Robust Land-Surface Parameterisation for Repeated Topographic Surveys in Dynamic Environments with Adaptive State-Space Models

Abstract

The proliferation of unmanned aerial vehicles has enabled cost-effective topographic surveys to be collected at high frequencies. However, terrain analyses rarely take advantage of the information provided by repeated observations. As a result, the ability to characterize the topographic surface and surface changes resulting from dynamic surface processes is undermined by the accumulation and propagation of uncertainty. Accurate surface model parameterisation benefits all derived local characteristics, such as surface slope and curvature. To address this, several advances in adaptive Kalman filtering were evaluated with respect to surface model coefficient estimation error, and the sensitivity to initial noise statistics was tested. A simple surface with exactly known parameters was simulated for a set of common geomorphological change regimes and survey temporal distributions. The results confirmed that all Kalman filters reduced error relative to a least-squares estimator under static conditions. Only adaptive filters outperformed a least-squares estimator under dynamic conditions, where average error was often reduced by approximately 50%, and up to 80%. However, adaptive Kalman filters exhibited up to a 40% increase in maximum error relative to a least-squares estimator in response to sudden surface changes, returning to lower error within 15–25 epochs. The adaptive Kalman filters were sensitive to the overestimation of measurement noise greater than two orders of magnitude from the true noise, resulting in degraded performance. Adaptive Kalman filters consistently and substantially reduced spatio-temporal coefficient error, which includes an estimate of local vertical displacement. The results demonstrated that adaptive Kalman filters address challenges related to the sensitivity of conventional Kalman filter performance to sub-optimal parameterisation, and they are robust estimators for both terrain analysis and surface change analysis when multiple surveys are available. Therefore, adaptive Kalman filters are well-suited for analyzing the local properties of topographic surfaces in general.

1. Introduction

Advances in remote sensing platforms, including Unmanned Aerial Vehicles (UAVs) and sensors, have increased the ability to collect high-resolution topographic data with flexible revisit times. Data collected from UAVs equipped with Light Detection And Ranging (LiDAR) sensors [1], cameras [2] and Interferometric Synthetic Aperture Radar (InSAR) [3] are readily processed into Digital Elevation Models (DEMs), and other elevation data sets [4]. While it is generally the case that survey capabilities are mission specific (e.g., [5,6]), it is now possible to survey several square kilometers within hours of time in the field [7]. Moreover, vertical and horizontal accuracy are frequently between 1 and 20 cm, with spatial resolutions between 5 and 50 cm, depending on the type of sensor, environmental conditions, and vegetation density, among other factors (e.g., [1,2,8,9,10,11,12]).

The properties of UAV-based topographic surveys are ideal for monitoring a wide variety of natural hazards [13]. The collection of multi-temporal topographic data also enables geomorphic change detection, which attempts to make inferences about the surface evolution based on observations of topographic change between surveys [14]. For example, InSAR is commonly used to infer motion from differential phase shifts in line-of-sight observations [15], and this information has proven useful for the validation of subsistence models [16]. For other sensors, a common practice to introduce the temporal dimension to elevation observations is through the subtraction of DEMs representing data collected at different times. This operation is called Difference of DEMs (DoD) when applied to raster elevation models, and is often used to calculate vertical or volumetric differences (i.e., the displacement of matter) resulting from geomorphological processes (e.g., [14,17,18,19]). This represents a dynamic analogue to the process–form relationship between geomorphological processes and the resulting landforms and landscape [20], where dynamic processes result in observable surficial changes over time.

However, a major problem with directly comparing multiple sets of observations is the inclusion and propagation of error and uncertainty. Several interdependent factors combine to make accounting for local uncertainty challenging [21]. In addition to artifacts [22], DEMs contain errors from interpolation, differentiation, and displacement [23]. It is well known that errors present in the source data are propagated to results and may be exaggerated by an analysis [24], and thus DEM subtraction, using two DEMs, suffers doubly. Minimizing error in surface models is important when calculating surface derivatives such as slope and various curvatures [25,26] because the sensitivity of these calculations to noise and error increase with order [27]. Kalman filters are a class of discrete-time state-space models that are well-suited to address noise and minimize uncertainty in dynamic systems using multiple observations. In fact, Kalman filters are optimal estimators if the noise statistical properties are known and Gaussian [28]. Non-linear, non-Gaussian state models such as particle filters and other sequential Monte Carlo methods may be used with better results because they make less assumptions about the data; however, this comes at the cost of increased computation [29]. However, the linear surface models used to extract spatial partial derivatives are compatible with conventional linear Kalman filter estimators.

Despite these advantages, Kalman filters have seen limited application to digital terrain analysis. DEM generation from InSAR data is a common application for Kalman filters because of the suitability for processing raw InSAR data [30,31]. Similar use cases are found in bathymetric applications where Kalman filters provide robust estimates of the sea floor position [32,33]. Alternatively, Kalman filters have also been applied to outlier detection and removal [34]. Wang [35] used a spatial Kalman filter (i.e., single time with neighboring observations) to remove outliers prior to surface differentiation. Similarly, Lawrence and Celestine [36] used Kalman filters to estimate elevation and its first-order spatial partial derivatives to smooth DEM values. Orti et al. [37] used Kalman filters to integrate multi-user topographic observations to improve gully delineation. However, the sensitivity to initial parameterisation of noise statistics, the difficulty in estimating these parameters, and the large impact they have on performance [38] are all major challenges that have limited the adoption of Kalman filters for terrain analysis. This problem is further complicated by the fact that many local surface models require an accurate characterization of local noise statistics. This is not feasible if these noise statistics vary spatially (e.g., [21,39]) or temporally. While automatic calibration algorithms exist (e.g., [40,41,42,43]), they require many epochs of data to search for the optimal solution. Winiwarter et al. [44] developed a spatio-temporal Kalman filter to compare LiDAR point clouds, reporting favourable noise reduction properties and also sensitivity to large and sudden surface changes. Thus, a performant filter must adapt not only to unknown and potentially variable noise statistics but also to a variety of local surface dynamics.

Rather than to denoise DEMs or remove artifacts, the purpose of this research is to improve the accuracy of local surface model coefficient estimates and associated temporal changes under static and dynamic geomorphological regimes. This research combines several advances in adaptive Kalman filtering into an algorithm that exploits the information provided by multiple topographic observations from repeat surveys. Several filter configurations are implemented, and the sensitivity of each filter to sub-optimal noise statistic parameterisation is evaluated using simulated surface models with exactly known values. Multiple dynamic regimes are simulated to assess how each filter responds to model violations (i.e., observations that are inconsistent with the model). Establishing filter performance characteristics in a controlled setting is necessary to understand estimator performance and develop confidence in all derived data.

2. Methods

2.1. Advances in State-Space Models and the Kalman Filter

State-space models estimate a set of state variables, which may be observed or unobserved, given observations over time. These models have widespread adoption in engineering and science to predict future states, estimate current states, or smooth previous states [45]. The general form of a discrete-time state-space model and the associated measurement model are

$$\begin{array}{ccc}\hfill {\mathit{x}}_k& ={\mathit{A}}_k{\mathit{x}}_{k-1}+{\mathit{B}}_k{\mathit{u}}_k+{\mathit{\omega }}_k,\hfill & \hfill {\mathit{\omega }}_k\sim \mathcal{N}\left(0,{\mathit{Q}}_k\right)\end{array}$$

(1)

$$\begin{array}{ccc}\hfill {\widehat{\mathit{z}}}_k& ={\mathit{C}}_k{\mathit{x}}_k+{\mathit{D}}_k{\mathit{u}}_k+{\mathit{\nu }}_k,\hfill & \hfill {\mathit{\nu }}_k\sim \mathcal{N}\left(0,{\mathit{R}}_k\right)\end{array}$$

(2)

where ${\mathit{x}}_k\in {\mathbb{R}}^{b\times 1}$ is the state vector with b states, ${\widehat{\mathit{z}}}_k\in {\mathbb{R}}^{a\times 1}$ is the output vector with a elements, ${\mathit{u}}_k$ is a control vector, ${\mathit{\omega }}_k$ and ${\mathit{\nu }}_k$ are noise vectors based on a multivariate normal distribution with process noise covariance ${\mathit{Q}}_k\in {\mathbb{R}}^{b\times b}$ and measurement noise covariance ${\mathit{R}}_k\in {\mathbb{R}}^{a\times a}$, respectively, ${\mathit{A}}_k\in {\mathbb{R}}^{b\times b}$ is the state transition matrix, which extrapolates the state vector to the next time, ${\mathit{B}}_k$ is the input matrix, ${\mathit{C}}_k\in {\mathbb{R}}^{a\times b}$ is the measurement matrix, which converts the state variables to outputs, ${\mathit{D}}_k$ is the feed-forward matrix, and $k\in \mathbb{N}$ is a discrete-time index.

The Kalman filter algorithm is a popular recursive implementation of a state-space model that minimizes estimate error [46] and has mathematical connections to Bayesian filtering [47]. The recursive property only requires that the current state and state uncertainty matrix ($\mathit{P}$) are stored, and they are updated as new information is collected. A Conventional Kalman filter (CKF) is a two-stage, optimal, linear, minimum variance-of-error filter that fuses a priori predictions with a posteriori corrections once observations are available at each time step. The linear equations of the Kalman filter are presented for a single epoch (k) below:

Stage 1:

Prediction

$$\begin{array}{cc}\hfill {\mathit{x}}_{k|k-1}& ={\mathit{A}}_k{\mathit{x}}_{k-1|k-1}+{\mathit{B}}_k{\mathit{u}}_k\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\mathit{P}}_{k|k-1}& ={\mathit{A}}_k{\mathit{P}}_{k-1|k-1}{\mathit{A}}_k^{\top }+{\mathit{Q}}_k\hfill \end{array}$$

(4)

Stage 2:

Measurement update

$$\begin{array}{cc}\hfill {\mathit{S}}_k& ={\mathit{C}}_k{\mathit{P}}_{k|k-1}{\mathit{C}}_k^{\top }+{\mathit{R}}_k\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {\mathit{K}}_k& ={\mathit{P}}_{k|k-1}{\mathit{C}}_k^{\top }{\mathit{S}}_k^{-1}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\mathit{x}}_{k|k}& ={\mathit{x}}_{k|k-1}+{\mathit{K}}_k\left({\mathit{z}}_k-{\mathit{C}}_k{\mathit{x}}_{k|k-1}\right)\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\mathit{P}}_{k|k}& =\left(\mathit{I}-{\mathit{K}}_k{\mathit{C}}_k\right){\mathit{P}}_{k|k-1}\hfill \end{array}$$

(8)

where ${\mathit{z}}_k\in {\mathbb{R}}^{a\times 1}$ is a vector of observations, ${\mathit{S}}_k\in {\mathbb{R}}^{a\times a}$ is the innovation covariance matrix, and ${\mathit{K}}_k\in {\mathbb{R}}^{b\times a}$ is the Kalman gain matrix.

Initial values for $\mathit{x}$, $\mathit{P}$, $\mathit{R}$, and $\mathit{Q}$ must be provided. $\mathit{x}$ can be set to zero and updated with the filter functions, or estimated in some way prior to Kalman filtering. A common practice to initialize $\mathit{P}$ involves creating a diagonal matrix with a large number (e.g., ${10}^3$) so that the first update stage favours incoming data, and it is subsequently updated to more appropriate values. Estimating $\mathit{R}$ and $\mathit{Q}$ are significantly more challenging, and both can strongly impact filter performance [28]. A common practice is to estimate these values from a combination of expert knowledge of the system and tuning [48]. $\mathit{Q}$ is particularly challenging to estimate because it represents an abstraction of a system with potentially unobservable components. Underestimating $\mathit{Q}$ limits error minimization, and overestimating $\mathit{Q}$ increases uncertainty such that incoming noisy measurements dominate the state estimate. In addition to optimization strategies that use large data sets to tune these parameters (e.g., [41,42,43,49]), several advances have been made to Kalman filter implementations to improve performance with sub-optimal noise statistic estimates.

2.1.1. Adaptive Filters

Several techniques have been developed to address the challenge of estimating state and measurement covariance. Adaptive filters are a popular solution that involves modulating various parameters to minimize the negative impact of sub-optimal initial estimates. Sage and Husa [50] developed a covariance-matching technique to evaluate the auto-covariance of state and measurement error vectors over a number of time lags. As the number of lags increases, the covariance estimates approach the true values at the expense of computational resources to store the residuals. An innovation-based approach was later developed to update covariance estimates in real-time [51,52]. Sorenson and Sacks [53] developed a fading memory filter as an alternative approach to reduce the effect of previous epochs on current estimates by using an adaptive factor to inflate state uncertainty. Yang et al. [54] developed this idea to adapt this factor based on a robust maximum-likelihood estimation. Subsequent research combined the adaptive factors to mix current covariance estimates with that of the previous epoch, thus addressing the limitations of computing the auto-covariance [55] with an adaptive factor [56].

This research employs adaptive Kalman filters based on an adaptive estimator [54,57]. An adaptive parameter $\alpha \in \mathbb{R}\cap \left[0,1\right]$ is used to minimize the impact of previous epochs, favouring recent observations in state estimates. Yang et al. [54] note that the adaptive estimator yields a variance-Weighted Least Squares (WLS) estimator when ${\alpha }_k=0$, the normal Kalman equations when ${\alpha }_k=1$, and balances the contributions of these estimators when $0<{\alpha }_k<1$. An adaptive estimator can be expressed using the following Kalman equations:

$$\begin{array}{cc}\hfill {\mathit{x}}_{k|k}& ={\mathit{x}}_{k|k-1}+{\overline{\mathit{K}}}_k\left({\mathit{z}}_k-{\mathit{C}}_k{\mathit{x}}_{k|k-1}\right)\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\mathit{P}}_{k|k}& =\left(\mathit{I}-{\overline{\mathit{K}}}_k{\mathit{C}}_k\right){\mathit{P}}_{k|k-1}/{\alpha }_k\hfill \end{array}$$

(10)

where $\overline{\mathit{K}}$ is an adaptive Kalman gain matrix:

$$\begin{array}{cc}\hfill {\overline{\mathit{K}}}_k& ={\displaystyle \frac{1}{{\alpha }_k}}{\mathit{P}}_{k|k-1}{\mathit{C}}_k^{\top }\left({\displaystyle \frac{1}{{\alpha }_k}}{\mathit{C}}_k{\mathit{P}}_{k|k-1}{\mathit{C}}_k^{\top }+{\mathit{R}}_k\right)\hfill \end{array}$$

(11)

The calculation of ${\alpha }_k$ is based on the discrepancy between the predicted state and an estimate based on current information using Equation (12). Yang and Gao [58] demonstrated that an approximately optimal adaptive factor quantifies the divergence between the state discrepancy relative to the system uncertainty. This can be evaluated by a learning score ${\upsilon }_k$ using Equation (13). The learning score is normalized using a descending three-segment function [54] based on a normal segment, a weight reduction segment, and an elimination segment defined by Equation (14).

$$\begin{array}{cc}\hfill \Delta {\mathit{x}}_k& ={\mathit{x}}_{k|k-1}-{\left({\mathit{C}}_k^{\top }{\mathit{R}}_k^{-1}{\mathit{C}}_k\right)}^{-1}{\mathit{C}}_k^{\top }{\mathit{R}}_k^{-1}{\mathit{z}}_{\mathit{k}}\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\upsilon }_k& ={\displaystyle \frac{\parallel \Delta \mathit{x}\parallel }{tr\left({\mathit{P}}_{k|k-1}\right)}}\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill \alpha \left(\upsilon ,c_1,c_2\right)& =\left\{\begin{array}{cc}1\hfill & \upsilon \le c_1\hfill \\ {{\displaystyle \frac{c_1}{\upsilon }}}^{{\left(\frac{c_2-\upsilon }{c_2-c_1}\right)}^2}\hfill & c_1<\upsilon \le c_2\hfill \\ 0\hfill & \upsilon >c_2\hfill \end{array}\right.\hfill \end{array}$$

(14)

where $c_1$ delineates the normal segment from the weight reduction segment and $c_2$ delineates the elimination segment. The lower limit defines the state discrepancy at which a standard Kalman filter is used and the upper limit defines the state discrepancy at which a Weighted Least Squares (WLS) estimator is used. Both upper and lower limits can be determined experimentally, or by using an appropriate probability distribution [59]. Alternatively, a two-segment version that asymptotically approaches zero can be used to control the initiation of adaptation using Equation (15).

$$\alpha \left(\upsilon ,c_1\right)=\left\{\begin{array}{cc}1\hfill & \upsilon \le c_1\hfill \\ {\left({\displaystyle \frac{c_1}{\upsilon }}\right)}^2\hfill & \upsilon >c_1\hfill \end{array}\right.$$

(15)

2.1.2. Covariance Estimation

The matrices ${\mathit{R}}_k$ and ${\mathit{Q}}_k$ can also be implemented to be estimated recursively. However, both matrices are interdependent, and varying both simultaneously may lead to unstable estimates [60]. Only an adaptive ${\mathit{Q}}_k$ is considered since measurement error data are often available for topographic applications. A common approach called covariance-matching uses a moving window [50] to approximate the true values (c.f., [51,52], for the derivation). Equation (16) is used to calculate the residual vector, which is averaged over several time lags in Equation (17), yielding an estimate of process noise covariance using Equation (18). Note that the estimate is based on the residual derived from the a posteriori estimate instead of the innovation, which is based on the a priori estimate. It is possible to use either the innovation or residual; however, the residual is generally regarded as the superior measure [52,60].

$$\begin{array}{cc}\hfill {\mathit{\delta }}_{\mathit{k}}& ={\mathit{z}}_k-{\mathit{C}}_k{\mathit{x}}_{k|k}\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill E\left[{\mathit{\delta }}_k{\mathit{\delta }}_k^{\top }\right]& ={\displaystyle \frac{1}{m}}\sum _{j=0}^{m-1}{\mathit{\delta }}_{k-j}{\mathit{\delta }}_{k-j}^{\top }\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\mathit{Q}}_k& =E\left[{\mathit{\delta }}_k{\mathit{\delta }}_k^{\top }\right]+{\mathit{K}}_{k}E\left[{\mathit{\delta }}_k{\mathit{\delta }}_k^{\top }\right]{\mathit{K}}_k\hfill \end{array}$$

(18)

where m is the number of time lags.

An adaptive parameter is used to blend previous and current process covariance estimates, serving a similar function as $\alpha$ in Section 2.1.1. This allows an on-line estimation scheme [55] that emulates a windowed approach while avoiding the computing resources required to store and analyze the windowed time lags using Equation (19). This can be combined with a covariance-scaling approach to inflate or deflate the process noise covariance [61] while preserving the internal structure of the process noise model (i.e., the noise model used to initialize ${\mathit{Q}}_0$). A noise variance parameter q is used to scale the process uncertainty model based on the ratio between the estimated and predicted process uncertainty matrices in Equation (20). Equation (21) applies the resulting scalar to the process noise matrix of the next epoch. Note that Equation (20) can be modified by taking the square root to smooth the multiplier, as suggested by Ding et al. [61]; however, this is not required.

$$\begin{array}{cc}\hfill {\widehat{\mathit{Q}}}_k& ={\alpha }_k{\mathit{Q}}_k+\left(1-{\alpha }_k\right)\left({\mathit{K}}_k{\mathit{\delta }}_k{\mathit{\delta }}_k^{\top }{\mathit{K}}_k\right)\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill q_{k+1}& =q_k{\displaystyle \frac{tr\left({\widehat{\mathit{Q}}}_k\right)}{tr\left({\mathit{Q}}_{\mathit{k}}\right)}}\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\mathit{Q}}_{k+1}& ={\mathit{Q}}_k\times q_{k+1}\hfill \end{array}$$

(21)

2.1.3. M-Type Robust Estimators

Huber [62]’s maximum likelihood-type estimators (M-type, or M-theory) are a class of robust estimators that are suitable when noise statistics are unknown [63]. The premise is that individual measurement error covariance is inflated to match the corresponding normalized residual. A robust M-type Kalman filter is simultaneously robust to outliers [54] and underestimated measurement noise covariance. This addresses common sources of non-systematic error in topographic data, such as non-terrain artifacts, by inflating the uncertainty of effected observations. The robust equivalent weight matrix ($\overline{\mathit{\rho }}$) of $\mathit{R}$ is given by Huber’s algorithm, which minimizes the cost function

$$\begin{array}{cc}\hfill J\left({\mathit{x}}_k\right)& =\sum _{i=1}^b\rho \left(r_i^{\prime }\right)\hfill \end{array}$$

(22)

using the score function

$$\begin{array}{cc}\hfill \rho \left(r_i^{\prime }\right)& =\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}{r}_i^{\prime 2},\hfill & \left|r_i^{\prime }\right|\le c_3\hfill \\ c_3\left|r_i^{\prime }\right|-{\displaystyle \frac{1}{2}}{c}_3^2,\hfill & \left|r_i^{\prime }\right|>c_3\hfill \end{array}\right.\hfill \end{array}$$

(23)

where $r_i^{\prime }$ is an element of the innovation vector standardized by the corresponding measurement variance (i.e., ${\mathit{R}}_{i,i}$) and $c_3$ is a constant usually set between 1.3 and 2.0 [54].

Setting the partial derivative of Equation (22) to zero yields the equivalent weight functions using Equation (24) for independent measurement covariance and Equation (25) for dependent measurement covariance [64].

$$\begin{array}{cc}\hfill \overline{\mathit{\rho }}\left\{i,i\right\}& =\left\{\begin{array}{cc}{\displaystyle \frac{1}{{\sigma }_{i,i}}},\hfill & \left|r_i^{\prime }\right|\le c_3\hfill \\ {\displaystyle \frac{c_3}{{\sigma }_{i,i}\left|r_{i,i}\right|}},\hfill & \left|r_i^{\prime }\right|>c_3\hfill \end{array}\right.\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill \overline{\mathit{\rho }}\left\{i,j\right\}& =\left\{\begin{array}{cc}{\displaystyle \frac{1}{{\sigma }_{i,j}}},\hfill & \left|r_i^{\prime }\right|\le c_3,\left|r_i^{\prime }\right|\le c_3\hfill \\ {\displaystyle \frac{c_3}{{\sigma }_{i,j}max\left(\left|r_i^{\prime }\right|,\left|r_j^{\prime }\right|\right)}},\hfill & \left|r_i^{\prime }\right|>c_3,\left|r_j^{\prime }\right|>c_3\hfill \end{array}\right.\hfill \end{array}$$

(25)

Finally, the robust measurement noise covariance matrix is obtained by inverting the equivalent weight matrix

$${\overline{\mathit{R}}}_k={\overline{\mathit{\rho }}}_k^{-1}$$

(26)

2.2. Robust Self-Adaptive Algorithm

The algorithm proposed by Yang et al. [54] builds upon the CKF described in Section 2.1, implementing a robust M-type Kalman filter by replacing ${\mathit{R}}_k$ with ${\overline{\mathit{R}}}_k$ and implementing the adaptive $\alpha$ parameter described in Section 2.1.1. Several modifications are made to improve the ability to parameterize the adaptive functions in the model. First, the learning score in Equation (13) is replaced with a related metric that accounts for correlated state estimate error, the Normalized Estimate Error Squared (NEES) in Equation (27). NEES is a ${\chi }^2$-distributed variable that normalizes the state discrepancy using the state uncertainty matrix [40]. Equation (28) provides a convenient means to select the upper threshold for Equation (14) based on a one-tailed probability ($\gamma$) to determine if the state estimate is sufficiently different from the current WLS estimate given the amount of uncertainty in the system, where the degrees of freedom is equal to the length of the state discrepancy vector. This function provides an intuitive and state length agnostic method to parameterize $c_1$ and $c_2$ based on probabilities ${\gamma }_1$ and ${\gamma }_2$, respectively. Similarly, thresholds can be parameterized using a two-tailed distribution where ${\gamma }_1=1-{\gamma }_2$. Using a small lower threshold increases the probability of $\alpha$-based inflation occurring, which in turn increases the state discrepancy required to trigger another adaptive response, while the upper threshold limits maximum error.

$$\begin{array}{cc}\hfill {\upsilon }_k& =\Delta {\mathit{x}}_k^{\top }{\mathit{P}}_{k|k-1}^{-1}\Delta {\mathit{x}}_k\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill c& =F_{{\chi }^2}^{-1}\left(\gamma ,dim\left(\Delta {\mathit{x}}_k\right)\right)\hfill \end{array}$$

(28)

where $F_{{\chi }^2}^{-1}$ is the inverse cumulative distribution function of the ${\chi }^2$ distribution.

Defining the elimination segment prevents large errors from over-inflating uncertainty, which approaches infinity as $\alpha$ approaches 0. However, adaptive filters encounter division by zero errors when $\alpha =0$ according to Equations (10) and (11). This is addressed by replacing the adaptive estimator in Equation (9) with the WLS estimator defined in Equation (29). Similarly, the adaptive uncertainty in Equation (10) is re-initialized to account for the decoupled state and state uncertainty using Equation (30) followed by CKF Equations (6) and (8). This modified version of the Yang et al. [54] algorithm will be referred to as the Adaptively-Robust Kalman filter (ARK) hereafter.

$$\begin{array}{cc}\hfill {\widehat{\mathit{x}}}_{k|k}& ={\left({\mathit{C}}_k^{\top }{\mathit{R}}_k^{-1}{\mathit{C}}_k\right)}^{-1}{\mathit{C}}_k^{\top }{\mathit{R}}_k^{-1}{\mathit{z}}_k\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill {\mathit{P}}_{k|k-1}& ={\mathit{A}}_k{\mathit{A}}_k^{\top }\hfill \end{array}$$

(30)

ARK was further modified to include the adaptive process noise covariance scaling approach described in Section 2.1.2. This version will be referred to as the Robust Self-Adaptive Kalman filter (RSAK) because $\mathit{Q}$ is able to self-select a more appropriate variance based on the $\alpha$ parameter. Algorithm 1 implements RSAK, which becomes ARK by omitting line 16. Note that the algorithm is shown for readability and is not optimized for efficiency.

Algorithm 1 Robust Self-Adaptive Kalman filter

Require:  $\mathit{P}$, $\mathit{R}$, $\mathit{Q}$, $\mathit{x}$, q, ${\gamma }_1$, ${\gamma }_2$, $c_3$ 1: $c_1\leftarrow F_{{\chi }^2}^{-1}\left({\gamma }_1,dim\left(\mathit{x}\right)\right)$ 2: $c_2\leftarrow F_{{\chi }^2}^{-1}\left({\gamma }_2,dim\left(\mathit{x}\right)\right)$ 3: for all k do 4:     $\Delta t\leftarrow t_k-t_{k-1}$ 5:     Update $\mathit{A},\mathit{Q}$ with $\Delta t, q$ 6:     CKF predict() function using Equations (3) and (4) 7:     function update($\mathit{z}$) 8:         $\overline{\mathit{R}}\leftarrow \overline{\mathit{\rho }}{\left(\mathit{z},\mathit{R},c_3\right)}^{-1}$ 9:         $\Delta \mathit{x}\leftarrow \mathit{x}-{\left({\mathit{C}}^{\top }{\overline{\mathit{R}}}^{-1}\mathit{C}\right)}^{-1}{\mathit{C}}^{\top }{\overline{\mathit{R}}}^{-1}\mathit{z}$ 10:         $\alpha \leftarrow \alpha \left(\Delta {\mathit{x}}^{\top }{\mathit{P}}^{-1}\Delta \mathit{x},c_1,c_2\right)$ 11:         if $\alpha >0$ then 12:            $\overline{\mathit{K}}\leftarrow \frac{1}{\alpha }\mathit{P}{\mathit{C}}^{\top }\left(\frac{1}{\alpha }\mathit{C}\mathit{P}{\mathit{C}}^{\top }+\overline{\mathit{R}}\right)$ 13:            $\mathit{x}\leftarrow \mathit{x}+\overline{\mathit{K}}\left(\mathit{z}-\mathit{C}\mathit{x}\right)$ 14:            $\mathit{P}\leftarrow \left(\mathit{I}-\overline{\mathit{K}}\mathit{C}\right)\mathit{P}/\alpha$ 15:            $\mathit{\delta }\leftarrow \mathit{z}-\mathit{C}\mathit{x}$ 16:            $q\leftarrow q\times tr\left(\alpha \mathit{Q}+\left(1-\alpha \right)\left(\overline{\mathit{K}}\mathit{\delta }{\mathit{\delta }}^{\top }\overline{\mathit{K}}\right)\right)/tr\left(\mathit{Q}\right)$ 17:         else 18:            $\mathit{x}\leftarrow {\left({\mathit{C}}^{\top }{\overline{\mathit{R}}}^{-1}\mathit{C}\right)}^{-1}{\mathit{C}}^{\top }{\overline{\mathit{R}}}^{-1}\mathit{z}$ 19:            $\overline{\mathit{K}}\leftarrow \mathit{A}{\mathit{A}}^{\top }{\mathit{C}}^{\top }\left(\mathit{C}\mathit{A}{\mathit{A}}^{\top }{\mathit{C}}^{\top }+\overline{\mathit{R}}\right)$ 20:            $\mathit{P}\leftarrow \left(\mathit{I}-\overline{\mathit{K}}\mathit{C}\right)\mathit{A}{\mathit{A}}^{\top }$ 21:         end if 22:     end function 23:     $k\leftarrow k+1$ 24: end for

2.3. Surface Model and Problem Formulation

Consider the elevation of a surface as the graph of a bivariate polynomial with monomial basis in Cartesian coordinates with finite degree d

$$z\left(x,y\right)={\beta }_0+{\beta }_{1}x+{\beta }_{2}y+{\beta }_3{x}^2+{\beta }_{4}xy+{\beta }_5{y}^2+\dots +{\beta }_c{y}^d$$

(31)

These quadratic models are deployed to analyze the surface geometry about a point based on the vector of model coefficients ($\mathit{\beta }$). Using a local coordinate system and differentiating a surface model with $d\le 2$ at the origin ($x=y=0$) yields the second-order spatial derivatives found in Equation (32). These spatial derivatives are used to characterize several surface properties, such as slope, aspect, and several curvatures [25] using differential geometry. Thus, the accurate estimation of the model coefficients directly contributes to more accurate characterization of local surface properties.

$$\begin{array}{cccccccccc}\hfill p& ={\displaystyle \frac{𝜕z}{𝜕x}}\hfill & \hfill q& ={\displaystyle \frac{𝜕z}{𝜕y}}\hfill & \hfill {\displaystyle \frac{r}{2}}& ={\displaystyle \frac{{𝜕}^{2}z}{𝜕x^2}}\hfill & \hfill s& ={\displaystyle \frac{{𝜕}^{2}z}{𝜕x𝜕y}}\hfill & \hfill {\displaystyle \frac{t}{2}}& ={\displaystyle \frac{{𝜕}^{2}z}{𝜕y^2}}\hfill \end{array}$$

(32)

Now consider the change in surface elevation in discrete-time separated by the time interval $\Delta t=t_k-t_{k-1}$. Because both polynomials belong to the same ring $z\in \mathbb{R}\left[X,Y\right]$, differences can be calculated by polynomial subtraction with Equation (33) and average rate of change can by calculated by scalar division with Equation (34), with both operating on coefficients and resulting a vector (${\mathit{\beta }}^{\prime }$) containing the spatio-temporal relationships. Note that the intercept of Equation (33) is an estimator of DoD.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill z_{\Delta }\left(x,y\right)& =z_k\left(x,y\right)-z_{k-1}\left(x,y\right)\hfill \\ \hfill & =\left({\beta }_{0,k}-{\beta }_{0,k-1}\right)+\left({\beta }_{1,k}-{\beta }_{1,k-1}\right)x+\dots +\left({\beta }_{c,k}-{\beta }_{c,k-1}\right)y^d\hfill \end{array}\end{array}$$

(33)

and the rate of change

$$\begin{array}{c}{z}^{\prime }\left(x,y\right)={\displaystyle \frac{\Delta z\left(x,y\right)}{\Delta t}}\end{array}$$

(34)

approximating the spatio-temporal partial derivatives

$$\begin{array}{ccccccccccc}\hfill {\displaystyle \frac{𝜕z}{𝜕t}}& \hfill & \hfill {\displaystyle \frac{{𝜕}^{2}z}{𝜕x𝜕t}}& \hfill & \hfill {\displaystyle \frac{{𝜕}^{2}z}{𝜕y𝜕t}}& \hfill & \hfill {\displaystyle \frac{{𝜕}^{3}z}{𝜕x^2𝜕t}}& \hfill & \hfill {\displaystyle \frac{{𝜕}^{3}z}{𝜕x𝜕y𝜕t}}& \hfill & {\displaystyle \frac{{𝜕}^{3}z}{𝜕y^2𝜕t}}\end{array}$$

The surface model is parameterised from a sufficiently large set of spatially distributed elevation observations sampled from a neighbourhood $\mathcal{L}$ about location ${\mathit{p}}_0={\left[x,y\right]}^{\top }$,

$$\left(z,\mathit{p}\right)\in \left\{\left(z_n,f\left({\mathit{p}}_0,{\mathit{p}}_n\right)\right)\right|{\mathit{p}}_n\in \mathcal{L}\}$$

(35)

where the function f centers the local coordinate system on ${\mathit{p}}_0$.

This can be expressed in matrix form by performing monomial basis expansion on the position observations

$$\begin{array}{c}{\mathsf{\varphi }}_d\left(\mathit{p}\right)=\left[\begin{array}{ccc}\prod {\mathit{p}}^0& \dots & \prod {\mathit{p}}^i\end{array}\right]\forall i\in {\mathbb{N}}^2|\sum i\le d\end{array}$$

(36)

such that

$$\begin{array}{c}\left[\begin{array}{c}{z}_0\\ ⋮\\ z_a\end{array}\right]=\left[\begin{array}{c}{\mathsf{\varphi }}_d\left({\mathit{p}}_0\right)\\ ⋮\\ {\mathsf{\varphi }}_d\left({\mathit{p}}_a\right)\end{array}\right]\left[\begin{array}{c}{\beta }_0\\ ⋮\\ {\beta }_c\end{array}\right],\mathit{z}=\mathsf{\Phi }\mathit{\beta }\end{array}$$

(37)

where $\mathit{z}\in {\mathbb{R}}^{a\times 1}$, $\mathsf{\Phi }\in {\mathbb{R}}^{a\times c}$, and $\mathit{\beta }\in {\mathbb{R}}^{c\times 1}$.

The coefficients of Equations (31) and (34) can be obtained by Ordinary Least Squares (OLS) regression using Equations (38) and (39), respectively. Note that Equation (39) requires that both sets of observations are coincident.

$$\begin{array}{c}\mathit{\beta }\approx {\left({\mathsf{\Phi }}^{\top }\mathsf{\Phi }\right)}^{-1}{\mathsf{\Phi }}^{\top }\mathit{z}\end{array}$$

(38)

$$\begin{array}{c}{\mathit{\beta }}^{\prime }\approx {\left({\mathsf{\Phi }}^{\top }\mathsf{\Phi }\right)}^{-1}{\mathsf{\Phi }}^{\top }\left({\displaystyle \frac{{\mathit{z}}_k-{\mathit{z}}_{k-1}}{\Delta t}}\right)\end{array}$$

(39)

Representing the surface model within the framework of the Kalman filter is achieved in Equation (40) by rearranging Equation (34) and expressing it in matrix form. The state vector $\mathit{x}$ now includes the surface and spatio-temporal coefficients (with length $b=2c$), and the state transition matrix $\mathit{A}$ is defined to extrapolate coefficients over the time interval. Equation (40) integrates the spatio-temporal coefficients over the time interval and adds them to the previous state to arrive at current model coefficients representing the mutated surface. The measurement matrix $\mathit{C}$ in Equation (41) is extended from Equation (37) to negate ${\mathit{\beta }}^{\prime }$ coefficients.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathit{x}}_k& ={\mathit{A}}_k{\mathit{x}}_{k-1}+{\mathit{\omega }}_k,{\mathit{\omega }}_k\sim \mathcal{N}\left(0,{\mathit{Q}}_k\right)\hfill \\ & =\left[\begin{array}{cc}\mathit{I}& \mathit{I}\Delta t_k\\ \mathbf{0}& \mathit{I}\end{array}\right]\left[\begin{array}{c}{\mathit{\beta }}_{k-1}\\ {\mathit{\beta }}_{k-1}^{\prime }\end{array}\right]+{\mathit{\omega }}_k\hfill \end{array}\end{array}$$

(40)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\widehat{\mathit{z}}}_k& ={\mathit{C}}_k{\mathit{x}}_k+{\mathit{\nu }}_k,{\mathit{\nu }}_k\sim \mathcal{N}\left(0,{\mathit{R}}_k\right)\hfill \\ & =\left[\begin{array}{cc}{\mathsf{\Phi }}_k& \mathbf{0}\end{array}\right]\left[\begin{array}{c}{\mathit{\beta }}_k\\ {\mathit{\beta }}_k^{\prime }\end{array}\right]+{\mathit{\nu }}_k\hfill \end{array}\end{array}$$

(41)

where $\mathit{I}\in {\mathbb{R}}^{c\times c}$ is an identity matrix multiplied by a scalar value. Note the use of block matrices regarding matrix dimensions from previous equations.

Modifications to the Kalman Update Function

Since the spatio-temporal coefficients are not used by the observation model, additional measures must be taken to estimate these coefficients when the WLS estimator is triggered (i.e., $\alpha =0$). The feed-forward matrix from Equation (2) is repurposed and set to ${\mathit{D}}_k=\left[\begin{array}{cc}\mathbf{0}& {\mathsf{\Phi }}_k\end{array}\right]$ to estimate the spatio-temporal coefficients using the difference between current and previous observations scaled by the elapsed time. Thus, Equation (29) is modified as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\widehat{\mathit{x}}}_{k|k}=& {\left({\mathit{C}}_k^{\top }{\overline{\mathit{R}}}_k^{-1}{\mathit{C}}_k\right)}^{-1}{\mathit{C}}_k^{\top }{\overline{\mathit{R}}}_k^{-1}{\mathit{z}}_k\hfill \\ \hfill & +{\left({\mathit{D}}_k^{\top }{\widehat{\mathit{R}}}_{k|k-1}^{-1}{\mathit{D}}_k\right)}^{-1}{\mathit{D}}_k^{\top }{\widehat{\mathit{R}}}_{k|k-1}^{-1}\left({\displaystyle \frac{{\mathit{z}}_k-{\mathit{z}}_{k-1}}{\Delta t_k}}\right)\hfill \end{array}\end{array}$$

(42)

where ${\widehat{\mathit{R}}}_{k|k-1}$ is a diagonal matrix

$$\begin{array}{c}{\widehat{\mathit{R}}}_{k|k-1}\{i,i\}=VAR\left({\overline{\mathit{R}}}_{i,i,k}\right)+VAR\left({\overline{\mathit{R}}}_{i,i,k-1}\right)-2COV\left({\overline{\mathit{R}}}_{i,i,k},{\overline{\mathit{R}}}_{i,i,k-1}\right)\end{array}$$

(43)

Note that this requires observations from the previous epoch, which introduces a dependency between the error terms from Equation (2) for both sets of observations. Equation (43) is simplified by assuming that the covariance term is 0, which results only in the addition of the robust measurement error variances from each epoch. While this assumption is dubious and unlikely to be true, this estimator is only used as an emergency correction to extreme model violations. Line 19 in Algorithm 1 was modified to use ${\widehat{\mathit{R}}}_{k|k-1}$ to reflect the elevated uncertainty.

The NEES score function, Equation (27), was modified to use only the observable components of the state vector and the associated uncertainty using Equation (44). This prevents models from being penalized for diverging from the rate of change estimates derived from Equation (42), which are expected to carry increased uncertainty. The additional noise present in the WLS rate of change estimate may impact $\alpha$ and unnecessarily inflate state uncertainty if the entire state vector is used.

$$\upsilon ={\left(\Delta {\mathit{x}}_i\right)}_{i\le c}^{\top }{\left({\mathit{P}}_{ij}\right)}_{\begin{array}{c}i\le c\\ j\le c\end{array}}^{-1}{\left(\Delta {\mathit{x}}_i\right)}_{i\le c}$$

(44)

2.4. Evaluation

The performance of Kalman filters in various configurations were evaluated using simulated surfaces. Local surface model coefficients are rarely known in practice, requiring simulations to establish exactly known state coefficients (${\tilde{\mathit{x}}}_k$). All simulations lasted 1000 epochs, and each simulation was run $n=500$ times to account for different random noise vectors added to observations. Random number generators were seeded based on the simulation index to ensure that all simulations received the same noise vectors. Filter performance is reported as the average euclidean distance between the estimated and true state vectors over all repeated simulations for a given epoch using Equation (45) and is referred to as error hereafter. All epoch metrics are averaged across the 500 simulations in the same manner as Equation (45).

$${ϵ}_k={\displaystyle \frac{1}{n}}\sum _{i=1}^n\parallel {\mathit{x}}_{k,i}-{\tilde{\mathit{x}}}_{k,i}\parallel$$

(45)

2.4.1. Filter Configurations

The OLS, CKF, ARK, and RSAK estimators were implemented with multiple models of process noise and compared. OLS is a conventional surface model estimator and CKF is a non-adaptive Kalman filter; both are included for reference. ARK and RSAK are both M-type robust estimators; however, ARK is only $\alpha$ adaptive, while RSAK is $\alpha$ and $\mathit{Q}$ adaptive. All estimators were initialized with the following parameters:

$$\begin{array}{cccccccccccccc}\hfill {\gamma }_1& =0.05\hfill & \hfill {\gamma }_2& =0.95\hfill & \hfill c_3& =1.5\hfill & \hfill {\mathit{x}}_0& =\mathrm{OLS}\left({\mathit{z}}_0\right)\hfill & \hfill {\mathit{P}}_0& =\mathit{I}\hfill & {\mathit{R}}_0& =\mathit{I}{r}_0^2& {\mathit{Q}}_0& =\mathit{Q}\left(\Delta t,q_0\right)\end{array}$$

where ${\gamma }_1$ is the opposite tail of ${\gamma }_2$, which is slightly less the values Yang and Xu [59] used to identify faulty observations, the initial state estimate ${\mathit{x}}_0$ is calculated with Equation (38) padded with zeros, $r_0$ is an initial estimate of measurement Root Mean Squared Error (RMSE) and is squared to represent error variance, $q_0$ is an initial estimate of process noise variance, and ${\mathit{Q}}_0$ is calculated with one of Equations (46)–(48).

All Kalman filters were initialized with first-order continuous white noise, which models the effect of zero-mean continuous variability in kinematic components [65] and is defined by Equation (48). Both ARK and RSAK were initialized with process uncertainty described by multiple continuous white noise model orders in addition to the first-order model due to the high sensitivity and variability of $\Delta t$. Zero-order continuous white noise models are less sensitive to $\Delta t$ and only affect the surface model coefficients, and are defined by Equation (47). Finally, Equation (46) was used to examine the effect of $\alpha$ only adaptation by ignoring process uncertainty (i.e., a null matrix). The noise model order is labelled as a suffix to each estimator. Additionally, the suffix ‘A’ was added to denote estimators configured with an asymptotic $\alpha$ function that use Equation (15).

$$\begin{array}{cc}\hfill {\mathit{Q}}^{\varnothing }\left(\Delta t,q\right)& =\mathbf{0}\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill \mathit{Q}\left(\Delta t,q\right)& =\left[\begin{array}{cc}\mathit{I}\Delta t& \mathbf{0}\\ \mathbf{0}& \mathbf{0}\end{array}\right]q\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill {\mathit{Q}}^{\prime }\left(\Delta t,q\right)& =\left[\begin{array}{cc}\mathit{I}{\displaystyle \frac{\Delta t^3}{3}}& \mathit{I}{\displaystyle \frac{\Delta t^2}{2}}\\ \mathit{I}{\displaystyle \frac{\Delta t^2}{2}}& \mathit{I}\Delta t\end{array}\right]q\hfill \end{array}$$

(48)

Two temporal distributions were used to simulate the effect of unevenly spaced surveys in units of days. $\Delta t_k$ was drawn from either a normal or uniform distribution based on an average of 183 days (i.e., biannual surveys). The normal distribution represents some variability in survey date, while the uniform distribution represents random temporal sampling, new knowledge of system activity, or opportunistic surveys. The parameters used for the normal and uniform distributions are given in Equations (49) and (50), respectively. This variability was added evaluate the impact of different process noise models, which are on dependent $\Delta t$ and affect covariance estimation and scaling. The time distributions are visualized in Figure 1A.

$$\begin{array}{cc}\hfill \Delta t_k^{\mathcal{N}}& \sim \mathcal{N}\left(183,15\right)\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill \Delta t_k^{\mathcal{U}}& \sim \mathcal{U}\left(30,336\right)\hfill \end{array}$$

(50)

Combinations of initial values for $r_0\in \{{10}^{-3},{10}^{-2},{10}^{-1},10\}$ and $q_0$∈ {${10}^{-24}$, ${10}^{-16}$, ${10}^{-8}$} were used to test the sensitivity of all filters to initial parameterisation using all cases and temporal distributions. This analysis was paired with a parameter space search on the ranges ${10}^{-1}<r_0<{10}^1$, ${10}^{-16}\le q_0\le {10}^0$, and $0.01<{\gamma }_1<0.58$ for a more targeted assessment. The parameter space search was conducted to identify the point at which the poor estimation of noise statistics began to degrade filter performance. All parameter space searches were performed using the uniform time distribution, the irregular case, and the true noise statistics.

2.4.2. Simulating a Simple Surface

A true state vector is defined to represent the temporal evolution of an initial surface model by a change model. A second-order surface model starts with the coefficients given in Equation (51). Dynamism was simulated by extrapolating the true state by a change model with coefficients given in Equation (52), or with zeros. The surface and change models were based on a rough field estimate. The observation model was based on a $5\times 5$ grid with $0.15$ m spacing. The true state was projected into observation space to generate true observations and was corrupted with noise drawn from a multivariate normal distribution, Equation (53), before use in the update function.

$$\begin{array}{cc}\hfill {\tilde{\mathit{\beta }}}_0& ={\left[\begin{array}{cccccc}130.0& 0.425& -0.725& 0.185& 0.095& 0.160\end{array}\right]}^{\top }\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill {\tilde{\mathit{\beta }}}^{\prime }& ={\left[\begin{array}{cccccc}-6.920e^{-4}& 1.500e^{-4}& 1.155e^{-5}& 1.230e^{-3}& 4.615e^{-4}& 2.650e^{-4}\end{array}\right]}^{\top }\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill {\mathit{z}}_k& ={\mathsf{\Phi }}_k{\tilde{\mathit{\beta }}}_k+{\mathit{\nu }}_k,{\mathit{\nu }}_k\sim \mathcal{N}\left(0,\mathit{I}{\tilde{r}}^2\right)\hfill \end{array}$$

(53)

where $\tilde{r}=0.01$ is a scalar representing one centimeter elevation RMSE to simulate DEM measurement noise with exactly known parameters.

The temporal application of the change model to the initial model was governed by four cases of surface dynamism. Each case represents a simplified temporal pattern designed to approximate dynamic surface processes and apply stress to each filter by deliberately violating the model with unpredictable, temporally non-linear changes. A small amount of process noise ($\tilde{q}$) with exactly known parameters was added to the state vector at every epoch.

Case 1:

Static models a perfectly static surface by never adding the change model or process noise to the state (i.e., $\tilde{q}=0$). Thus, the only change apparent to the filter is from observation noise. This case acts as a control by eliminating all surface change and isolating the effects of repeated observations.

Case 2:

Periodic models period processes such as seasonal erosion and deposition by scaling the change model continuously between $\left[-0.25,0.75\right]$ using Equation (54). Note that this imparts a periodic trend based on the epoch rather than the time difference. A first-order white process noise vector with variance $\tilde{q}={10}^{-16}$ was added at every epoch.

Case 3:

Irregular models a surface that starts and stops changing intermittently, mimicking some landslide behaviors. This is implemented by selecting five random ‘start’ epochs, which then received constant change for 25 epochs before returning to static. A first-order white process noise vector with variance $\tilde{q}={10}^{-16}$ was added at every epoch.

Case 4:

Catastrophe models a surface that experiences a natural disaster, such as a large mass movement. This is implemented as a single epoch where 5 m elevation is subtracted and the remaining coefficients are randomized. Two catastrophic events are simulated 365 epochs apart, and the second event was subjected to and additional 25 epochs of constant change to simulate sustained effects following a large disturbance. Note that the catastrophe represents a highly non-linear impulse error signature with non-Gaussian properties. A first-order white process noise vector with $\tilde{q}={10}^{-16}$ was added at every epoch.

$${\tilde{\mathit{\beta }}}_k^{\prime }={\tilde{\mathit{\beta }}}^{\prime }\times \left(0.5cos\left({\displaystyle \frac{2\pi }{365}}k+2\pi -arccos\left({\displaystyle \frac{-0.25}{0.5}}\right)\right)+0.25\right)$$

(54)

Each case is visualized in Figure 1B using the uniform temporal distribution. 3D visualizations of the initial surface, an epoch of change, and a catastrophe are presented in Figure 1C–E.

2.4.3. Spatial Analysis

A preliminary assessment of a spatially distributed array of Kalman filters was conducted on a small 500 × 500 cell DEM with a spatial resolution of 0.15 m, collected with UAV-LiDAR near Atsuma, Hokkaido, Japan (42.780°N, 142.020°E). A field of spatially auto-correlated noise was generated using a turning bands simulation [66] with five iterations and an auto-correlation range of 75 m. All noise fields were centered and scaled such that the mean was zero and the standard deviation was 0.1 m. This experiment simulates a static surface where the initial uncorrupted DEM was assumed to be true, and 50 epochs of corrupted observations were analyzed using 5 × 5 cell observation windows. The ARK-∅A filter was compared to an OLS estimator to eliminate the choice of initial process noise due, which does not apply a static surface. All ARK-∅A filters were initialized with $r_0=1.0=\mathit{I}$, and the state was initialized with an OLS estimate. An example of a single epoch of this process is visualized in Figure 2.

The estimated elevation (i.e., the intercept ${\mathit{\beta }}_0$) for each of the 250,000 filters was recorded at each epoch and compared to the true DEM. Additionally, the spatial distributions of minimal curvature were generated for a qualitative assessment. Minimal curvature ($k_{min}$) is the curvature of a principal section with the lowest value of curvature [25,26] and can be calculated from the spatial partial derivatives defined in Equation (32) using Equation (55). Minimal curvature values tend to be large when the local geometry resembles a channel, which skews negatively when linear errors are abundant [27].

$$\begin{array}{cc}\hfill k_{min}=& -{\displaystyle \frac{\left(1+q^2\right)r+2pqs+\left(1+p^2\right)t}{2\sqrt{{\left(1+p^2+q^2\right)}^3}}}\hfill \\ \hfill & -\sqrt{{\left({\displaystyle \frac{\left(1+q^2\right)r+2pqs+\left(1+p^2\right)t}{2\sqrt{{\left(1+p^2+q^2\right)}^3}}}\right)}^2-{\displaystyle \frac{rt-s^2}{{\left(1+p^2+q^2\right)}^2}}}\hfill \end{array}$$

(55)

3. Results

3.1. Aggregate Performance

The average error values for all cases, $r_0$ and $q_0$ excluding the individual parameter space search results, were aggregated, and the error distributions are summarized in Figure 3. The rows record the error for the $\mathit{\beta }$ and ${\mathit{\beta }}^{\prime }$ coefficients and the columns record the temporal distributions. The increased measurement noise due to DEM subtraction is evident when comparing the OLS model error distributions between the $\mathit{\beta }$ and ${\mathit{\beta }}^{\prime }$ coefficients. This is particularly apparent for the uniform distribution (Figure 3B,D). The filters with null or zero-order process noise models tended to perform better than the first-order filters. Interestingly, the ARK-∅ and ARK-∅A filters achieved low error across all simulations by $\alpha$ adaptation alone, independently of any process noise model. While the error distributions for the $\mathit{\beta }$ coefficients were relatively similar for OLS and Kalman filters, all Kalman filters had much lower ${\mathit{\beta }}^{\prime }$ coefficient error.

3.2. Sensitivity to Initial Parameters

The simulation results were summarized as the average and maximum state error over all epochs using the normal time distributions for the static, periodic, irregular, and catastrophic cases in

Table 1. The average and maximum state error ($ϵ$) for all epochs of the normal distribution, for all $r_0$, $q_0$, and the static case. For reference, the OLS method had an average state error of 0.082 and a maximum state error of 0.086 for all simulations. The true values are $\tilde{r}={10}^{-2}$ and $\tilde{q}={10}^{-16}$. Values exceeding the comparable OLS value are marked in bold.

		Average $\mathit{ϵ}$					Maximum $\mathit{ϵ}$
		${\mathit{r}}_{\mathbf{0}}$	${\mathbf{10}}^{-\mathbf{3}}$	${\mathbf{10}}^{-\mathbf{2}}$	${\mathbf{10}}^{-\mathbf{1}}$	${\mathbf{10}}^{\mathbf{1}}$	${\mathbf{10}}^{-\mathbf{3}}$	${\mathbf{10}}^{-\mathbf{2}}$	${\mathbf{10}}^{-\mathbf{1}}$	${\mathbf{10}}^{\mathbf{1}}$
	${\mathit{q}}_{\mathbf{0}}$
CKF-1	${10}^{-24}$		0.032	0.034	0.034	0.044	0.081	0.081	0.081	0.081
	${10}^{-16}$		0.023	0.014	0.013	0.039	0.081	0.081	0.081	0.081
	${10}^{-8}$		0.081	0.076	0.060	0.023	0.086	0.085	0.081	0.081
ARK-∅			0.036	0.024	0.017	0.044	0.081	0.081	0.081	0.081
ARK-∅A			0.032	0.026	0.018	0.044	0.081	0.081	0.081	0.081
ARK-0	${10}^{-24}$		0.036	0.024	0.017	0.044	0.081	0.081	0.081	0.081
	${10}^{-16}$		0.036	0.024	0.017	0.044	0.081	0.081	0.081	0.081
	${10}^{-8}$		0.031	0.023	0.016	0.044	0.081	0.081	0.081	0.081
RSAK-0	${10}^{-24}$		0.025	0.022	0.016	0.044	0.081	0.081	0.081	0.081
	${10}^{-16}$		0.025	0.022	0.016	0.044	0.081	0.081	0.081	0.081
	${10}^{-8}$		0.026	0.022	0.016	0.044	0.081	0.081	0.081	0.081
ARK-1	${10}^{-24}$		0.036	0.024	0.017	0.044	0.081	0.081	0.081	0.081
	${10}^{-16}$		0.036	0.024	0.017	0.039	0.081	0.081	0.081	0.081
	${10}^{-8}$		0.071	0.070	0.058	0.023	0.085	0.087	0.081	0.081
RSAK-1	${10}^{-24}$		0.055	0.051	0.031	0.044	0.081	0.081	0.081	0.081
	${10}^{-16}$		0.055	0.051	0.031	0.039	0.081	0.081	0.081	0.081
	${10}^{-8}$		0.071	0.070	0.058	0.023	0.084	0.087	0.081	0.081
RSAK-1A	${10}^{-24}$		0.056	0.052	0.032	0.044	0.081	0.081	0.081	0.081
	${10}^{-16}$		0.057	0.052	0.032	0.039	0.081	0.081	0.081	0.081
	${10}^{-8}$		0.071	0.070	0.058	0.023	0.085	0.087	0.081	0.081

,

Table 2. The average and maximum state error ($ϵ$) for all epochs of the normal distribution, for all $r_0$, $q_0$, and the periodic case. For reference, the OLS method had an average state error of 0.082 and a maximum state error of 0.086 for all simulations. The true values are $\tilde{r}={10}^{-2}$ and $\tilde{q}={10}^{-16}$. Values exceeding the comparable OLS value are marked in bold.

		Average $\mathit{ϵ}$					Maximum $\mathit{ϵ}$
		${\mathit{r}}_{\mathbf{0}}$	${\mathbf{10}}^{-\mathbf{3}}$	${\mathbf{10}}^{-\mathbf{2}}$	${\mathbf{10}}^{-\mathbf{1}}$	${\mathbf{10}}^{\mathbf{1}}$	${\mathbf{10}}^{-\mathbf{3}}$	${\mathbf{10}}^{-\mathbf{2}}$	${\mathbf{10}}^{-\mathbf{1}}$	${\mathbf{10}}^{\mathbf{1}}$
	${\mathit{q}}_{\mathbf{0}}$
CKF-1	${10}^{-24}$		20.025	22.214	22.829	24.831	32.283	36.760	38.167	41.761
	${10}^{-16}$		0.277	2.254	4.938	21.605	0.521	4.127	9.957	34.786
	${10}^{-8}$		0.081	0.076	0.060	0.277	0.086	0.085	0.081	0.521
ARK-∅			0.082	0.076	0.051	1.244	0.103	0.096	0.081	1.873
ARK-∅A			0.079	0.075	0.051	1.231	0.100	0.095	0.081	1.853
ARK-0	${10}^{-24}$		0.082	0.076	0.051	1.244	0.103	0.096	0.081	1.873
	${10}^{-16}$		0.082	0.077	0.051	1.244	0.103	0.097	0.081	1.873
	${10}^{-8}$		0.080	0.076	0.052	1.244	0.099	0.096	0.081	1.874
RSAK-0	${10}^{-24}$		0.115	0.117	0.178	1.645	0.166	0.170	0.274	2.501
	${10}^{-16}$		0.115	0.117	0.178	1.645	0.164	0.170	0.274	2.501
	${10}^{-8}$		0.116	0.117	0.178	1.645	0.165	0.166	0.274	2.501
ARK-1	${10}^{-24}$		0.082	0.077	0.051	1.244	0.101	0.096	0.081	1.873
	${10}^{-16}$		0.083	0.076	0.051	1.244	0.103	0.097	0.081	1.875
	${10}^{-8}$		0.072	0.071	0.058	0.277	0.087	0.085	0.081	0.521
RSAK-1	${10}^{-24}$		0.064	0.063	0.063	2.908	0.083	0.083	0.111	5.327
	${10}^{-16}$		0.064	0.063	0.063	2.909	0.083	0.085	0.111	5.327
	${10}^{-8}$		0.072	0.071	0.058	0.277	0.087	0.085	0.081	0.521
RSAK-1A	${10}^{-24}$		0.064	0.063	0.063	2.889	0.081	0.083	0.105	5.290
	${10}^{-16}$		0.065	0.063	0.063	2.889	0.082	0.082	0.105	5.290
	${10}^{-8}$		0.072	0.071	0.058	0.277	0.086	0.085	0.081	0.521

,

Table 3. The average and maximum state error ($ϵ$) for all epochs of the normal distribution, for all $r_0$, $q_0$, and the irregular case. For reference, the OLS method had an average state error of 0.082 and a maximum state error of 0.086 for all simulations. The true values are $\tilde{r}={10}^{-2}$ and $\tilde{q}={10}^{-16}$. Values exceeding the comparable OLS value are marked in bold.

		Average $\mathit{ϵ}$					Maximum $\mathit{ϵ}$
		${\mathit{r}}_{\mathbf{0}}$	${\mathbf{10}}^{-\mathbf{3}}$	${\mathbf{10}}^{-\mathbf{2}}$	${\mathbf{10}}^{-\mathbf{1}}$	${\mathbf{10}}^{\mathbf{1}}$	${\mathbf{10}}^{-\mathbf{3}}$	${\mathbf{10}}^{-\mathbf{2}}$	${\mathbf{10}}^{-\mathbf{1}}$	${\mathbf{10}}^{\mathbf{1}}$
	${\mathit{q}}_{\mathbf{0}}$
CKF-1	${10}^{-24}$		9.534	10.482	10.794	11.977	16.003	17.279	17.518	18.446
	${10}^{-16}$		0.292	0.737	2.387	10.440	1.776	3.942	7.229	16.802
	${10}^{-8}$		0.081	0.076	0.061	0.292	0.086	0.085	0.126	1.776
ARK-∅			0.052	0.045	0.026	0.696	0.117	0.117	0.084	2.111
ARK-∅A			0.050	0.045	0.028	0.687	0.168	0.134	0.115	2.054
ARK-0	${10}^{-24}$		0.052	0.045	0.026	0.696	0.118	0.117	0.084	2.111
	${10}^{-16}$		0.052	0.045	0.026	0.696	0.116	0.116	0.084	2.111
	${10}^{-8}$		0.050	0.046	0.025	0.697	0.117	0.116	0.084	2.113
RSAK-0	${10}^{-24}$		0.049	0.047	0.025	1.135	0.116	0.117	0.084	3.378
	${10}^{-16}$		0.049	0.047	0.025	1.135	0.117	0.117	0.084	3.378
	${10}^{-8}$		0.049	0.047	0.025	1.135	0.118	0.117	0.084	3.378
ARK-1	${10}^{-24}$		0.053	0.045	0.026	0.696	0.117	0.116	0.084	2.111
	${10}^{-16}$		0.052	0.045	0.025	0.703	0.117	0.116	0.084	2.160
	${10}^{-8}$		0.076	0.076	0.062	0.292	0.345	0.341	0.313	1.776
RSAK-1	${10}^{-24}$		0.061	0.057	0.037	0.547	0.216	0.160	0.134	3.189
	${10}^{-16}$		0.063	0.057	0.037	0.547	0.227	0.158	0.135	3.189
	${10}^{-8}$		0.076	0.076	0.062	0.292	0.344	0.342	0.313	1.776
RSAK-1A	${10}^{-24}$		0.070	0.068	0.053	0.534	0.299	0.289	0.266	3.127
	${10}^{-16}$		0.069	0.067	0.053	0.534	0.297	0.287	0.266	3.127
	${10}^{-8}$		0.076	0.076	0.062	0.292	0.343	0.342	0.313	1.776

and

Table 4. The average and maximum state error ($ϵ$) for all epochs of the normal distribution, for all $r_0$, $q_0$, and the catastrophe case. For reference, the OLS method had an average state error of 0.082 and a maximum state error of 0.086 for all simulations. The true values are $\tilde{r}={10}^{-2}$ and $\tilde{q}={10}^{-16}$. Values exceeding the comparable OLS value are marked in bold.

		Average $\mathit{ϵ}$					Maximum $\mathit{ϵ}$
		${\mathit{r}}_{\mathbf{0}}$	${\mathbf{10}}^{-\mathbf{3}}$	${\mathbf{10}}^{-\mathbf{2}}$	${\mathbf{10}}^{-\mathbf{1}}$	${\mathbf{10}}^{\mathbf{1}}$	${\mathbf{10}}^{-\mathbf{3}}$	${\mathbf{10}}^{-\mathbf{2}}$	${\mathbf{10}}^{-\mathbf{1}}$	${\mathbf{10}}^{\mathbf{1}}$
	${\mathit{q}}_{\mathbf{0}}$
CKF-1	${10}^{-24}$		2.919	4.347	4.913	5.021	9.179	11.181	11.713	11.730
	${10}^{-16}$		0.119	0.373	1.008	3.018	3.624	4.484	6.535	9.182
	${10}^{-8}$		0.081	0.077	0.062	0.119	0.096	0.243	0.912	3.624
ARK-∅			0.044	0.034	0.021	1.335	0.128	0.128	0.121	3.043
ARK-∅A			N/A	0.035	0.022	0.325	N/A	0.131	0.131	1.932
ARK-0	${10}^{-24}$		0.044	0.034	0.021	1.335	0.128	0.128	0.121	3.043
	${10}^{-16}$		0.044	0.034	0.021	1.335	0.128	0.128	0.121	3.043
	${10}^{-8}$		0.040	0.034	0.020	1.335	0.128	0.128	0.121	3.043
RSAK-0	${10}^{-24}$		0.037	0.034	0.020	1.386	0.128	0.128	0.121	3.247
	${10}^{-16}$		0.037	0.034	0.020	1.386	0.128	0.128	0.121	3.247
	${10}^{-8}$		0.037	0.035	0.020	1.386	0.128	0.128	0.121	3.247
ARK-1	${10}^{-24}$		0.044	0.034	0.021	1.335	0.128	0.128	0.121	3.043
	${10}^{-16}$		0.043	0.034	0.020	1.335	0.128	0.128	0.121	3.043
	${10}^{-8}$		0.072	0.072	0.059	0.096	0.258	0.257	0.230	1.449
RSAK-1	${10}^{-24}$		0.055	0.054	0.045	0.189	0.246	0.247	0.248	1.906
	${10}^{-16}$		0.055	0.054	0.045	0.189	0.246	0.246	0.248	1.906
	${10}^{-8}$		0.058	0.057	0.049	0.169	0.248	0.248	0.261	1.867
RSAK-1A	${10}^{-24}$		0.054	0.052	0.036	0.082	0.169	0.165	0.126	1.092
	${10}^{-16}$		0.054	0.052	0.036	0.082	0.170	0.164	0.126	1.092
	${10}^{-8}$		0.055	0.052	0.037	0.060	0.170	0.167	0.126	0.965

, respectively. These tables provide a coarse impression of the impact of $r_0$ and $q_0$ choice combinations on filter performance. In general, all Kalman filters achieved lower average error than OLS. However, maximum error for all adaptive Kalman filters was commonly near or slightly higher than OLS. Otherwise, relative to the OLS estimator, the adaptive Kalman filters reduced average error by approximately 50% for most simulations (Table 1, Table 2, Table 3 and Table 4), up to approximately 80% (Table 1). All filters performed particularly poorly when noise was greatly overestimated (i.e., $r_0=10$), and the performance of CKF at very low process noise (i.e., $q_0={10}^{-24}$) was extremely poor. The RSAK-0 filter also performed particularly poorly for the periodic case (Table 2), and both RSAK-1 and RSAK-1A had notably large maximum errors for the irregular and catastrophe cases (Table 3 and Table 4). The choice of $q_0$ had minimal impact on average and maximum error for all adaptive Kalman filters when it was accurately estimated or underestimated, and approached the OLS estimator performance when it was overestimated. Typical maximum error for the adaptive Kalman filters was 30% or 40% greater than the OLS maximum error when $r_0\le \tilde{r}$ and were roughly equivalent when $r_0={10}^{-1}$ (Table 1, Table 2, Table 3 and Table 4). Note that ARK-∅A performance data are not available for the catastrophe case with the lowest noise parameter settings due to fatal matrix inversion errors.

The $r_0$ parameter space was searched to find the value at which $r_0$ began to negatively effect filter error. Simulations used the uniform time distribution and the irregular case, using $q_0=\tilde{q}$ to assess the impact of $r_0$ values between ${10}^{-1}\le r_0\le {10}^1$ on filter performance (Figure 4). The null and zero-order noise models began to diverge in performance at $r_0\ge 1.51$ and $r_0\ge 0.81$, respectively, with RSAK-0 demonstrating a particularly clear divergence (Figure 4A,C). A smooth degradation in performance with increasing $r_0$ for ARK-∅A contrasts with the sudden jumps observed in the bounded ARK-∅ variant (Figure 4A,B). The first-order filters had different points of divergence, with ARK-1 diverging smoothly at $r_0\ge 1.51$ and with RSAK-1 diverging suddenly at $r_0\ge 0.81$ (Figure 4D,E). These results establish that adaptive filter performance was stable when the $r_0$ estimate is less than approximately two orders of magnitude from the true measurement noise.

The $q_0$ parameter space was searched to find the value at which $q_0$ began to negatively effect error. Simulations used the uniform time distribution and irregular case, using $r_0=\tilde{r}$ to assess the impacts of $q_0$ values between ${10}^{-16}\le q_0\le {10}^0$ on filter performance (Figure 5). The zero-order noise model showed that error remained consistently low when $q_0\le {10}^{-4}$, and then suddenly increased to approximately the same error as the OLS estimator (Figure 5A). All three first-order noise model filters showed a smooth transition from low error to matching OLS estimator performance when $q_0\ge {10}^{-10}$ (Figure 5B,C). All three first-order noise model filters also showed an error impulse at $q_0={10}^{-8}$. Greatly overestimating $q_0$ slightly elevated the error relative to the OLS estimator for all tested filters; however, this effect on RSAK-0 was less pronounced.

The ${\gamma }_1$ parameter space was searched to find the value at which ${\gamma }_1$ began to negatively effect the error. Simulations used the uniform time distribution and irregular case, using $r_0=\tilde{r}$, $q_0=\tilde{q}$, and ${\gamma }_2=0.95$ to assess the impacts of ${\gamma }_1$ values between $0.01\le {\gamma }_1\le 0.58$ on filter performance (Figure 6). The ARK-∅A and RSAK-1 filters both experienced increasingly large error impulses as ${\gamma }_1$ increased. The other filters showed relatively low error irrespective of the noise model used. However, during long periods of static surface dynamics (e.g., $600<k<800$), some filters achieved the lowest error with ${\gamma }_1=0.05$ while others achieved it with ${\gamma }_1=0.58$. In general, lower ${\gamma }_1$ reduced the maximum error and increased the minimum error, effectively narrowing the error envelope around the WLS estimate used in the state discrepancy calculation (e.g., Figure 6C).

3.3. Response to Case Dynamics

The average state error (${ϵ}_k$) is plotted as a function of epoch and shows how the performance of each filter reacted to the dynamics characterized by each case. Figure 7 shows the results for the uniform time distribution, $r_0={10}^{-3}$ and $q_0={10}^{-24}$ as an example. Other simulation results follow a similar pattern, and performance can be inferred by cross-referencing with Section 3.2. Recall that only the measurement noise vector is added to the static case, and that no change or process noise is added. The error for each filter converged on a lower limit defined by the model noise parameters, which is lowest for CKF and RSAK-0 and highest for RSAK-1 and RSAK-1A (Figure 7A). The periodic case demonstrated an inability for RSAK-0 to track the constant periodic change, while only RSAK-1 and RSAK-1A were able to consistently outperform OLS (Figure 7B). The irregular case in general, and the inset image for the catastrophe case, all exemplify how each filter responded to sudden change (Figure 7C,D). RSAK-1 required fewer epochs to transition from inferior to superior performance relative to OLS than all other filters (approximately 12 epochs for RSAK-1 and 30 epochs for the other filters, Figure 7D). The asymptotic filters exhibited large error impulses, while all of the other filters exhibited comparable maximum error across cases, with the notable exception of RSAK-1 on the second catastrophic event (Figure 7C,D). The differential performance based on process noise model order demonstrates the trade-off between error minimization and responsiveness through the noise parameters.

Figure 8 shows the average $\mathit{\beta }$ and ${\mathit{\beta }}^{\prime }$ error for the irregular and catastrophe cases and all filters except CKF where $r_0={10}^{-1}$ and $q_0={10}^{-16}$. The WLS estimator and state reset (i.e., $\alpha =0$) can be observed truncating the error of both components to the same error as the OLS estimator (Figure 8). This pattern of error truncation was only observed for the $\mathit{\beta }$ coefficients when $r_0={10}^{-1}$, as documented in Table 4. The close relationship between both sets of coefficients is demonstrated as the small error in the ${\mathit{\beta }}^{\prime }$ coefficients is extrapolated into larger $\mathit{\beta }$ coefficients. However, the non-linear temporal signature of the catastrophe is observed as a very large ${\mathit{\beta }}^{\prime }$ impulse error in the asymptotic filters, despite the absence of a similar error signature in the $\mathit{\beta }$ component. Otherwise, all non-asymptotic adaptive filters yielded a lower ${\mathit{\beta }}^{\prime }$ error than the OLS estimator.

3.4. Mode of Adaptation

The ${\alpha }_k$ and $q_k$ values were recorded for the normal time distribution with $r_0=\tilde{r}$ and $q_0=\tilde{q}$ to demonstrate the mode by which the ARK and RSAK filters adapted to the irregular case. Figure 9A shows that non-asymptotic filters responded to the initiation and cessation of change with a WLS state reset, triggered by the $\alpha =0$ condition. RSAK-1 and RSAK-1A were the only filters that consistently retained $\alpha$ values near 1. All other filters maintained $\alpha$ values near 0.95, irrespective of the process noise model. Figure 9B,C record the $q_k$ multiplier used to scale the $\mathit{Q}$ matrix for the RSAK-0 and RSAK-1 models, where both filters adjusted the original value immediately at the first epoch. RSAK-0 estimated $q_k$ values around ${10}^{-7}$ and was resilient to the added observation noise vector (observed as a small confidence interval), but fluctuated over time. RSAK-1 stabilized on $q_k\approx {10}^{-11}$, and a greater variance between simulations was observed. However, neither filter accurately estimated the true process noise.

3.5. Spatial Distributions

A 2D array of ARK-∅A and OLS filters was constructed to analyze a small DEM. The time series of predicted elevation error relative to the true surface is shown in Figure 10A with solid lines. The OLS estimate error was consistently equivalent to the standard deviation of the added noise. The ARK-∅A error demonstrated an immediate reduction in estimation error from 0.1 m by 0.04 m (a reduction of approximately 40%), where it varied from epoch 10 onward. Both OLS and ARK-∅A increased the negative skew of the minimal curvature; however, OLS increased the negative skew by a larger margin than ARK-∅A, Figure 10A. The spatial distributions of minimal curvature in Figure 10B1–D1,B2–D2 show that the spatial pattern of minimal curvature remained negatively affected by spatially auto-correlated noise throughout the experiment. However, upon closer inspection, the ARK-∅A distribution in Figure 10D2 showed a substantially lower impact of the linear artifacts than the OLS distribution in Figure 10D1.

4. Discussion

The use of Kalman filters for topographic analysis compared favourably to OLS estimators when multiple surveys are available. While OLS estimators do not risk degraded performance due to poor parameterisation, they do not account for measurement or system noise nor estimate uncertainty. Kalman filters demonstrated the potential to lower error relative to OLS; however, they risk severe error if parameterisation is sub-optimal or error is non-linear, resulting in unreliable state estimates, as reported elsewhere [38,44]. CKF exemplified this behavior with some parameterisations yielding a lower error than OLS estimators, but often producing much higher errors. All adaptive filters were effective at modeling the dynamics defined by all cases tested and are superior to OLS if the surface can be assumed static. The adaptive Kalman filters generally outperformed OLS and CKF across all parameterisations for dynamic cases, reducing the average error by up to 80%. Maximum errors were contained within approximately 30% greater than OLS by the WLS state reset, addressing the propensity of Kalman filters to become unreliable in response to model violations. These patterns remain true for both time distributions, with the only notable difference being a slightly increased error variance for the uniform distribution and a higher maximum error, exceeding OLS by approximately 40%. While adaptive Kalman filters do not guarantee superior performance compared to OLS estimators, the potential for error reduction is greater than the potential error increase.

All adaptive filters outperformed OLS by a large margin when estimating spatio-temporal coefficients (Figure 3). The extrapolation of the ${\mathit{\beta }}^{\prime }$ coefficients strongly penalizes errors, while DEM subtraction incorporates multiple measurement noise vectors (Figure 3D). This is likely related to the larger innovation variance systematically deflating the Kalman gain, reducing the impact of measurements on the subsequent state estimates, allowing a faster error reduction in the ${\mathit{\beta }}^{\prime }$ components. Thus, the state estimate relies more on the predicted observations based on ${\mathit{\beta }}^{\prime }$ rather than on subsequent measurements. However, the use of asymptotic filters for highly non-linear dynamics, such as the catastrophe case, should be approached with caution due to the large impulse error in ${\mathit{\beta }}^{\prime }$ components and lack of WLS reset (e.g., Figure 8D). While adding second-order temporal coefficients (i.e., acceleration) may improve the response to and recovery from non-linear change, it may result in extreme sensitivity to noise and irregular temporal spacing. Analyzing the local spatial properties of surface change has not been considered in terrain analysis. A similar idea has been explored in the related field of surface metrology, where measuring geometric deviations between surfaces is of interest [67]. The spatio-temporal coefficients may provide information for characterizing local surface change, such as vertical displacement or surface geometry. These data are complementary to non-local surface change analyses (e.g., translation) derived from InSAR or other methods (e.g., [30,68]).

Parameter selection is largely dependent on the desired error envelope. Filters were relatively sensitive to choice in $r_0$, favoring overestimation by one order of magnitude, but not more than two orders of magnitude (Figure 4 and Table 1, Table 2, Table 3 and Table 4). Slightly overestimated $r_0$ relative to the true noise consistently yielded the lowest average error. Overestimating $q_0$ forced the estimator to converge on the current estimate by increasing uncertainty to favor incoming observations (Figure 5). All RSAK filters immediately adjusted the $q_0$ value, rendering the initial estimate somewhat irrelevant. However, there is a theoretical upper limit, because a large enough $q_0$ may inflate uncertainty in Equation (4) such that no observed state discrepancies are large enough to yield $\alpha <1$ after normalization, preventing adaptation from occurring. Thus, all adaptive filters should be initialized with greatly underestimated $q_0$. The error envelope is strongly effected by ${\gamma }_1$, where low ${\gamma }_1$ thresholds allow the current WLS estimate to guide adaptation and control most model divergences. This diverges from the notion of optimal devised by Yang and Gao [58], where a much larger margin of error is recommended before initiating adaptation. This discrepancy is likely due to the frequent violation of model assumptions by the various surface dynamics, which reflects a theoretical incompatibility with temporally non-linear dynamics. However, the results demonstrated that ${\gamma }_2$ acts as a guard against excessively large error due to non-linear surface dynamics by truncating error at the WLS estimate (e.g., Figure 8). However, there is no obvious solution to guarantee that Kalman filter error is below WLS error because Kalman filters do not typically operate with awareness of true values. Instead, error for complex dynamics can only be contained within a margin about the WLS estimate defined by $\gamma$.

The choice of filter and noise model determines the reaction time to a dynamic event. For the static case, RSAK-0 and CKF had the lowest minimum error and RSAK-1 and RSAK-1A had the highest, with all ARK filters performing in between (e.g., Figure 7A,C). However, RSAK-0 performed very poorly in the periodic case, and therefore cannot be considered reliable for dynamic environments. The irregular and catastrophe cases show that RSAK-1 recovers in the fewest epochs from a state reset, requiring approximately 10 epochs to achieve lower error than OLS compared to the approximately 25 epochs required by the other adaptive filters (Figure 7D). However, the performance discrepancy between RSAK-1 and ARK-1 for the catastrophe case reveals unpredictable performance driven by the $\mathit{Q}$ estimation, suggesting that ARK filters are more reliable. Alternatively, ARK-∅ and ARK-∅A were able to achieve comparable performance with the other filters without the complications of designing the process noise model. This suggests that, at least in the tested cases, $\alpha$-based adaptation alone can effectively absorb process noise. As a result, ARK-∅ balances several performance trade-offs and represents a suitable filter for general analysis (i.e., without a priori process knowledge). The asymptotic filters, especially ARK-∅A, have a smooth adaptive function that may preserve spatial continuity when mapping spatial distributions by avoiding a threshold-dependent state reset. However, the absence of the state reset permits very large impulse error for the ${\mathit{\beta }}^{\prime }$ components of the catastrophe case (Figure 8) and caused fatal linear algebra errors when $r_0={10}^{-3}$. In practice, catastrophic events such as landslides or earthquakes are temporally well-defined, so the error impulse for asymptotic filters can be partially addressed by manually re-initializing the filter when an event occurs at the cost of reintroducing noise.

Topographic surveys likely have spatially variable measurement uncertainty resulting from interactions between slope and point density [69,70], slope and horizontal position error and vertical noise [14,39,71,72], or the inclusion of DEM artifacts such as those derived from vegetation and point classification (e.g., [73]). Performance was somewhat sensitive to the choice of $r_0$, requiring an accurate estimate by a margin between one and two orders of magnitude before causing unstable performance (Figure 4). The total survey measurement error is the sum of a reasonably well-known set of sources [74,75,76]. Total error estimates for LiDAR surveys typically span a single order of magnitude (e.g., [75,77,78]), although many factors contribute to the actual magnitude. This suggests that the sensitivity of the adaptive Kalman filters to $r_0$ falls within common measurement error ranges, and that the adaptive filters are therefore suitable for application on LiDAR-derived topographic data.

The inclusion of non-terrain data as surface measurements remains a pervasive challenge known to severely degrade data quality [79,80] and is compounded when comparing multiple surveys. These errors should be considered as a separate issue from other sources of measurement uncertainty because they occur at much larger scales and tend not to have systematic causes (e.g., LiDAR point mis-classification). While several algorithms address this issue (e.g., [81,82,83]), none guarantee that all artifacts are addressed. Yang and Xu [59] shows that a three-segment M-type robust function can be used to nullify faulty or extremely unexpected measurements, which could identify and remove artifacts. This observation space strategy mirrors the state reset used in this research to mitigate unexpected state estimates. Thus, the current configuration of M-type robust estimation provides an adaptive mechanism to address locally varying measurement noise, provides a potential mechanism to address artifacts through the $c_3$ parameter, and can be extended to implement the three-segment function used by Yang and Xu [59]. Using larger windows can also apply some smoothing to mitigate the influence of artifact errors on the surface coefficients while simultaneously implementing multiscale topographic characterization. However, models based on large neighborhoods may interact negatively with the M-type robust function by inflating uncertainty due to topographic roughness rather than measurement noise. Newman et al. [84] showed that the RMSE of a quadratic surface model increases with neighborhood size, increasing the probability of artificial measurement uncertainty inflation from a poor model fit over large or rough areas (i.e., truncation error). Other methods for multiscale terrain analysis, such as Gaussian smoothing [85], smooth excess surface roughness, which may improve performance in a multiscale context.

The similar performance between the normal and uniform time distributions suggest that the adaptive Kalman filters are suitable for imprecise survey temporal distributions (Figure 3). Even with recent technological advances in UAV platforms, mission planning for topographic surveys is subject to a wide variety of constraints that impose limitations on when surveys are actually conducted. While these discrete-time filters are appropriate for most topographic survey mission designs and field campaigns, some technologies, like terrestrial LiDAR, have been used to conduct continuous monitoring with high time resolution (e.g., minute to hour scale intervals, [86,87]). These shorter time intervals may leverage continuous-time variants of the Kalman filter (e.g., [88]). Otherwise, the survey temporal distribution must be sufficiently short to resolve the rate of motion for the measured surface to minimize the generalization of temporal dynamics. The large error impulses observed for the catastrophe case (Figure 7D and Figure 8B,D) reinforce the poor suitability of Kalman filters for highly non-linear surface dynamics, which is consistent with other conclusions [44]. While the state reset provides a partial solution, it may be more suitable to reformulate the problem for use in Extended Kalman filters, Unscented Kalman filters, or particle filters, all of which are more robust to non-linear and non-Gaussian data [89,90].

The Kalman filter has considerably higher memory requirements compared to the OLS or finite difference methods commonly used in terrain analysis due to the additional uncertainty matrices and larger set of state coefficients. A DEM or point cloud consists of millions of locations that must be processed for a single epoch before the addition of new data. Given the large memory requirement of a Kalman filter, any spatially distributed implementation will likely be too large to fit into memory and will likely require a database memory management using permanent storage (e.g., [91]). However, there are several properties that are favourable to managing these limitations. Several matrices, such as $\mathit{A}$, $\mathit{C}$ for regularly gridded raster data, and an independent version of $\mathit{R}$, can be shared between multiple locations to reduce memory and storage requirements. The presented implementation in Algorithm 1 uses $\Delta t$, $r_k$, and $q_k$ scalars to reconstruct the extrapolation and covariance matrices at the beginning of each epoch to minimize persistent data requirements. As a result, only the state vector containing model coefficients, the state uncertainty matrix, which records local coefficient uncertainty, and the scalar $q_k$ must be saved between epochs. This also raises the issue of measurement independence and spatial auto-correlation, especially over short distances, and is expected for interpolated DEMs. The results of the spatial analysis in Figure 10 demonstrate that Kalman filters are capable of minimizing even highly spatially auto-correlated noise, reducing the error of all elevation estimates by approximately 40%. While it is possible to encode spatial auto-correlation in the measurement noise matrix [92], the $\widehat{\mathit{R}}$ function currently only returns a diagonal matrix. Future research is required to conduct a more rigorous evaluation on spatially auto-correlated noise in order to further improve Kalman filter performance.

5. Conclusions

True surface parameters are rarely exactly known in practice. Thus, the quality of surface characterization is dependent on estimator quality with little ability to validate parameter estimates. This research sought to evaluate the potential for adaptive Kalman filters to exploit information from multiple topographic surveys to improve surface model coefficient estimates. The results showed that adaptive Kalman filters are generally able to reduce error relative to OLS estimators by 50–80%. However, error was elevated by approximately 40% for up to 25 epochs when an unexpected change in the dynamic regime occurred.

The combination of adaptive filters and M-type robust estimators addressed several issues limiting the application of Kalman filters to topographic analysis simultaneously. Many of the tested adaptive Kalman filter configurations achieved favourable performance and were relatively insensitive to initial parameterisation. Process noise estimation in combination with adaptation yielded the fastest error recovery times; however, performance was not reliable. Instead, robust filter performance was achieved by relying on early $\alpha$ only adaptation with small ${\gamma }_1$ thresholds and an underestimated or null $q_0$. The ARK-∅ and its asymptotic variant ARK-∅A were able to achieve consistently favourable performance while bypassing process noise entirely, representing a general option for surface modeling and mapping, respectively. Measurement noise was insensitive to overestimation between one and two orders of magnitude from the true variance, achieving the lowest error when $r_0$ was overestimated by one order of magnitude. Despite comparable performance to current methods, highly non-linear surface dynamics remain a challenge for adaptive filters.

Adaptive Kalman filters have several implications for terrain analysis beyond this research. Primarily, the results show that adaptive Kalman filters reliably reduce the influence of noise on surface parameter estimates, resulting in lower error. This increases the accuracy of local surface characteristics derived from the surface coefficients, which is particularly valuable when calculating higher-order surface derivatives, or conversely, minimizing the negative impact of noise and artifacts. Moreover, the spatio-temporal coefficients provide novel information with which local surface dynamics can be interrogated from a time series of surface observations, which may be useful for providing validation data for other models (e.g., volumetric change) or characterizing the impact of various dynamic processes on the Earth’s surface. The ability to achieve low error for static and dynamic landscapes over vastly different time distributions is highly valuable for terrain analysis in general. Several limitations must be overcome to apply this method in a spatially distributed fashion. Memory management is particularly important given the relatively large state vector and uncertainty matrices. Subsequent research must address these limitations and assess the spatial distribution of multiple independent model errors.