similar approach here. Assuming that ε in Eq. () is Gaussian with mean 0 and covariance matrix R, we generate perturbed data by zj∼N(z,R), j=1,…,Nz. Each perturbed datum leads to a perturbed feature fj=F(zj) and we compute the covariance Rf=1Nz-1∑j=1Nz(fj-f)(fj-f)T. We then use η∼N(0,Rf) as our noise model for the feature-based problem in Eq. ().

Note that the rank of the covariance Rf is min⁡{dim(f),Nz-1}. For high-dimensional features, the rank of Rf may therefore be limited by the number of perturbed observations and features we generate, and this number depends on the computational requirements of the feature extraction. We assume that Nz is larger than the dimension of the feature, which is the case if either the computations to extract the features are straightforward, or the feature is low-dimensional.

One may also question why η should be Gaussian. In the same vein, one may wonder why ε in Eq. () should be Gaussian, which is routinely assumed. We do not claim that we have answers to such questions, but we speculate that if the feature does indeed constrain some parameters, then assuming a unimodal likelihood is appropriate and, in this case, a Gaussian assumption is also appropriate.

Feature-based data assimilation requires that one defines and selects relevant features. In principle, much of the machine learning technology can be applied to extract generic features from data. For example, one can define F by the PCA, or singular value decomposition, of the data and then neglect small singular values and associated singular vectors. As a specific example, suppose that the data are measurements of a time series of M data points of an n-dimensional system. In this case, the function MF consists of the following steps: (i) simulate the model; and (ii) compute the singular value decomposition (SVD) of the n×M matrix containing the data. The feature f in Eq. () may then be the first d largest singular values and associated right and left singular vectors (see the example in Sect.  for more detail). In practice, relevant features may often present themselves. For example, in Lagrangian data assimilation, coherent structures (and their SVDs) are a natural candidate, as explained by . In Sect.  we present several examples of “intuitive” features, constructed using physical insight, and discuss what numerical methods to use in the various situations.

The choice of the feature suggests the numerical methods for the solution of the feature-based problem. One issue here is that, even with our simplifying assumption of additive (Gaussian) noise in the feature, evaluating a feature-based likelihood can be noisy. This happens in particular when the feature is defined in terms of averages over solutions of stochastic or chaotic equations. Due to limited computational budgets, such averages are computed using a small sample size. Thus, sampling error is large and evaluation of a feature-based likelihood is noisy, i.e., evaluations of the feature-based likelihood, even for the same set of parameters θ and feature-data f, may lead to different results, depending on the state of the random number generator. This additional uncertainty makes it difficult to solve some feature-based problems numerically by Monte Carlo. However, one can construct a numerical framework for computing maximum a posteriori estimates using derivative-free optimization methods that are robust to noise, e.g., global Bayesian optimization . We will specify these ideas in the context of a numerical example with the Kuramoto–Sivashinsky (KS) equation in Sect. .

A natural question is the following: under what conditions should I consider a feature-based approach? There are three scenarios which we discuss separately before we make connections between the three scenarios using the concept of “effective dimension”.

We illustrate that a data assimilation problem with fewer data points can be as useful as one with significantly more, but redundant, data points. We consider a mass–spring–damper system d2xdt2+2ζωdxdt+ω2x=h(t-5), where t≥0 is time, ζ>0 is a viscous damping coefficient, ω>0 is a natural frequency and h(τ) is the “step-function”, i.e., h(τ)=0 for τ<0 and h(τ)=1 for τ≥0. The initial conditions of the mass–spring–damper system are x(0)=0, dx/dt(0)=0. The parameters we want to estimate are the damping coefficient ζ and the natural frequency ω, i.e., θ=(ζ,ω)T. To estimate these parameters we use a uniform prior distribution over the box 0.5,4×0.5,4 and measure the displacement x(t) every Δt=0.5 time units (starting with a measurement at t=0). The duration of a (synthetic) experiment is τ=MΔt and we consider experiments of durations between τ=25 to τ=250 time units, with M=51 to M=501 data points. The data of an experiment of duration τ=MΔt are thus zi=x(iΔt)+vi,vi∼N(0,10-3),i=0,…,M. Writing z={z0,…,zM}, we obtain the likelihood lτ(z|θ)∝exp⁡-12∑i=0M103(zi-x(iΔT))2. The likelihood and the uniform prior distribution define the posterior distribution pτ(θ|z)=1Cτlτ(z|θ)ifθ∈0.5,4×0.5,4,0otherwise, where Cτ is a normalization constant. Data of an experiment of duration τ=40 is shown in Fig. a. These synthetic data are generated with “true” parameters ζ=1.5 and ω=1. With these parameters the oscillator is “overdamped” and reaches its steady state (lim⁡t→∞x(t)=1) quickly. We anticipate that data collected after t≈25 is redundant in the sense that the same displacement is measured again and again. This suggests that the posterior distributions of experiments of duration τ=iΔt and τ=jΔt are approximately equal to each other, provided that i,j>50/Δt. In other words, a data assimilation problem with M=101 or M=251 data points may have “roughly the same” posterior distribution and, consequently, lead to similar estimates.

We investigate this idea by solving data assimilation problems with experiment durations between τ=25 and τ=225. We compare the resulting posterior distributions p25,…,p225 to the posterior distribution p250, corresponding to an experiment of duration τ=250. We use the Kullback–Leibler (KL) divergence, DKL(p^0||p^1) of two distributions to measure “how far” two distributions are from one another. For two k-dimensional Gaussians p0=N(m0,P0) and p1=N(m1,P1), the KL divergence is given by DKL(p0||p1)=12(Tr(P1-1P0)+(m1-m0)TP1-1m1-m0-k+log⁡detP1detP0). Note that DKL(p0||p1)=0 if the two distributions are identical and a large DKL(p0||p1) suggest that p0 and p1 are quite different. Computing the KL divergence for non-Gaussian distributions is numerically more challenging and here were are content to measure the distance of two distributions by the KL divergence of their Gaussian approximations. We thus compute Gaussian approximations to the posterior distributions p25,…,p250, by computing the posterior mode θ∗ (by Gauss–Newton optimization) and the Hessian H of the negative logarithm of the posterior distribution at the mode. We then define the Gaussian approximation by 8pτθ|zM≈p^τ(θ|zM)=N(θ∗,H-1), and use DKL(p^250||p^τ) to measure the distance of p250 and pτ.

Each experiment is in itself a random event because the measurement noise is random. The KL divergence between the various posterior distributions is, thus, also random and we address this issue by performing 1000 independent experiments and then average the KL divergences. Our results are shown in Fig. b. We plot the average KL divergence, as well as “error bars” based on the standard deviation, as a function of the experiment duration and note an exponential decrease of KL divergence with experiment duration or, equivalently, number of data points used for parameter estimation. Thus, as we increase the number of data points, the posterior distributions get closer, as measured by this KL divergence, to the posterior distribution with M=501 data points. In other words, we obtain very similar posterior distributions with M=101 or M=501 data points. This indicates that the steady state data can be ignored because there is little additional information in these data. These results suggest that the data can be compressed without significant loss of information about the parameters. One could, for example, define a feature by simply neglecting data collected after t>30. This feature would lead to parameter estimates almost identical to those obtained using the full data set.

We now consider a feature that compresses the data into two numbers. The first component of our feature is the average of the last 50 data points. This average is directly related to the natural frequency since lim⁡t→∞x(t)=1/ω2. The second component of the features is the slope of a linear fit to the seven data points collected after t=5, i.e., after the step is applied.

The covariance matrix R of the assumed Gaussian noise η (see Eq. ) is obtained using the perturbed observation approach as described in Sect. . We generate 103 perturbed data sets to compute R and find that the off-diagonal elements are small compared to the diagonal elements. We thus neglect the correlation between the two components of the feature, but this is not essential. Altogether the feature-based likelihood is given by lF(f|θ)∝exp⁡-12f-FM(θ)TR-1f-FM(θ),

where FM represents the following two computations: (i) simulate the oscillator with parameters θ for τ time units, and (ii) compute the feature, i.e., the average steady state value and slope, as described above. Together with the uniform prior distribution, we obtain the feature-based posterior distribution pF(θ|f)=1CFlF(f|θ)ifθ∈0.5,4×0.5,4,0otherwise, where CF is a normalization constant.

We solve this feature-based problem for an experiment of duration τ=40 by implicit sampling (see Sect. ) using Ne=103 samples. From these samples we compute ρ≈1.07, i.e., almost all samples are effective samples. Results are illustrated in Fig. c, where we plot trajectories corresponding to 50 samples of θ=(ζ,ω) of the feature-based posterior distribution. We note that the trajectories are all “near” the data points. For comparison, we also solve the data assimilation problem without using features and compute p40 (see Eq. ), also by implicit sampling with Ne=103 samples. We find that ρ≈1.38 in this case. We note that the feature-based posterior distribution is different from the “classical” one. This can be seen by comparing the clouds of trajectories in Fig. a and c. The wider cloud of trajectories indicates that the feature does not constrain the parameters as much as the full data set. The relaxation induced by the feature-based approach, however, also results in the feature-based approach being slightly more effective in terms of the number of effective samples.

Finally, we show triangle plots of the posterior distribution p40 and the feature-based posterior distribution in Fig. d–g. A triangle plot of the feature-based posterior distribution pF consists of histograms of the marginals pF(ζ|f) and pF(ω|f), plotted in blue in Fig. d and e, and a histogram of pF(θ|f) in Fig. (f). A triangle plot of the posterior distribution p40(θ|z0,…,z80) is shown in Fig. d, e and f. Specifically, we plot histograms of the marginals p40(ζ|z0,…,z80) and p40(ω|z0,…,z80) in purple in Fig. d and g and we plot a histogram of the posterior distribution p40(θ|z0,…,z80) in Fig. e. We find that the marginals pF(ω|f) and p40(ω|z0,…,z80) are nearly identical, which indicates that the feature constrains the frequency ω nearly as well as the full data set. The damping coefficient ζ is less tightly constrained by our feature, which results in a wider posterior distribution pF(ζ|f) than p40(ζ|z0,…,z80). A more sophisticated feature that describes the transient behavior in more detail would lead to different results, but our main point is to show that even our simple feature, which neglects most of the data, leads to useful parameter estimates.

We consider the Lotka–Volterra (LV) equations dxdt=αx-βxy,dydt=-γy+δxy, where t is time, α,β,γ,δ>0 are parameters, and x and y describe “prey” and “predator” populations. Our goal is to estimate the four parameters in the above equations as well as the initial conditions x0=x(0), y0=y(0), i.e., the parameter vector we consider is θ=(α,β,γ,δ,x0,y0)T. Since we do not have prior information about the parameters, we choose a uniform prior distribution over the six-dimensional cube [0,10]6.

We use the lynx and hare data of the Hudson's Bay Company to define a likelihood. The data set covers a period from 1897 to 1935, with one data point per year. Each data point is a number of lynx furs and hare furs, with the understanding that the number of collected furs is an indicator for the overall lynx or hare population. We use data from 1917 to 1927, because the solution of the LV equations is restricted to cycles of fixed amplitude and the data during this time period roughly has that quality. We scale the data to units of “104 hare furs” and “103 lynx furs” (so that all numbers are order one). We use this classical data set here, but predator–prey models have recently also been used in low-dimensional cloud models that can represent certain aspects of large eddy simulations . However, the sole purpose of this example is to demonstrate that the feature-based approach is robust enough for use with “real” data (rather than the synthetic data used in example 1).

We define a feature f by the first (largest) singular value and the first left and right singular vectors of the data. The feature vector f thus has dimension 14 (we have 2×11 raw data points). We compute the noise η for the feature-based likelihood using the “perturbed observation” method as above. We generate 10 000 perturbed data sets by adding realizations of a Gaussian random variable with mean of zero and unit covariance to the data. The resulting sample covariance matrix serves as the matrix Rf in the feature-based likelihood. Note that our choice of noise on the “raw” data is somewhat arbitrary. However, as stated above, the main purpose of this example is to demonstrate our ideas, not to research interactions of lynx and hare populations.

We use the MATLAB implementation of the affine invariant ensemble sampler to solve the feature-based data assimilation problem; see and . We use an ensemble size Ne=12 and each ensemble member produces a chain of length ns=8334. We thus have N = 100 008 samples. Each chain is initialized as follows: we first find the posterior mode using Gauss–Newton optimization. To do so, we perform an optimization with different starting points and then choose the optimization result that leads to the largest feature-based posterior probability. The initial values for our ensemble of walkers are 12 draws from a Gaussian distribution whose mean is the posterior mode and whose covariance is a diagonal matrix with elements (0.02,0.02,0.02,0.02,0.2,0.2). We disregard the first 2500 steps of each chain as “burn-in” and compute an average IACT of 735, using the methods described in . We have also performed experiments with larger ensembles (Ne=12 is the minimum ensemble size for this method), and with different initializations of the chains, and obtained similar results. We have also experimented with the overall number of samples (we used up to 106 samples) and obtained similar results.

We show a triangle plot of the feature-based posterior distribution, consisting of histograms of all one- and two-dimensional marginals, in Fig. .

We observe that there is strong correlation between the parameters α,β,γ,δ, but less so between these parameters and the initial conditions. A reason for the strong correlations between the parameters is that only combinations of the parameters define the solution of the differential equation (after nondimensionalization). Perhaps most importantly, we find that the feature-based posterior distribution constraints the parameters well, especially compared to the prior distribution which is a hyper-cube with sides of length 10.

We plot the trajectories of the LV equations corresponding to 100 samples of the feature-based posterior distribution in Fig. .

We note that the trajectories pass near the 22 original data points (shown as orange dots in Fig. ). The fit of the lynx population is particularly good, but the trajectories of the hare populations do not fit the data well. For example, all model trajectories bend downwards towards the end of the cycle, but the data seem to exhibit an upward tendency. However, this inconsistency is not due to the feature-based approach. In fact, we obtain similar solutions with a “classical” problem formulation. The inconsistency is due to the limitations of the LV model, which is limited to cycles, whereas the data are not cyclic. Nonetheless, our main point here is that the feature-based approach is sufficiently robust that it can handle “real” data and “simple” models. We also emphasize that this data assimilation problem is not difficult to do using the “classical” approach, i.e., without using features. This suggests that this problem is of category (i) in Sect. .

We consider the Earth's magnetic dipole field over timescales of tens of millions of years. On such timescales, the geomagnetic dipole exhibits reversals, i.e., the North Pole becomes the South Pole and vice versa. The occurrence of dipole reversals is well documented over the past 150 Myr by the “geomagnetic polarity timescale” , and the dipole intensity over the past 2 Myr is documented by the Sint-2000 and PADM2M data sets . Several low-dimensional models for the dipole dynamics over the past 2 Myr have been created; see, e.g., , , , , and . We consider two of these models and call the model of the P09 model and the one of the B13 model. The B13 model is the stochastic differential equation (SDE) 9dx=f(x)dt+g(x)dW, where t is time in Myr, x describes the dipole intensity and where W is Brownian motion (see for details). The functions f and g are called the drift- and diffusion coefficients and in , f is a spline and g a polynomial whose coefficients are computed using PADM2M. We use the same functions f and g as described in . The P09 model consists of an SDE of the form () for a “phase”, x, with f(x)=α0+α1sin⁡(2x), g(x)=0.2|α1|, α1=-185Myr-1, α0/α1=-0.9. The dipole is computed from the phase x as D=Rcos⁡(x+x0), where x0=0.3 and R=1.3 defines the amplitude of the dipole.

In both models, the drift, f, represents known, or “resolved” dynamics and the diffusion coefficient g, along with Brownian motion W, represents the effects of turbulent fluid motion of the Earth's liquid core. The sign of the dipole variable defines the dipole polarity. We take the negative sign to mean “current configuration” and a positive sign means “reversed configuration”. A period during which the dipole polarity is constant is called a “chron”. The P09 and B13 models exhibit chrons of varying lengths; however, the mean chron duration (MCD) is fixed. With the parameters cited above the models yield an MCD on the same order of magnitude as the one observed over the past 30 Myr. Simulations of the B13 and P09 model are illustrated in Fig. , where we also show the last 100 Myr of the geomagnetic polarity timescale.

The geomagnetic polarity timescale shows that the Earth's MCD varies over the past 150 Myr. For example, there were 125 reversals between today and 30.9 Myr ago (MCD≈0.25 Myr), 57 reversals between 30.9 and 73.6 Myr ago (MCD≈0.75 Myr), and 89 between 120.6 Myr ago and 157.5 Myr ago (MCD≈0.41 Myr) . The B13 and P09 models exhibit a constant MCD and, therefore, are valid over periods during which the Earth's MCD is also constant, i.e., a few million years. We modify the B13 and P09 models so that their MCD can vary over time, which makes the models valid for periods of more than 100 Myr. The modification is a time-varying, piecewise constant parameter θ(t) that multiplies the diffusion coefficients of the models. The modified B13 and P09 models are thus SDEs of the form 10dx=f(x)dt+θ(t)g(x)dW. We use feature-based data assimilation to estimate the value of θ(t) such that the modified B13 and P09 models exhibit similar MCDs as observed in the geomagnetic polarity timescale over the past 150 Myr. Note that straightforward application of data assimilation is not successful in this problem. We tried several particle filters to assimilate the geomagnetic polarity timescale more directly into the modified B13 and P09 models. However, we had no success with this approach because the data contain only information about the sign of the solution of the SDE.

The feature we extract from the geomagnetic polarity timescale is the MCD, which we compute by using a sliding window average over 10 Myr. We compute the MCD every 1 Myr, so that the “feature data”, f1,…,f149, are 149 values of MCD. We obtain these 149 values by using the geomagnetic polarity timescale (starting at 157.53 Myr ago) and a 10 Myr averaging window. For the first data point, f1, we use slightly less than 10 Myr of data (from 157.53 to 148 Myr ago). The averaging window is always “left to right”, i.e., we average from the past to the present. For the last few data points (f144…f149), the averaging is not centered and uses 10 Myr of data “to the left”.

The geomagnetic polarity timescale and the MCD feature are shown in Fig. .

We note that the averaging window of 10 Myr is too short during long chrons, especially during the “cretaceous superchron” that lasted almost 40 Myr (from about 120 to 80 Myr ago). We set the MCD to be 250 Myr whenever no reversal occurs within our 10 Myr window. This means that the MCD feature has no accuracy during this time period, but indicates that the chrons are long.

To sequentially assimilate the feature data, we assume that the parameter θ(t) is piecewise constant over 1 Myr intervals and estimate its value θk=θ(k⋅1Myr), k=-147,…,0 based on the feature fk and our estimate of θk-1. The feature fk and the modified B13 and P09 models are connected by the equation 11fk=MF(θk)+ηk, which defines the feature-based likelihood and where MF are the computations required to compute the MCD for a given θk. These computations work with a discretization of the modified P09 and B13 SDEs using a 4th-order Runge–Kutta scheme for the deterministic part (f(x)dt), and an Euler–Maruyama scheme of the stochastic part (θ(t)g(x)dW). The time step is 1 kyr. For a given θk, we perform a simulation for a specified number of years and compute MCD based on this run. All simulations are initialized with zero initial conditions (but the precise value of the initial conditions is not essential because it is averaged out over the relatively long simulations) and are performed with a fixed value for θk. The value of θk determines the duration of a simulation, since small values of θk require longer simulations because the chrons tend to become longer. Specifically, we perform a simulation of 300 Myr if θk<0.7, of 100 Myr if 0.7≤θk<1, of 50 Myr if 1≤θk<1.6 and of 20 Myr if θk≥1.6. Note that computation of MCD, in theory, requires an infinite simulation time. We choose the above simulation times to balance a computational budget, while at the same time our estimates of MCD are reliable enough to avoid large noise during feature-based likelihood evaluations.

For the modified B13 model we add one more step. The numerical solutions of this model tend to exhibit short chrons (a few thousand years) during a “proper reversal,” i.e., when the state transitions from one polarity (+1) to the other (-1), it crosses zero several times. On the timescales we consider, such reversals are not meaningful and we filter them out by smoothing the numerical solutions of the modified B13 model by a moving average over 25 kyr. In this way, the chrons we consider and average over have a duration of at least tens of thousands of years.

We investigate how to choose the random variable η in Eq. (), which represents the noise in the feature, by performing extensive computations. For each model (B13 and P09), we choose a grid of θ values that lead to MCD that we observe in the geomagnetic polarity timescale. The θ grid is different for the B13 and P09 model because the dependency of MCD on θ is different for both models and because computations with P09 are slightly faster. For both models, a small θ leads to reversal being rare, even during 300 Myr simulations. We choose to not consider θ smaller than 0.3, again for computational reasons and, as explained above, our simulations and computations lose accuracy during very long chrons such as the cretaceous superchron. Thus, the “actual” θ during a period with large MCD may be smaller than the lower bound we compute; however, we cannot extract that information from the feature data and the computational framework we construct. This means that if the upper or lower bounds of θ are achieved, all we can conclude is that θ is small (large), perhaps smaller (larger) than our assumed lower (upper) bound, which leads to MCDs that are longer (shorter) than what we can actually compute with our model and bounded model parameters.

For each value of θ on our grid, we perform 100 simulations and for each run compute average MCD. The mean and standard deviation of average MCD, computed from these simulations, are shown in Fig. . We occasionally observe large standard deviations for small θk because only a few reversals may occur during these runs, which makes estimates of the standard deviations unreliable (see above). In this case, we assign a maximum standard deviation of 2.5 Myr.

We base our feature-error model ηk on this graph and pick ηk to be a zero-mean Gaussian with a standard deviation σk that we read from the graph as illustrated by Fig. , i.e., for a given fk, we use the standard deviation we computed for the nearest point on our MCD–θ grid.

A feature fk defines ηk and then Eq. () defines a feature-based likelihood. We define a prior distribution by the Gaussian p0,k(θk)=N(θ‾k-1,σ02), where σ0=0.1 and where θ‾k-1 is the mean value we computed at the previous time, k-1 (we describe what we did for the first time step k=1 below). This results in the feature-based posterior pk(θk|fk)∝exp⁡-12σk2fk-MF(θk)2-12σ02θ‾k-1-θk2. We draw 100 samples from this posterior distribution by direct sampling with a proposal distribution qk(θk)=N(μk,σq), where σq=0.05 and where μk is based on the MCD–θ graph shown in Fig. , i.e., we choose μk to be the θ value corresponding to the MCD value fk we observe. We have experimented with other values of σq=0.05 and found that how σq is chosen is not critical for obtaining the results we present. We repeat this process for all but the very first of the features fk. For the first step, k=1, we set the prior distribution equal to the proposal distribution.

Our results are illustrated in Fig. . Figure a and b show 100 samples of the posterior distributions pk(θk|fk) as a function of time, as well as their mean. The panel on the right shows results for the modified B13 model, the panel on the left shows results for the modified P09 model. We note that, for both models, θk varies significantly over time. The effect that a time-varying θ has on the MCD of the modified B13 and P09 models is illustrated in Fig. c and d, where we plot 100 features generated by the modified P09 and B13 models using the 100 posterior values of θk shown in the top row. We note a good agreement with the recorded feature (shown in black). This is perhaps not surprising, since we use the feature data to estimate parameters, which in turn reproduce the feature data. However, this is a basic check that our data assimilation framework produces meaningful results.

We further illustrate the results of the feature-based data assimilation in Fig. , where we plot the geomagnetic polarity timescale as well as the dipole of the modified B13 and P09 models, generated by using a sequence θk, drawn from the feature-based posterior distributions. We note that the modified models exhibit a time-varying MCD and that superchrons (chrons longer than 10 Myr) appear in both models at (roughly) the same time as on the Earth.

The advantage of the feature-based approach in this problem is that it allows us to calibrate the modified B13 and P09 models to yield a time-varying MCD in good agreement with the data (geomagnetic polarity timescale), where “good agreement” is to be interpreted in the feature-based sense. Our approach may be particularly useful for studying how flow structure at the core affects the occurrence of superchrons. A thorough investigation of what our results imply about the physics of geomagnetic dipole reversals will be the subject of future work. In particular, we note that other choices for the standard deviation σ0, that defines expected errors in the feature, are possible and that other choices will lead to different results. If one wishes to use the feature-based approach presented here to study the Earth's deep interior, one must carefully choose σ0. Here we are content with showing how to use feature-based data assimilation in the context of geomagnetic dipole modeling.

We consider the Kuramoto–Sivashinsky equation ∂ϕ∂t=-θ∇2ϕ-∇4ϕ+|∇ϕ|2, where t∈[0,T], the spatial domain is a two-dimensional square [x,y]∈[0,10π]×[0,10π] and the boundary conditions are periodic. Here ∇=∂/∂x,∂/∂y and θ is the parameter we want to estimate. We use a uniform prior distribution over [0,5]. As in earlier examples, our focus is on formulating likelihoods and our choice of prior is not critical to the points we wish to make when illustrating the feature-based techniques. The initial condition of the KS equation is a Gaussian random variable, which we choose as follows. We simulate the KS equation for T “time” units starting from uniformly distributed Fourier coefficients within the unit hypercube (see a few sentences below for how these simulations are done). We pick T large enough so that ϕ(x,y,T) varies smoothly in space. We repeat this process 100 times to obtain 100 samples of solutions of the KS equation. The resulting sample mean and sample covariance matrix of the solution at time T define the mean and covariance of the Gaussian which we use as a random initial condition below.

For computations we discretize the KS equation by the spectral method and exponential time differencing with δt=0.005/θ. For a given θ, we then compute ϕ in physical space by Fourier transform and interpolation onto a 256×256 grid. The solution of the KS equation depends on the parameter θ in a way that a typical spatial scale of the solution, i.e., the scale of the “valleys and hills” we observe, increases as θ decreases, as illustrated by Fig. , where we show snapshots of the solution of the KS equation after 2500 time steps for two different choices of the parameter θ.

The data are 100 snapshots of the solution of the KS equation obtained as follows. For a given θ, we draw an initial condition from the Gaussian distribution (see above) and simulate for 2500 time steps. We save the solution on the 256×256 grid every 50 time steps. We repeat this process, with another random initial condition drawn from the same Gaussian distribution, to obtain another 50 snapshots of the solution. The 100 snapshots constitute a data set with a total number of more than 6 million points.

The feature we extract from the data is as follows. We interpolate the snapshots onto a coarser 64×64 grid and use the coarsened snapshots to compute a covariance matrix. Then we compute the largest eigenvalues of the covariance matrix and compute a linear approximation to the logarithm of the eigenvalues (using least squares). The slope and intercept of this line constitutes the feature. An example is shown in Fig. .

We choose this feature because the parameter θ defines the spatial scale of the solution (see above) and this scale is connected to the length scale of a covariance function of a Gaussian process approximation of the solution. The length scale of the Gaussian process in turn defines the exponential decay of the eigenvalues of its associated covariance matrix and this decay is what we capture by our feature. In simple terms, the larger the length scale, the faster the decay of the eigenvalues.

It is important to note that the feature we construct does not depend on the initial conditions. This is the main advantage of the feature-based approach. Using the feature, rather than the trajectories, enables estimation of the parameter θ without estimation of initial conditions. With a likelihood based on the mismatch of model and data, one has to estimate the parameter θ and the initial conditions, which makes the effective dimension of the problem large, so that the required computations are substantial. Most importantly, estimating the initial condition based on a mismatch of model output and data is difficult because the KS equation is chaotic. For these reasons, the feature-based approach makes estimation of the parameter θ feasible. Note that the feature has also reduced the effective dimension of the problem (see Sect. ) because the number of parameters to be estimated has been reduced from the number of modes (2562) to 1. The price to be paid for this reduction in (effective) dimension is that the feature-based approach does not allow us to compute trajectories that match the data point-wise.

The feature-based likelihood is defined by the equation 12f=MF(θ)+η,η∼N(0,R), where f=F(z) is the feature computed from the data, R is a 2×2 covariance matrix (see below) and where MF is shorthand for the following computational steps for a given parameter θ: i.

Draw random initial conditions and obtain 100 snapshots of the solution of the KS equation with parameter θ.

Finally, we need to choose a covariance matrix R. The perturbed observation approach (see Sect. ) is not useful here. If we assume that we collect data with measurement errors that are uncorrelated in space and time (adding an isotropic Gaussian to each snapshot), then this noise has no effect on the overall spatial scale of the solution and, thus, will not correctly reflect the uncertainty of the feature. The largest source of uncertainty in the feature is sampling error due the small number of snapshots we use for computing the GP approximation. We can decrease the effects of this noise by using more snapshots; however, this increases the computational cost. In addition, this uncertainty due to sampling error makes feature-based likelihood evaluations noisy, i.e., for a fixed θ and feature f, different runs of our simulations may lead to different likelihoods. This rules out Monte Carlo sampling for numerical solution of the data assimilation problem.

We address these issues by using a variational approach and compute an a posteriori estimate of θ, i.e., we estimate θ by maximizing the function g(θ)=exp⁡-12(f-MF(θ))′R-1(f-MF(θ))ifθ∈0,5,0otherwise, which is proportional to the feature-based posterior distribution. This will lead to a point estimate for θ that leads to solutions that are compatible with the data. For point estimates, the covariance R is not so essential. We set this covariance R to be a diagonal matrix with diagonal entries R11=2.25, R22=0.0625. These values are chosen to reflect a relatively large amount of uncertainty in the feature and to balance the different scales of the two components of the feature. However it is important to note that our approach does not allow us to draw conclusions about the uncertainty of our parameter estimate, for which we would require approximations of the posterior distribution. This may not be ideal; however, in view of the computational difficulties, a point estimate is the best we can provide.

We need to decide on a numerical method for solving the optimization problem. Since the function g is noisy and computationally expensive, we cannot compute its derivatives. Global Bayesian optimization (see, e.g., ) is a derivative-free method for optimization in exactly that setting, i.e., when the function to be optimized is computationally expensive to evaluate and noisy. The basic idea of GBO is to model the function g(θ) by a Gaussian process and then to carefully choose additional points for evaluation of the function to improve the GP model. The maximizer of the mean of the GP model is then used to approximate the maximizer of the (random) function g(θ). We first explain how to build an initial GP model for the function g(θ) and then describe how to improve on the model given function evaluations. For more details about GBO, see or references therein.

A GP model for g(θ) consists of the mean and covariance functions

μ(θ)=μ(constant mean function),C(θ,θ′)=σ2exp⁡(θ-θ′)2/L2, where μ,σ,L>0 are “hyperparameters” which we must define. To acknowledge the fact that g(θ) is noisy, we add another hyperparameter, s>0, such that the covariance at the “observed points” θobs is given by C(θobs,θobs′)+s (see Sect. 3.3.5 of ). We define the hyperparameters based on a small number of model function evaluations. Specifically, we evaluate g at three points within [0,5] generated by a Sobol sequence, which is a space-filling sequence of quasi-random points. This procedure suggests evaluation of the function at the boundaries and “in the middle” (see Fig. a). Given these three points (θi,g(θi)), i=1,2,3, we maximize the “log marginal likelihood”, which describes the probability of the three function evaluations (θi,g(θi)) (see Sect. 3.3.6 of ). This optimization is computationally inexpensive because it does not involve evaluating g or solving the KS equation. We use an interior-point method (MATLAB's “fmincon”) to carry out the optimization and enforce the bounds 0≤L≤1, 0.3≤σ2≤1, 0≤s≤0.1, 0≤μ≤2. This results in a crude approximation of g. We update this initial GP by the three function evaluations we already have, i.e., we recompute the mean μ and the covariance C, given these three function evaluations. The result is the GP illustrated in Fig. a, where we show the mean (blue) and 200 samples (turquoise) of the updated GP, along with the three sample points (purple dots). Note that the GP model does not reflect the fact that g(θ) is nonnegative. However, GBO is not easily modified to optimize nonnegative functions.

To improve our GP model of g(θ) we wish to evaluate the function at additional points and we use the “expected improvement” criterion to determine these points. Expected improvement suggests points for additional evaluations of g(θ) using a trade-off between where the function is already known to be large and where the function is unknown (see Sect. 3.4.1 of ). This led to good results for our problem; however, more advanced methods, e.g., knowledge gradient, may improve overall performance of the algorithm. We stopped the optimization when the integrated expected improvement is below a threshold (10-4 in our case). With this setup, we evaluated g(θ) 5 more times and computed the maximizer of g(θ) to be θ∗=3.29, which is near the parameter value we used to generate the feature data (θtrue=3.38).

The updated GP model is illustrated in the right panel of Fig. , where we show the mean (blue), the initial and additional points where g(θ) is evaluated (purple and red dots respectively) and 100 realizations of the updated GP model (turquoise). We also show 100 realizations of g(θ), obtained by evaluating g(θ) repeatedly over a grid of 100 equally spaced points. We note that the GP accurately describes the function and our confidence in the function for θ>2.5, where most of the function evaluations took place. The uncertainty is large for θ<2.5, which could be reduced by additional function evaluations. In summary, the feature-based approach, combined with an appropriate numerical technique for optimizing noisy functions, is successful in estimating a parameter of a chaotic partial differential equation.