1 Introduction

Understanding and quantifying wildfire behaviors is of interest to the fire management and research communities for numerous reasons. Fire managers are often asked to predict how large an incident will grow, how long it will continue, or when rapid growth may threaten public safety or hamper firefighter effectiveness. Fire researchers analyze how these behaviors are influenced by the environment, monitor long-term impacts, and assess existing scientific theories. Regardless of the audience, an often-valuable piece of information is a complete and accurate record of wildfire growth. Fire managers use this information as a training aid, decision-making guide, and prediction tool (Alexander and Thomas 2003) and it is also used by the research community for a variety of purposes including the validation and improvement of spread models (Andrews et al. 2007; Alexander and Thomas 2003; Taylor et al. 2013), estimation of wildfire emissions (Veraverbeke et al. 2015; Turquety et al. 2007; Mangeon et al. 2015; Lavoué et al. 2008) and associated health effects (Moeltner et al. 2013), relating meteorology to extreme growth (Billmire et al. 2014), and identifying correlated wildfire behaviors (Birch et al. 2014). Important quantities such as final burn area (Turetsky et al. 2004), area burned per day (Billmire et al. 2014; Birch et al. 2014; Turner et al. 1994) and the classification of high and low growth days (Finney et al. 2009) are easily calculated from this kind of information. Records of growth throughout an incident’s lifetime are relatively rare in comparison to other kinds of information such as size, occurrence and duration, but are sometimes available from a combination of written and spatially explicit sources.

Written growth records may come from case studies, historical accounts, administrative records (Taylor et al. 2013), newspaper reports (Johnston et al. 2006), and social media services (De Longueville et al. 2009). In the United States, the incident status summary (ICS 209) details information regarding natural disasters occurring on federally owned lands and are a rich source of written growth records for wildfire incidents. ICS 209 reports provide near-daily updates to agency managers who make decisions at broad geographic scales regarding the planning, allocation, and prioritization of resources. Each report contains 47 blocks of information including incident name, date and time, coordinates, management strategy, fuel types, values-at-risk, firefighting resources, estimated costs, containment dates, and incident size. While sometimes possible to construct a complete daily wildfire growth record from ICS 209 reports, they are not compiled for every incident and may be based on rough estimates of burn area.Footnote 1

Information similar to that contained in written growth records can often be derived from other spatially explicit growth records, which have become increasingly available with technological improvements in burn area mapping. Prior to these advances, spatially explicit growth records had to come from hand-drawn maps based on observations from ground-level (Petersen 2014; Callister et al. 2016), a highly subjective and unreliable process. The use of aerial surveys and later Global Positioning Systems alleviated these limitations somewhat by allowing mappers to quickly and sometimes more accurately map the burn perimeters (Kasischke et al. 2002; Conese and Bonora 2005). Infrared imaging systems that detect wildfires through darkness and smoke (Hirsch 1965) dramatically improved the accuracy of aerial imagery and are incorporated into a variety of sophisticated technologies and products widely used in modern wildfire perimeter mapping (Allison et al. 2016). Satellite instrumentation was first used to monitor wildfire activity in the 1970’s (Taylor et al. 2013) and many other space-based instruments have been launched since then, taking regular samples of burn area and active fire detections globally (Joyce et al. 2009). Satellite coverage is currently sufficient to monitor daily fire activity across North America (McNamara et al. 2004), providing a useful spatially explicit growth record when alternative sources are incomplete (Veraverbeke et al. 2014; Parks 2014) or nonexistent (Zhang et al. 2011).

Observation errors, missing data, and disagreement among sources suggest that no single data source can be trusted as authoritative and we can often improve data quality by integrating information across multiple data sources into a unified, but still uncertain estimate of the growth curve (Magnani and Montesi 2010). We explore the problem of reconstructing growth curves from incomplete and corrupted data in Sect. 2, where we present two state space models for reconciling data from multiple data sources into a coherent growth curve. In Sect. 3, we apply the models to wildfire growth data to assess the relative fit and computational requirements of both models. Section 4 presents the results of a sensitivity analysis that compares the effects of substituting priors. Section 5 closes the paper with a discussion of the potential applications of both models as data reconciliation tools and suggests future research directions.

2 Model assumptions

We address the problem of reconstructing growth curves from incomplete and corrupted data through the use of state space models, which have separate components to describe the underlying process and the observations. Specifically, we assume the actual burn area over time, \({\mathbf {x}=\{}X_{0},X_{1}\ldots ,X_{T-1}\}\), is the underlying process and follows deterministically from a growth model that accepts parameters in the vector \({\theta }\), and we assume the observations from data source \(i\in \{1,2,\ldots ,N\}\), denoted \(\mathbf {y}_{i}=\{Y_{i0},Y_{i1}, \ldots ,Y_{i T-1}\}\), are noisy realizations of the underlying process and are related via probability distributions that accept parameters in the vector \(\gamma _{\mathrm {i}}\) (Godsill et al. 2004).

Given known aspects of wildfire behavior and growth records, we assume the growth model to be discrete-time, non-decreasing, non-negative, and sigmoidal. The standard Beverton–Holt difference equation, described by Eq. (1), follows these constraints and we assume it to be a reasonable model of the process.

$$\begin{aligned} X_{t}=\frac{r_{t-1}X_{t-1}}{1+(r_{t-1}-1)\,X_{t-1}}. \end{aligned}$$

The resulting growth curve is controlled by the inherent growth parameters, \(r_{t}\) and the initial condition, \(X_{0}\) (Beverton and Holt 1957; De La Sen 2008), and if \(0<X_{0}< 1\) and \(r_{t}\ge 1\), then the process will be discrete, begin at size \(X_{0}\), and approach size one according to a sigmoidal function.

We present two versions of the standard Beverton–Holt difference equation: the constant growth parameter (CGP) model and the non-constant growth parameter (NGP) model. The CGP model uses only one inherent growth parameter to describe the distribution of growth over the wildfires’ lifetime, assuming an underlying process that is a discrete-time analog of the logistic equation (Berezansky and Braverman 2004). The NGP model has a time-varying inherent growth parameter, producing curves that better describe processes which deviate from the simple sigmoid growth curve. The NGP model allows time-varying growth by letting the inherent growth parameter follow the difference equation described in Eq. (2),

$$\begin{aligned} r_{t}=1+(r-1)\omega _{t}. \end{aligned}$$

Here the shifted inherent growth parameter is multiplied by lognormal noise, \(\omega _{t}\), that has geometric mean 1 and geometric standard deviation \(\tau \). Note that for a process of length T, the last value of the inherent growth parameter, \(r_{T-1}\), is arbitrary and we set \(X_{T-1}=1\). Also note that the number of parameters in \({\theta }\) depends on the growth model under consideration, with the CGP model accepting \(\theta _{CGP}=(X_{0},r)\) and the NGP model accepting \(\theta _{NGP}=(X_{0},r_{0:T-1})\).

The observation equations can relate the actual burn area to our data through deterministic or stochastic means, with rescaling procedures being an example of the former and regression models the latter. We assume a stochastic relationship between the process and observations, with multiplicative rather than additive observation errors due to the presumed nonnegativity of burn area data and the use of proportional errors in existing validation studies (Kolden et al. 2012; Sparks et al. 2015). We also assume that observation errors are larger during periods of high-growth than in low-growth and weigh the observations according to Eq. (3).

$$\begin{aligned} Y_{it}\sim \mathrm {Lognormal}(\ln (K_{i}X{}_{t}), \sigma \sqrt{X_{t}/X_{t-1}}). \end{aligned}$$

Here \(K_{i}\) is the final burn area parameter for data source i and rescales the process. The observation error parameter, \(\sigma \), represents the geometric standard deviation of observations generated from a wildfire that is not growing. The observation equations have the same structure in the CGP and NGP models with a final burn area parameter specific to each data source and a common observation error parameter.

The parameter values are not known with certainty, therefore, we describe them with probability distributions, or priors, that represent our beliefs regarding their values.

$$\begin{aligned}&X_{0}\sim \mathrm {U}((1+r^{1/2})^{-1}, (1+r^{T-3/2})^{-1}). \end{aligned}$$
$$\begin{aligned}&m\sim \mathrm {Beta}(2.37,1.65). \end{aligned}$$
$$\begin{aligned}&\sigma \sim \mathrm {U}(3.219\times 10^{-7},1.175). \end{aligned}$$
$$\begin{aligned}&\mathrm {ln}(K/404)\sim \mathrm {Gen.Gamma}(0.396,0.298,2.357). \end{aligned}$$
$$\begin{aligned}&\tau \sim \mathrm {Lognormal}(-6.827,5.214). \end{aligned}$$

The distribution shown in Eq. (4) constrains the possible growth curves so that the largest daily size increment occurs during the reconstructed burn area time series. The CGP model forces the time step of the largest daily size increment, \(\arg \max _{t}(X_{t}-X_{t-1})\), to an integer, \(t_{max}\ge 1\), by initializing at \(X_{0}=(1+r^{(t_{max}-1/2)})^{-1}\). Assuming that the largest daily size increment occurs within the observation window implies that \(1\le t_{max}\le T-1\), or equivalently, \((1+r^{1/2})^{-1}<X{}_{0}<(1+r^{T-3/2})^{-1}\), which form the limits of a uniform distribution.

The inherent growth parameter prior is shown in Eq. (5) and is elicited by fitting to independent data of peak growth, \(m=\max _{t\in \left\{ 0,1,\ldots ,T-2\right\} }(X_{t+1}-X_{t})/K\), which under the CGP model, is related to the inherent growth parameter via \(r=(m+1)^{2}/(m-1)^{2}\). Peak growth estimates come from 2013 ICS 209 recordsFootnote 2 of large (\(>404\) ha) wildfires occuring within the continental United States during the years 2002-2013. The beta distribution is fit via maximum-likelihood and adequately describes the peak growth data as confirmed upon visual inspection of the QQ-plot.Footnote 3

The observation error parameter prior shown in Eq. (6) places finite bounds on the range of possible errors using a uniform distribution. The upper bound represents a scenario where a multiplicative error of a factor of 10 or greater occurs in 1 in 20 observations. For context, the largest overestimate of burn area in a survey of eight relevant ICS 209 reports was by a factor of 1.8, with values less than 1.05 being more common. The lower bound represents a scenario where a multiplicative error of 1-1/1585000 or greater happens occurs in 1 in 20 observations, which suggests the observations of a Yellowstone sized wildfire are largely accurate to within an acre.

Equation (7) describes the final burn area parameter prior, which is a generalized gamma distributionFootnote 4 fit to transformed burn area (BA) data from the same ICS 209 records as the peak growth prior. The generalized gamma distribution is fit via maximum-likelihood and adequately describes the distribution of the transformed burn area data as confirmed via visual inspection of the QQ-plot. In some cases it is natural to use an unscaled final burn area parameter prior, which equals one with probability one, to represent a scenario in which the observations are simply noisy realizations of the underlying process.

In the NGP model, we require a so-called noise prior to describe the process noise, \(\tau \), which we construct using two extreme scenarios to bound the distribution’s central 90th percentile. To set the lower bound on the noise prior, we consider near identical levels of peak growth, \(m_{1}\) and \(m_{2}\), convert them to \(r_{1}-1\) and \(r_{2}-1\) and find the the multiplicative difference. The 5th percentile of the noise prior then corresponds to a scenario in which the incremental difference of the inherent growth parameter is of that scale or greater occurs with probability 0.002, or about once every 500 time steps. We calculate the upper percentile using a similar process, but with very dissimilar levels of peak growth, representing the scenario where process variability is extremely high. Specifically, the 95th percentile of the noise prior corresponds to a scenario in which the incremental difference in the inherent growth parameter of that scale or greater happens with probability 0.2, or about every 5 time steps. The quantities \(m_{1}=1/1{,}585{,}000\) and \(m_{2}=1/1{,}585{,}001\) are used to calculate the lower bound and \(m_{1}=1/1000\) and \(m_{2}=1/1{,}585{,}001\) for the upper bound, resulting in the final noise prior described in Eq. (8).

3 Application

3.1 Data and computational methods

To illustrate the application of the state space models, we reconstruct wildfire growth curves from 13 incidents from the 2014 wildfire season (Table 1) using \(N=2\) data sources: burn area estimates from GeoMAC wildfire perimeters.Footnote 5 and cumulative hotspot detects from the Hazard Mapping System (HMS).Footnote 6

For each incident k, there are two observation vectors of length \(T_{k}\), where \(T_{k}\) is the number of days between the incident’s first and last perimeter plus a six day buffer period to capture information outside the lifetime of GeoMAC measurements. One observation vector is populated with burn area estimates extracted from the “area” feature of GeoMAC perimeters, retaining only the largest perimeter when two exist on the same day. The other observation vector is populated with the percentage of the total HMS hotspot detects occurring within the incident boundary, where we define the incident boundary to be the largest perimeter with an 8-kilometer buffer. Note that the priors in the previous section are fit to data that are independent of those used in this application.

Both the CGP and NGP models are fit using JAGS software with the runjags package in R (R Development Core Team 2008) on a MacBook Pro with a 2.7 GHz Intel Core i7 processor. We first compute an initial Markov chain Monte Carlo (MCMC) using three parallel chains with a nominal sample size of 1000, thinning interval of 100, burn-in period of 10,000 and adaptive phase of 10,000. Convergence is monitored visually and by calculating the potential scale reduction factor of the range of the central 90% of the marginal posteriors (Brooks and Gelman 1998). If neccesary, the simulations are continued in batches of 1000 iterations until the maximum potential scale reduction factor falls below 1.01 (Gelman and Shirley 2011). We assess the relative fit of the reconstruction models by using the expected log Bayes factor, \(E[2\mathrm {ln}(\mathrm {Pr}(\mathbf {x},{\theta }_{NGP}, {\gamma _{\mathrm {1:2}}}|\mathbf {y}_{\mathrm {1:2}}) /\mathrm {Pr}(\mathbf {x}, {\theta }_{CGP},{\gamma }_{\mathrm {1:2}}|\mathbf {y}_{\mathrm {1:2}}))]\) (Kass and Raftery 1995) and assess model quality using the root mean square error (RMSE), mean absolute error (MAE), mean absolute percent error (MAPE), and mean bias error (MBE) between the GeoMAC observations and the median reconstructed growth curve (Cruz and Alexander 2013).

Table 1 Computational requirements of both NGP and CGP models

3.2 Computational requirements and fit

The computational requirements of the CGP model are substantially less than the NGP model in terms of the number of Gibbs iterations required, time required and sampling efficiency (Table 1), with even the slowest CGP model reconstruction completing in far less time than any NGP reconstruction. In the NGP model, longer duration incidents require more Gibbs iterations, more time to converge, and fewer Gibbs iterations per unit time compared to shorter duration incidents. In CGP models, longer duration incidents took more time to converge, fewer Gibbs iterations per unit time, but did not require substantially more iterations to converge. In general, applying the CGP model will require at most a few minutes to converge, while the NGP model takes between 13 min to 9 h.

With the exception of the Somers fire, the increased computational requirements of the NGP models are compensated by noticeable improvements in fit to observed data, as evidenced by the log Bayes factor scores (Table 2), goodness-of-fit statistics (Table 3), and graphical comparisons of both models (Fig. 1). While the NGP model provides a superior fit compared to the CGP model, the overall accuracy varies across incidents. For instance, the range of MAPE statistics under the NGP model are quite variable, with high accuracy in the Snag Canyon (10%), King (14%), and Eiler (16%) wildfires, but poor goodness-of-fit in other cases such as Carlton (9109%), South Fork (1532%), and Buzzard (1498%).

Table 2 Log Bayes factor, sample size and GeoMAC burn area
Table 3 Goodness-of-fit summaries
Fig. 1
figure 1

Observed burn area estimates from GeoMAC and the Hazard Mapping System are shown as square and circular points, respectively. The median and central 90th percentile of the marginal posterior distributions of the growth curve estimates are shown under the constant growth parameter (blue) and non-constant growth parameter (red) model. Letters correspond to the wildfire incidents in Table 2

4 Sensitivity analysis

Multiple defensible priors can often be applied to the same problem for reasons such as differences in philosophies regarding their purpose in Bayesian analysis, variation in elicitation techniques, and diverse individual levels of uncertainty (Spiegelhalter et al. 2004). Given that priors are both subjective and influential on the final results, it is valuable to determine how changes to the priors may influence our burn area time series reconstructions. To that end, we perform a sensitivity analysis to gauge how the choice of priors influences the results from the Johnson Bar fire, which we reconstruct multiple times under multiple model configurations.

Five of these configurations exchange the final burn area prior, two of which are distributions based on simple assumptions about the bounds of the parameter, while the remaining three are fit to burn area data. The first distribution is called the lower-truncated inverse-uniform (LTIU) prior, \(404\times K^{-1}\sim \mathrm {U(0,1)}\), which sets no upper limit on final burn area but does set a lower limit of 404 ha. Strict upper and lower limits are imposed using the bounded inverse-uniform (BIU) prior, \(K^{-1}\sim U(6.41\times 10^{-5},404^{-1})\), which constrains the burn area to be between 404 ha and the size of the 1988 Yellowstone wildfires. The remaining final burn area parameter priors are based on burn area data from the monitoring trends in burn severity projectFootnote 7 dataset, which includes 9050 fires larger than 404 ha across the conterminous U.S. and from the years 1984–2010. The lower-truncated inverse-beta (LTIB) prior, \(404\times K^{-1}\sim \mathrm {Beta}(0.964,1.152)\) and the bounded inverse-beta (BIB) prior \(404\times (K^{-1}-6.41\times 10^{-5})/(1/404-6.41\times 10^{-5}) \sim \mathrm {Beta}(0.958,1.149)\), are maximum-likelihood estimates of the transformed burn area data and similarly, the Pareto prior, \(K\sim \mathrm {Pareto}(404,1.13)\), is fit to the untransformed data.

Four of these configurations exchange the original peak growth prior with an alternative distribution. The four alternatives are called the uniform (U), mode-zero triangle (MZT), mode-one triangle (MOT) and arcsine priors, each being a special case of the beta distribution: \(m_{U}~\sim \mathrm {Beta}(1,1),\) \(m_{MZT}\sim \mathrm {Beta}(1,2)\) , \(m_{MOT}\sim \mathrm {Beta}(2,1)\), and \(m_{Arcsine}\sim \mathrm {Beta}(0.5,0.5)\) (Fig. 2).

Fig. 2
figure 2

Boxplots of each model parameter’s marginal posterior for each reconstruction model and fire. The left shows process component parameters incuding the inherent growth parameter (top); initial conditions (middle) and the process noise (bottom). The right shows the observation component parameters including final burn area (top) and the observation error (bottom). Outliers are omitted. Letters correspond to fires in Table 2

We also propose three new configurations for observation error priors and noise priors, which we elicit using slight variations of the techniques used to form the original distributions. We originally construct the observation error parameter prior by finding the uniform distribution with endpoints corresponding to specific error exceedance scenarios. We propose alternative priors for the observation error parameter by repeating this process using different exceedance probabilities for the extreme observation error variability scenario, generating a new upper-bound on the uniform distribution. Specifically, the upper-truncation points are recalculated under small (0.01) medium (0.1) and high (0.5) exceedance probabilities, resulting in the alternative upper limits of 0.89, 1.40 and 3.41. Similar to the elicitation of new observation error priors, the alternative noise priors assume that the central 90th percentile of the original distribution represents a different amount of probability mass. Specifically, in our three alternative noise priors, we reinterpret the original bounds as the central 99th, 95th and 50th percentile of a lognormal distribution: \(\tau _{small}\sim \mathrm {Lognormal}(-6.827,3.330)\), \(\tau _{middle}\sim \mathrm {Lognormal}(-6.827,4.376)\), and \(\tau _{large}\sim \mathrm {Lognormal}(-6.827,12.716)\). Depending on the reconstruction model, we reanalyze the Johnson Bar fire using either 12 or 15 new configurations, where each new configuration is identical to the original, except that the prior for one of the five parameters is exchanged with an alternative. All sensitivity analysis computations use the same hardware and software as Sect. 3.

The computational requirements are largely invariant to the majority of substitutions, with the exception of the MOT prior in the NGP model, which converges in about 20–30% of the iterations and time compared to the original configuration. For the CGP model, the largest response occurs in the final burn area parameter when substituting the LTIU distribution, increasing the posterior mean by 1%. In the NGP model, the largest response occurs in the noise parameter when substituting the MOT distribution, reducing the posterior mean by 2% (Fig. 3). The Johnson bar growth curve reconstructions are fairly robust changes in the priors, with the most dramatic responses occurring early in the time series, where little or no data are available. The largest change to the growth curve is associated with substituting the arcsine distribution, which inflates the initial condition parameter by about 1%. The sensitivity to prior substitutions decreased shortly after the ignition, reducing to near-zero levels in the last days of the reconstruction (Fig. 4).

Fig. 3
figure 3

Boxplots of four model parameter’s marginal posterior with the original and new prior configurations, under the non-constant growth parameter (left panels) and constant growth parameter (right panels) models. The x-axis labels denote the type of prior substitution

Fig. 4
figure 4

The ratio of the non-constant growth parameter model’s median burn area time series reconstruction under the original (Fig. 1j) and alternative prior configurations. Values greater than one represent burn area estimates that are larger in the new prior configuration than in the original

5 Conclusions

Our two reconstruction models represent novel methods of improving the quality of burn area time series and have a number of desirable features. Our growth model generates reconstructions that have behavior consistent with the underlying process: non-decreasing, non-negative, and sigmoidal. The priors incorporate additional information from independent historical growth records, providing the reconstructions with a guide of typical wildfire growth curve characteristics and uncertainty. These state space models are also attractive because they incorporate multiple data sources, reflecting the uncertainty in the growth curve and also permitting information borrowing when needed. They are easily modifiable to accommodate a variety of other data generating and growth processes beyond those explored here. The state space approach is also ideal because the model output is fairly easy to interpret as the probability distribution of likely growth curves given the observed data and known fire behavior.

The NGP model is particularly well-suited for capturing daily variation in fire growth, but the computational requirements are much higher than in the CGP model. If the computational requirements are too burdensome in a given application, a couple of mitigation strategies are available. For instance, the marginal convergence tends to be slower towards the end of the time series and the introduction of an extinguishment parameter would lessen parameter space redundancy and by extension could increase speed. The use of better hardware and more efficient sampling algorithms, like Hamiltonian MCMC, are also obvious strategies for reducing this burden (Neal 2011). In some cases, such as with short duration fires, the NGP model may not be needed and strategic use of the CGP model can substantially reduce overall time and computational requirements.

These reconstruction models and variations of them, have a number of potential applications in fire management and research. A wildfire growth curve database could be organized using these reconstruction models, where all available data are aggregated to produce high quality growth curve estimates with uncertainty. By relaxing the constraints on the initial conditions as to allow peak growth events beyond the range of observations, the reconstruction models can be modified to simulate the future growth of not yet extinguished fires. Future iterations of the model could also incorporate environmental covariates into the reconstruction estimates, describe spatially explicit wildfire growth processes and behaviors, and explore how the models can be useful for other non-wildfire applications. Although we explore the sensitivity of the models to perturbations in the prior choice in Sect. 4, the flexibility of the state space models suggests that many other structural changes could also be applied, influencing the parameter estimates in unknown ways. Other model customizations could include changes to the process error structure, the use of additional variables for other behaviors such as extinguishment, changing the observation error structure, adding or omitting data, and using alternative growth models.

In closing, our reconstruction models offer a natural way of integrating prior knowledge and data from multiple sources into a single coherent estimate of the underlying growth curve that includes estimates of associated uncertainties. Data quality issues are handled elegantly, managing missing and erroneous data without discarding potentially useful information. We do not recommend one of the two models over the other, but wish to highlight the unique circumstances in which each approach may be most beneficial. The NGP model is particularly well-suited for describing the daily variation in wildfire growth and improves the quality of the growth curves based on satellite and ground-based observations alone. However, the relatively high computational requirements of the NGP model suggest that the CGP may be appropriate in situations where processing time is a constraint. While further sensitivity analysis is recommended, the results of the Johnson bar fire are robust to a range of prior substitutions, suggesting that under typical applications of this model configuration, prior sensitivity is not likely a serious issue. We recommend that future research explore the potential of these models as data reconciliation tools, as well as how these models may be modified to meet other research and management needs.