Skewness of the Prior Through Position Errors and Its Impact on Data Assimilation

Abstract

Uncertainty in the position of a feature is a ubiquitous influence on data assimilation (DA) in geophysical applications. This chapter explores the properties of distributions arising from the uncertainty of the location of a flow feature. It is shown that distributions arising from phase uncertainty have surprisingly complex, non-Gaussian characteristics. These non-Gaussian characteristics are explored from an ensemble DA perspective in which the skewness (third-moment) is shown to be a significant contributor to the state-estimates obtained through Bayesian state estimation. Idealized examples, as well as an example in a real tropical cyclone using a state-of-the-art numerical weather prediction model, will be shown.

Appendices

Appendix 1: Matrix and Notation Definitions

The prior error covariance matrix is extended to include the higher moments:

$$\mathbf{\hat{P}}_{f} = \frac{{\mathbf{\hat{Z}\hat{Z}}}^{T}} {K - 1} = \left [\begin{array}{cc} \mathbf{P}_{f} & \mathbf{T}_{f} \\ \mathbf{T}_{f}^{T}&\mathbf{F}_{f} -\mathbf{p}_{f}\mathbf{p}_{f}^{T}\\ \end{array} \right ],$$

(7.44)

whose square root form may be approximated with an ensemble as

$$\mathbf{\hat{Z}} = \left [\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \boldsymbol{\epsilon }_{1} \quad & \boldsymbol{\epsilon }_{2} \quad &\cdots \quad & \boldsymbol{\epsilon }_{K} \\ \boldsymbol{\epsilon }_{1} \otimes \boldsymbol{\epsilon }_{1} -\mathbf{p}_{f}\quad &\boldsymbol{\epsilon }_{2} \otimes \boldsymbol{\epsilon }_{2} -\mathbf{p}_{f}\quad &\cdots \quad &\boldsymbol{\epsilon }_{K} \otimes \boldsymbol{\epsilon }_{K} -\mathbf{p}_{f}\\ \quad \end{array} \right ]$$

(7.45)

and the vectorized covariance matrix, p _f  = vec(P _f ), is an N ²-vector constructed from the concatenation of the N columns of P _f and whose organization follows that of the Kronecker product “ ⊗ ”, and K is the ensemble size.

The extended observation operator takes the following form:

$$\mathbf{\hat{H}} = \left [\begin{array}{c} \mathbf{H}\\ \mathbf{H} _{2}\\ \end{array} \right ] = \left [\begin{array}{c} \mathbf{H}\\ \mathbf{H} \otimes \mathbf{H}\\ \end{array} \right ].$$

(7.46)

Note that for nonlinear observation operators one would not use a linearized form of the operator. Instead, the correct procedure is to operate the nonlinear observation operator on each member of the ensemble and then perform linear or nonlinear regression on this new distribution of predicted prior observations against the state variables needing update (Houtekamer and Mitchell 2001).

The covariance matrices in the extended state-space takes the form:

$$\displaystyle\begin{array}{rcl} & & \left \langle \boldsymbol{\epsilon }_{f}\mathbf{{\hat{v}}^{{\prime}T}}\right \rangle = \left [\begin{array}{ccc} \mathbf{P}_{ f}{\mathbf{H}}^{T}&\mathbf{T}_{ f}\mathbf{H}_{2}^{T}&\cdots \\ \end{array} \right ],\end{array}$$

(7.47)

$$\displaystyle\begin{array}{rcl} & & \left \langle \mathbf{{\hat{v}}^{{\prime}}{\hat{v}}^{{\prime}T}}\right \rangle = \left \langle {\mathbf{\hat{v}\hat{v}}}^{T}\right \rangle -\left \langle \mathbf{\hat{v}}\right \rangle \left \langle {\mathbf{\hat{v}}}^{T}\right \rangle ,\end{array}$$

(7.48)

$$\displaystyle\begin{array}{rcl} & & \left \langle {\mathbf{\hat{v}\hat{v}}}^{T}\right \rangle = \left [\begin{array}{ccc} \mathbf{HP}_{f}{\mathbf{H}}^{T} + \mathbf{R}& \mathbf{HT}_{ f}\mathbf{H}_{2}^{T} &\cdots \\ \mathbf{H}_{2}\mathbf{T}_{f}^{T}{\mathbf{H}}^{T} &\mathbf{H}_{2}\mathbf{F}_{f}\mathbf{H}_{2}^{T} + \mathbf{A} + \mathbf{B} + \mathbf{C} + \mathbf{R}_{4} & \cdots \\ \vdots & \vdots & \ddots\\ \end{array} \right ]\end{array}$$

(7.49)

$$\displaystyle\begin{array}{rcl} & & \left \langle \mathbf{\hat{v}}\right \rangle \left \langle {\mathbf{\hat{v}}}^{T}\right \rangle = \left [\begin{array}{ccc} \mathbf{0}& \mathbf{0} &\cdots \\ \mathbf{0}&\left \langle {\mathbf{v}}^{2}\right \rangle \left \langle {\mathbf{v}}^{2T}\right \rangle &\cdots \\ \vdots & \vdots & \ddots\\ \end{array} \right ],\end{array}$$

(7.50)

and 0 is the p ×p zero matrix. The matrices in (7.17) are defined as:

$$\displaystyle\begin{array}{rcl} \mathbf{P}_{f} = \left \langle \boldsymbol{\epsilon }_{f}\boldsymbol{\epsilon }_{f}^{T}\right \rangle ,& &\end{array}$$

(7.51a)

$$\displaystyle\begin{array}{rcl} \mathbf{T}_{f} = \left \langle \boldsymbol{\epsilon }_{f}\boldsymbol{\epsilon }_{f}^{2T}\right \rangle ,& &\end{array}$$

(7.51b)

$$\displaystyle\begin{array}{rcl} \mathbf{F}_{f} = \left \langle \boldsymbol{\epsilon }_{f}^{2}\boldsymbol{\epsilon }_{ f}^{2T}\right \rangle ,& &\end{array}$$

(7.51c)

$$\displaystyle\begin{array}{rcl} \mathbf{R} = \left \langle \boldsymbol{\epsilon }_{o}\boldsymbol{\epsilon }_{o}^{T}\right \rangle ,& &\end{array}$$

(7.52a)

$$\displaystyle\begin{array}{rcl} \mathbf{R}_{4}& = \left \langle \boldsymbol{\epsilon }_{o}^{2}\boldsymbol{\epsilon }_{o}^{2T}\right \rangle ,&\end{array}$$

(7.52b)

$$\displaystyle\begin{array}{rcl} & & \mathbf{A} = \left \langle \boldsymbol{\epsilon }_{o}^{2}\boldsymbol{\epsilon }_{ f}^{2T}\right \rangle \mathbf{H}_{ 2}^{T} + \mathbf{H}_{ 2}\left \langle \boldsymbol{\epsilon }_{f}^{2}\boldsymbol{\epsilon }_{ o}^{2T}\right \rangle ,\end{array}$$

(7.53)

$$\displaystyle\begin{array}{rcl} & & \mathbf{B} = \mathbf{R} \otimes \mathbf{HP}_{f}{\mathbf{H}}^{T} + \mathbf{HP}_{ f}{\mathbf{H}}^{T} \otimes \mathbf{R},\end{array}$$

(7.54)

$$\displaystyle\begin{array}{rcl} & & \mathbf{C} = \left \langle \left (\boldsymbol{\epsilon }_{o}\boldsymbol{\epsilon }_{f}^{T}{\mathbf{H}}^{T}\right ) \otimes \left (\mathbf{H}\boldsymbol{\epsilon }_{ f}\boldsymbol{\epsilon }_{o}^{T}\right )\right \rangle + \left \langle \left (\mathbf{H}\boldsymbol{\epsilon }_{ f}\boldsymbol{\epsilon }_{o}^{T}\right ) \otimes \left (\boldsymbol{\epsilon }_{ o}\boldsymbol{\epsilon }_{f}^{T}{\mathbf{H}}^{T}\right )\right \rangle ,\end{array}$$

(7.55)

and H ₂ = H ⊗ H is the matrix operator that takes an N ² vector into the p ² predictor space and copious use of the identity, $(\mathbf{H}\boldsymbol{\epsilon }_{f}) \otimes (\mathbf{H}\boldsymbol{\epsilon }_{f}) = (\mathbf{H} \otimes \mathbf{H})(\boldsymbol{\epsilon }_{f} \otimes \boldsymbol{\epsilon }_{f})$, has been made.

In (7.51a), P _f is a square matrix listing the necessary second moments of the prior distribution. In (7.51b), T _f is a rectangular matrix listing the necessary third moments of the prior distribution. In (7.51c), F _f is a square matrix listing the necessary fourth moments of the prior distribution. In (7.52a), R ₄ is a square matrix listing the necessary fourth moments of the observation likelihood. Note that even when the observation error covariance matrix, R, is diagonal R ₄ is not diagonal. The matrices A, B, andC are sparse, square matrices that represent various combinations of observation error covariances and forecast error covariances.

The covariance matrix of squared innovations is

$$\displaystyle\begin{array}{rcl} \boldsymbol{\Pi } = \mathbf{H}_{2}\mathbf{F}_{f}\mathbf{H}_{2}^{T} + \mathbf{A} + \mathbf{B} + \mathbf{C} + \mathbf{R}_{ 4} -\mathbf{H}_{2}\mathbf{T}_{f}^{T}{\mathbf{H}}^{T}{\left (\mathbf{HP}_{ f}{\mathbf{H}}^{T} + \mathbf{R}\right )}^{-1}\mathbf{HT}_{ f}\mathbf{H}_{2}^{T} -\left \langle {\mathbf{v}}^{2}\right \rangle \left \langle {\mathbf{v}}^{2T}\right \rangle.& &\end{array}$$

(7.56)

Appendix 2: Non-Gaussian Phase Uncertainty from Variable Shear Flows

To isolate the effects of phase uncertainty we focus on the one-dimensional advection equation:

$$\frac{\partial p} {\partial t} + c\left (x\right )\frac{\partial p} {\partial x} = 0,$$

(7.57)

where p = p(x, t) is the wavefield being advected at the speed c(x). We attach to (7.57) an unbounded domain in x as well as a localized initial condition such as the functions (7.42), which was our prototype for the surface pressure field of members of a prior distribution of tropical cyclones. By the method of characteristics we know that the solutions to (7.57) move away from their initial location according to,

$$\frac{\mathit{dX}} {\mathit{dt}} = c(X),$$

(7.58)

where X = X(t) is the location of, say, the minimum central pressure of the function (7.42). We may immediately note that even though equation (7.57) is linear in the amplitude of the disturbance, equation (7.58) may be non-linear if the function c(x) is non-linear. Hence, the motion of the location of the minimum central pressure may evolve non-linearly and therefore even if the initial distribution of phase uncertainty is Gaussian it still may evolve into a non-Gaussian phase distribution owing to the non-linearity in (7.58). Two examples follow:

• (1) Linear Shear

In the case where the shear is a linear increasing function of distance from the origin, i.e.

$$c(x) = c_{0} \frac{x} {L},$$

(7.59)

where c ₀ is a characteristic phase speed and L is characteristic length scale. By inserting (7.59) into (7.58) and solving finds

$$X(t) = X_{0}\exp \left (\frac{c_{0}t} {L} \right ),$$

(7.60)

where X ₀ is the initial location of the minimum central pressure. In Sect. 7.3 the parameter X ₀ was denoted as $\varphi$ and was normally distributed. Notice that if the location of the minimum central pressure is normally distributed with mean $\bar{x}$ and variance σ ², then at a later time, t, the phase distribution will be normal with

$$X \sim N\left (\bar{x}\exp \left (\frac{c_{0}t} {L} \right ){,\sigma }^{2}\exp \left (\frac{2c_{0}t} {L} \right )\right ).$$

(7.61)

Hence, linear shear produces disturbances that move away from their initial location exponentially with time. Nevertheless, disturbances in linear shear preserve the Gaussian character of their initial phase uncertainty.

• (2) Quadratic Shear

In the case where the shear is a quadratic function of distance from the origin, i.e.

$$c(x) = c_{0}{\left ( \frac{x} {L}\right )}^{2},$$

(7.62)

the solution to (7.58) is

$$\displaystyle\begin{array}{rcl} X(t) = \frac{L} { \frac{L} {X_{0}} -\frac{c_{0}t} {L} }.& &\end{array}$$

(7.63)

In this case, the phase locations of the disturbances move away from their initial locations faster than exponential with time. This can be seen by the fact that disturbances in linear shear flow approach infinity, i.e. X → ∞, as t → ∞, but disturbances in quadratic shear flow reach infinity in finite-time $[t_{\infty } = {L}^{2}/(c_{0}X_{0})]$. Note that in the limit of small time, i.e., $t \ll {L}^{2}/(c_{0}X_{0})$ then equation (7.63) behaves approximately as

$$X(t) \approx X_{0} + \frac{c_{0}t} {{L}^{2}} X_{0}^{2},$$

(7.64)

which is obviously non-Gaussian even when X ₀ is normally distributed and becomes increasingly non-Gaussian as time goes on.

Hence, the structure of the shear flow the disturbances are being advected within will determine whether the resulting phase distribution will be Gaussian or non-Gaussian at some time later. Moreover, even though (7.57) is an equation linear in the amplitude of the disturbance, its characteristics maybe be non-linear, which could lead to non-Gaussian distributions. This implies that it is not sufficient to simply note that the physical system (7.57) is linear in amplitude in order to assess whether a Kalman filter will be optimal or not. The correct condition is that both the model and its characteristics must be linear and the initial distribution one draws from must also be Gaussian. Given the severity of these conditions it would appear that it is unlikely that the phase distributions of actual flow features found in nature would always maintain a normally distributed character. Rather, it would seem more likely that the atmosphere would evolve through time and flow configurations in which a Gaussian initial phase distribution would be altered to have non-Gaussian characteristics.