Spatially dependent flood probabilities to support the design of civil infrastructure systems

Conventional flood risk methods typically focus on estimation at a single location, which can be inadequate for civil infrastructure systems such as road or railway infrastructure. This is because rainfall extremes are spatially dependent; to understand overall system risk, it is necessary to assess the interconnected elements of the system jointly. For example, when designing evacuation routes it is necessary to understand the risk of one part of the system failing given that another region is flooded or exceeds the level at which evacuation becomes necessary. Similarly, failure of any single part of a road section (e.g., a flooded river crossing) may lead to the wider system’s failure (i.e., the entire road becomes inoperable). This study demonstrates a spatially dependent intensity–duration–frequency (IDF) framework that can be used to estimate flood risk across multiple catchments, accounting for dependence both in space and across different critical storm durations. The framework is demonstrated via a case study of a highway upgrade comprising five river crossings. The results show substantial differences in conditional and unconditional design flow estimates, highlighting the importance of taking an integrated approach. There is also a reduction in the estimated failure probability of the overall system compared with the case where each river crossing is treated independently. The results demonstrate the potential uses of spatially dependent intensity–duration–frequency methods and suggest the need for more conservative design estimates to take into account conditional risks.


Introduction 29
Methods for quantifying the flood risk of civil infrastructure systems such as road and rail networks 30 require considerably more information compared to traditional methods that focus on flood risk at a 31 point. For example, the design of evacuation routes requires the quantification of the risk that one part 32 of the system will fail at the same time that another region is flooded or exceeds the level at which 33 evacuation becomes necessary. Similarly, a railway route may become impassable if any of a number 34 of bridges are submerged, such that the 'failure probability' of that route becomes some aggregation of 35 the failure probabilities of each individual section. Successful estimation of flood risk in these systems 36 therefore requires recognition both of the networked nature of the civil infrastructure system across a 37 spatial domain, as well as the spatial and temporal structure of flood-producing mechanisms (e.g. storms 38 and extreme rainfall) that can lead to system failure (e.g., Leonard  One way to estimate such flood probabilities is to directly use information contained in historical 41 streamflow data. For example, annual maximum streamflow at two locations might be assumed to 42 follow a bivariate generalized extreme value distribution (Favre et al., 2004;Wang, 2001; Wang et al.,43 2009), which can then be used to estimate both conditional probabilities (e.g. the probability that one 44 river is flooded given that the other river level exceeds a specified threshold) and joint probabilities 45 (e.g. the probability that one or both rivers are flooded). Several frameworks have been demonstrated 46 based directly on streamflow observations, including functional regression (Requena et al., 2018), 47 multisite copulas (Renard and Lang, 2007), and spatial copulas (Durocher et al., 2016). However, in 48 many instances continuous streamflow data are unavailable or insufficient at the locations of interest, 49 or the catchment conditions have changed such that historical streamflow records as unrepresentative 50 of likely future risk. For these situations, rainfall-based methods are often more appropriate. 51 There are two primary classes of rainfall-based methods to estimate flood probability. The first uses 52 continuous rainfall data (either historical or generated) to compute continuous streamflow data using a 53 rainfall-runoff model ( whereas many catchments respond at sub-daily timescales. This is likely to be because the capacity of 60 space-time rainfall models to simulate the statistics of sub-daily rainfall remains a challenging research 61 problem (Leonard et al., 2008), although one approach is to exploit the relative abundance of data at 62 the daily scale, then apply a downscaling model to reach sub-daily scales (Gupta and Tarboton, 2016). 63 Continuous simulation is receiving ongoing attention and increasing application, yet there remain 64 limitations when applying these models in many practical contexts. 65 The second rainfall-based method proceeds by applying probability calculations on rainfall, to construct 66 'Intensity-Duration-Frequency' (IDF) curves, which are then translated to a runoff event of equivalent 67 probability either via empirical models such as the rational method to estimate peak flow rate 68 (Kuichling, 1889;Mulvaney, 1851), or via event-based rainfall-runoff models that are able to simulate 69 the full flood hydrograph (Boyd et al., 1996;Chow et al., 1988;Laurenson and Mein, 1997). Regional 70 frequency analysis is one type of method to estimate IDF values, where the precision of at-site estimates 71 is improved by pooling data from sites in the surrounding region (Hosking and Wallis, 1997). These 72 methods can be combined with spatial interpolation methods to estimate parameters for any ungauged 73 location of interest (Carreau et al., 2013). To determine an effective mean depth of rainfall over a 74 catchment with the same exceedance probability as at a gauge location, the pointwise estimate of 75 extreme rainfall is multiplied by an areal reduction factor (ARF) (Ball et al., 2016). However, such 76 methods do not account for information on the spatial dependence of extreme rainfall-whether for a 77 single storm duration, or for the more complex case of different durations across a region (Bernard, 78 1932;Koutsoyiannis et al., 1998). The underlying independence assumption prevents these approaches 79 from being applied to estimate conditional or joint flood risk at multiple points in a catchment or across 80 several catchments, as would be required for a civil infrastructure system.  Having obtained the IDF estimates for individual locations in Fig. 1, the next step is commonly to 141 convert this to spatial IDF maps by interpolating results between gauged locations. Figure 2 shows 142 hypothetical IDF maps from individual sites, with a separate spatial contour map usually provided for 143 each storm burst duration. In a conventional application the respective maps are used to estimate the 144 magnitude of extreme rainfall over catchments for a specified time of concentration. The IDF estimates 145 are combined with an areal reduction factor (ARF) to determine the volume of rainfall over a region 146 (since rainfall is not simultaneously extreme at all locations over the region). However, because the 147 spatial dependence was broken in the IDF analysis, the ARFs come from a separate analysis and are an 148 attempt to correct for the broken spatial relationship within a catchment (Bennett et al., 2016a). Lastly, 149 the rainfall volume over the catchment is combined with a temporal pattern (i.e. the distribution of the 150 rainfall hyetograph within a single 'storm burst') and input to a runoff model to simulate flood-flow at 151 a catchment's outlet. Where catchment flows can be considered independently this process has been 152 acceptable for conventional design, but because this process does not account for dependence across 153 durations and across a region, it is not possible to address problems that span multiple catchments, as 154 with civil infrastructure systems.

159
The process in Fig. 1 breaks out the dependence of the observed rainfall, which makes the conventional 160 approach unable to analyse the dependence of flooding at two or more separate locations. Instead, this 161 paper advocates for spatially dependent IDF estimates that are developed by retaining the dependence 162 of observed rainfall in the estimation of extremal rainfall. By applying spatially dependent IDF 163 estimates to a rainfall-runoff model, it becomes possible to represent the dependence of flooding 164 between separate locations. 165

Case study and data 166
The region chosen for the case study is in the mid north coast region of New South Wales, Australia. 167 This region has been the focus of a highway upgrade project and has an annual average daily traffic 168 volume on the order of 15,000 vehicles along the existing highway. The upgrade traverses a series of 169 coastal foothills and floodplains for a total length of approximately 20 km. The project's major river 170 crossings consist of extensive floodplains with some marsh areas. 171 The case study has five main catchments that are numbered in sequence in durations is expected to be lower than across a single duration, since short-and long-rain events are 179 often driven by different meteorological mechanisms (Zheng et al., 2015). However some spatial 180 dependence is still likely to be present, given that extremal rainfall in the region is strongly associated 181 with 'east coast low' systems off the eastern coastline, whereby extreme hourly rainfall bursts are often 182 embedded in heavy multi-day rainfall events.  The black circles in Fig. 3 represent the sub-daily rain stations used for this study. There were seven

Methodology 196
This section describes the method used to estimate the conditional and joint probabilities of streamflow 197 for civil infrastructure systems based on rainfall extremes, with the sequence of steps illustrated in Fig.  198 4. The overall aim is to estimate rainfall exceedance probabilities and corresponding flow estimates that 199 account for dependence across multiple catchments. The generalized Pareto distribution (GPD) is used 200 as the marginal distribution to fit to observed rainfall for all durations at each location (Section 4.1). An 201 extremal dependence model is required to evaluate conditional and joint probabilities. Here, an inverted 202 max-stable process is used with dependence not only in space but also in duration (Section 4.2). The

Marginal model for rainfall 210
This study defines extremes as those greater than some threshold . For large , the distribution of 211 conditional on > may be approximated by the generalized Pareto distribution (GPD) (Pickands, The selection of the appropriate threshold involves a trade-off between bias and variance. A threshold 218 that is too low leads to bias because the GPD approximation is poor. A threshold too high leads to high 219 variance because of a small number of excesses. Two diagnostic tests are used to determine the 220 appropriate threshold : the mean residual life plot and the parameter estimate plot (Coles, 2001; 221 Davison and Smith, 1990). These methods use the stability property of a GPD, so that if a GPD is valid 222 for all excesses above , then excesses of a threshold greater than should also follow a GPD (Coles,

Dependence model for spatial rainfall 228
Consider rainfall as a stationary stochastic process associated with a location and a specific 229 duration (the notation is simplified from ( ) to ). An important property of dependence in the 230 extremes is whether or not two variables are likely/unlikely to co-occur as the extremes become rarer, 231 as this can significantly influence the estimate of frequency for flood events of large magnitude. This 232 is referred to as asymptotic dependence/independence, respectively. For the case of asymptotic 233 independence, the dependence structure becomes weaker as the extremal threshold increases, which is 234 defined as lim →∞ { 1 > | 2 > } = 0 for all 1 ≠ 2 . The spatial extent of a rainfall event with 235 asymptotically independent extremes will diminish as its rarity increases. This study uses an 236 asymptotically independent model, of which there are multiple types including the Gaussian copula 237 The dependence structure of the inverted max-stable process is represented by the pairwise residual tail 242 dependence coefficient (Ledford and Tawn, 1996). For a generic continuous process for a given 243 duration and associated with a specific location , the empirical pairwise residual tail dependence 244 coefficient for each pair of locations ( 1 , 2 ) is 245 (2) 246 The value of ∈ (0,1] indicates the level of extremal dependence between 1 and 2 (Coles et al., 247 1999), with lower values indicating lower dependence. An example of how to calculate the residual tail 248 dependence coefficient is provided in Appendix A for a sample dataset. To estimate the dependence 249 structure of an inverted max-stable model, the theoretical residual tail dependence coefficient function 250 is fitted to its empirical counterpart. Here the residual tail dependence coefficient function is assumed 251 to only depend on the Euclidean distance between two locations ℎ = ‖ 1 − 2 ‖. The theoretical 252 residual tail dependence coefficient function for the inverted Brown-Resnick model is given as: 253 where Φ is the standard normal cumulative distribution function, ℎ is the distance between two 255 locations, and (ℎ) belongs to the class of variograms (ℎ) = ‖ℎ‖ ⁄ for > 0 and ∈ (0,2). The 256 model is fitted to the empirical residual tail dependence coefficient by modifying parameters and 257 until the sum of squared errors is minimized. 258 The inverted max-stable process is fitted to the observations by minimizing the sum of the squared 259 errors of the residual tail dependence coefficients. When the extreme rainfall at location 1 and 2 are 260 of different durations, the dependence is less than when the extremes are of the same duration.

Simulation based estimation of areal and joint rainfall 282
The dependence model specification in the previous section enables the calculation of joint and 283 conditional probabilities (Appendix B). Therefore, in addition to traditional IDF return level maps that 284 are based on independence between locations and durations, it is possible to account for the coincidence 285 of rainfall within the region. Current design procedures using IDF estimates are event-based and rely 286 on ancillary steps to reconstruct elements of the design storm that were broken during the estimation 287 procedure. One critical element is the areal reduction factor (ARF), which the dependence model can 288 also be used to estimate. ARFs are used to adjust rainfall at a point (such as the centroid of a catchment) 289 to an effective mean rainfall over the catchment with equivalent probability of exceedance (Ball et al.,290 extrapolate them for long return periods from observations with just 35 years of record for this study. 292 To deal with this difficulty and to analyse the asymptotic behaviour of ARFs, Le et al. (2018a) proposed 293 a framework to simulate ARFs using the same inverted-max stable process model adopted here. The advantage is that the use of empirical distributions guarantees that the marginal distributions of 307 simulated rainfall below the threshold match the observed marginal distributions. There may be a 308 drawback by forcing the simulated rainfall to have the same extremal dependence structure for both 309 parts below and above the threshold, which may not be true for non-extreme rainfall. However, the 310 dependence structure of non-extreme rainfall contributes insignificantly to extreme events (Thibaud et 311 al., 2013) and is unlikely to affect the results. 312 For calculating ARFs, the simulation is implemented separately for spatial rainfall of 36 and 9 hrs 313 duration. ARFs are calculated for each duration and different return periods, which can be found in the 314 supplementary material (Fig. S1 and S2). Figure S1 and S2 provide relationships between ARFs and 315 area (in km 2 ) for different return periods for the case study catchments simulated using the inverted 316 Brown-Resnick process over equally sized grid points. The relationships are interpolated to obtain the 317 ARFs for each subcatchment. 318 The recommended approach for estimating the overall failure probability of a system is demonstrated 319 by considering a hypothetical traffic system with multiple river crossings at locations. If there is a one-320 to-one correspondence between extreme rainfall intensity over a catchment and flood magnitude, the 321 overall failure probability will be approximately equal to the probability that there is at least one river 322 crossing whose contributing catchment has rainfall extremes exceeding the design level, which can be 323

Transforming rainfall extremes to flood flow 334
To estimate flood flow from rainfall extremes, the Watershed Bounded Network Model (WBNM) 335 (Boyd et al., 1996), is employed. WBNM calculates flood runoff from rainfall hyetographs that 336 represent the relationship between the rainfall intensity and time (Chow et al., 1988). It divides the 337 catchment into subcatchments, allowing hydrographs to be calculated at various points within the 338 catchment, and allowing the spatial variability of rainfall and rainfall losses to be modelled. It separates 339 overland flow routing from channel routing, allowing changes to either or both of these processes, for 340 example in urbanised catchments. The rainfall extremes are estimated at the centroid of the catchment, 341 and are converted to average spatial rainfall using the simulated ARFs described in Section 4.3. Design 342 rainfall hyetographs are used to convert the rainfall magnitude to absolute values through the duration 343 of a storm following standard design guidance in Australia (Ball et al., 2016). 344 Hydrological models (WBNM) for the case study area were developed and calibrated in previous 345 studies (WMAWater, 2011). Hydrological model layouts for the Bellinger, Kalang River, Nambucca, 346 Warrell and Deep Creek catchments can be found in the supplementary material (Fig. S3 to S5). 347

Evaluation of model for space-duration rainfall process 349
A GPD with an appropriate threshold was fitted to the observed rainfall data for 36 hr and 9 hr durations, 350 and the Brown-Resnick inverted max-stable process model was calibrated to determine the spatial 351

dependence. 352
Analysis of the rainfall records led to the selection of a threshold of 0.98 for all records as reasonable 353 across the spatial domain and the GPD was fitted to data above the selected threshold. Figure 5 shows 354 QQ plots of the marginal estimates for a representative station for two durations (36 and 9 hr). Overall 355 the quality of fitted distributions is good and plots for all other stations can be found in the 356 supplementary material (Fig. S6 and S7).  The inverted max-stable process across different durations was calibrated to determine dependence 361 parameters. The theoretical pairwise residual tail dependence coefficient function between two 362 locations ( 1 and 2 ) was calculated based on Eq. (3) and Eq. (4), and the observed pairwise residual 363 tail dependence coefficient was calculated using Eq. (2). Figure 6 shows the pairwise residual tail 364 dependence coefficients for the Brown-Resnick inverted max-stable process versus distance. The black 365 points are the observed pairwise residual tail dependence coefficients, while the red lines are the fitted 366 pairwise residual tail dependence coefficient functions. A coefficient equal to 1 indicates complete 367 spatial dependence, and a value of 0.5 indicates complete spatial independence. The top-left panel 368 shows the dependence between 36 hr extremes across space, with the distance h = 0 corresponding to 369 "complete dependence". It also shows the dependence decreasing with increasing distance. Figure 6  370 indicates that the model has a reasonable fit to the observed data given the small number of dependence 371 parameters. Although the theoretical coefficient (red line) does not perfectly match at long distances, 372 the main interest for this case study is in short distances, including at ℎ = 0 for the case of dependence 373 between two different durations at the same location. 374 The remaining panels of Fig. 6 show the dependence of 36 vs. 9 hr extremes, 36 vs. 6 hr extremes, and 375 36 vs. 3 hr extremes, with the latter two duration combinations not being used directly in the study but 376 nonetheless showing the model performance across several durations. As expected, the dependence 377 levels are weaker compared with 36 vs. 36 hr extremes at the same distance, especially at zero distance. 378 This is expected, as extremes of different durations are more likely to arise from different storm events 379 compared to storms of the same duration. 380

Estimating conditional rainfall return levels and corresponding conditional flows for evacuation 386 route design 387
The recommended approach for estimating conditional rainfall extremes is demonstrated by considering 388 a hypothetical evacuation route across location 2 , given a flood occurs at location 1 , evaluated using 389 Eq. (B.4). This approach is applied to a case study of the Pacific Highway upgrade project that contains 390 five main river crossings (from Fig. 3). For evacuation purposes, we need to know "what is the 391 probability that a bridge fails only once on average every times (e.g., = 10 for a one in 10 chance 392 conditional event) when a neighbouring bridge is flooded?" This section provides the conditional 393 estimates for two pairs of neighbouring bridges in the case study that have the shortest Euclidean 394 distances, i.e. pairs ( 1 , 2 ) and ( 2 , 3 ). The comparisons of unconditional and conditional maps are 395 given in Fig. 7 and Fig. 8, and the corresponding unconditional and conditional flows are given in Fig.  396

397
The left panel of Fig. 7  respectively. The relative difference between the conditional and unconditional return levels is only 413 1.45 times, compared with 1.74 times for the case in Fig. 7. This is because the pair of locations in Fig.  414 8 has a longer distance than those in Fig. 7, so that the dependence level is weaker. Moreover, the 415 location pair in Fig. 8 was analysed for different durations (between 36 and 9 hr extremes), which has 416 weaker dependence than the case of the equivalent durations in Fig. 7 (between 36 and 36 hr), based on 417 indicating that the peak conditional flow at the river crossings is almost 2.0 and 1.7 times higher than 438 the unconditional flow for the two catchments, respectively. This difference is a direct result of the 439 conditional event having a higher rainfall magnitude than the unconditional event: given that there is 440 an extreme event nearby, it is more likely for an extreme event to occur at a nearby location. If a bridge 441 design were to take into account this extra criterion for the purposes of evacuation planning it would 442 require the design to be at a higher level. probability. The worst case is complete independence where extremes do not happen together unless by 451 random chance; this means the failure probability of the highway is much higher than that for individual 452 river crossings. Taking into account the real dependence, there are some extremes that align and it seems 453 from Fig. 10 that this is a relatively weak effect. As an example from Fig. 10, to design the highway 454 with a failure probability of 1% annual exceedance probability (AEP), we would have to design each 455 individual river crossing to a much rarer AEP of 0.25% (see green lines in Fig. 10).

Discussion and Conclusions 465
Hydrological design that is based on IDF estimates has conventionally focussed on separate estimation 466 at single locations. Such an approach can lead to the misspecification of wider system risk of flooding 467 since weather systems exhibit dependence in space, time and across storm durations, which can lead to 468 the coincidence of extremes. A number of methods have been developed to address the problem of 469 antecedent moisture within a single catchment, by accounting for the temporal dependence of rainfall 470 at locations of interest through loss parameters or sampling rainfall patterns (Rahman et al., 2002). 471 However, there have been fewer methods that account for the spatial dependence of rainfall across 472 multiple catchments, due in part to the complexity of representing the effects of spatial dependence in 473 risk calculations. Different catchments can have different times of concentration, so spatial dependence 474 may also imply the need to consider dependence across different durations of extreme rainfall bursts. 475 Recent and ongoing advances in modelling spatial rainfall extremes provide an opportunity to revisit 476 the scope of hydrological design. Such models include a max-stable model fitted using a Bayesian 477 hierarchical approach (Stephenson et al., 2016), max-stable and inverted max-stable models (Nicolet et  The spatial rainfall was simulated using an asymptotically independent model, which was then used to 495 estimate conditional and joint rainfall extremes. An empirical method was obtained from the framework 496 of Le et al. (2018b) to make an asymptotically independent model-the inverted max-stable process-497 able to capture the spatial dependence of rainfall extremes across different durations. The fitted residual 498 tail dependence coefficient function showed that the model can capture the dependence for different 499 pairs of durations. For our example, the highest ratio of the one in 10 chance conditional event (in 500 considering the effect of a 20-year event rainfall occurring at the conditional location) to the 10-year 501 unconditional event was 1.74, for the two catchments having the strongest dependence (Fig. 7). The 502 corresponding conditional flows were then estimated using a hydrological model WBNM and shown 503 to be strongly related to the ratio of conditional and unconditional rainfall extremes (Fig. 9). 504 The joint probability of rainfall extremes for all catchments and for all possible pairs of catchments in 505 the case study area was estimated empirically from a set of 10,000 years of simulated rainfall extremes, 506 repeated 100 times to estimate the average value. The results showed that there were differences in the 507 failure probability of the highway after taking into account the rainfall dependence, but the effect was 508 not as emphatic as with the case of conditional probabilities. The difference in the failure probability 509 became weaker as the return period increased, which is consistent with the characteristic of 510 asymptotically independent data (Ledford and Tawn, 1996;Wadsworth and Tawn, 2012). A 511 relationship was demonstrated (Fig. 10) to show how the design of the overall system to a given failure 512 probability requires the design of each individual river crossing to a rarer extremal level than when each 513 crossing is considered in isolation. For the case study example, it would be necessary to design each of 514 the five bridges to a 0.25% AEP event in order to obtain a system failure probability of 1%. 515 There is a need to reimagine the role of intensity-duration-frequency relationships. Conventionally they 516 have been developed as maps of the marginal rainfall in a point-wise manner for all locations and for a 517 range of frequencies and durations. The increasing sophistication of mathematical models for extremes, 518 computational power and interactive graphics abilities of online mapping platforms means that analysis 519 of hydrological extremes could significantly expand in scope. With an underlying model of spatial and 520 duration dependence between the extremes, it is not difficult to conceive of digital maps that 521 dynamically transform from the marginal representation of extremes to the corresponding 522 representation conditional extremes after any number of conditions are applied. This transformation is 523 exemplified by the differences between left and right panels in Fig. 7 and Fig. 8. Enhanced IDF maps 524 would enable a very different paradigm of design flood risk estimation, breaking away from analysing 525 individual system elements in isolation and instead emphasizing the behaviour of entire system. 526

Appendix A. Calculation of empirical tail dependence coefficient 527
To illustrate how Eq. (2) in the manuscript is calculated, consider a set of = 10 observed values at 528 the two locations: 1 and 2 (see Table A1). First, 1 and 2 are converted to empirical cumulative 529 probability estimates via the Weibull plotting position formula = ( + 1) ⁄ where is ranked index 530 of a data point giving 1 and 2 (see Table A1). 531 Table A1. Observed data 1 and 2 and corresponding empirical cumulative probabilities 1 and 2 . where 1 is the return period (in number of observations for 36 hr rainfall) corresponding to the return 559 level 1 . It is also noted that in this paper 1 and 2 were taken as threshold exceedances, so the return 560 period 1 should be in the number of observations, which is equivalent to a 1 243 ⁄ −year return period 561 because there are 243 observations for 36 hr rainfall in a year. 562 The probability that there is at least one location that has an extreme event exceeding a given threshold 563 can be calculated based on the addition rule for the union of probabilities, as: 564 where is the number of locations. 567 For the case of dependent variables, the joint probability for only two locations { 1 > 1 , 2 > 2 } 568 can be easily obtained from the bivariate CDF for inverted max-stable process in Eq. (B.2). However, 569 for the case of multiple locations (five different locations for this paper), it is difficult to derive the 570 formula for this probability because there are dependences between extreme events at all locations. So 571 this probability is empirically calculated from a large number of simulations of the dependent model 572 (see the description of the simulation procedure for an inverted max-stable process in Section 4.3). 573 For the case that all of events are independent, the joint probability for independent variables is broken 574 down as the product of the marginals, and the conditional probability is equivalent to the marginal 575 probability. When applying Eq. (B.5) for independent variables, the joint probability is therefore 576 calculated by ( 1 > 1 , … , > ) = ( 1 > 1 ) … ( > ). 577

Acknowledgments 578
The lead author was supported by the Australia Awards Scholarships (AAS) from Australia 579 Government. A/Prof Westra was supported by Australian Research Council Discovery grant 580 DP150100411. We thank Mark Babister and Isabelle Testoni of WMA Water for providing the 581 hydrologic models for the case study; and Leticia Mooney for her editorial help in improving this 582 manuscript. The rainfall data used in this study were provided by the Australian Bureau of Meteorology, 583 and can be obtained from the corresponding author. 584