Characterization of uncertainty in maximum tidal elevation near Bangladesh coastline due to uncertain sea level rise

The Ganges‐Brahmaputra‐Meghna delta is vulnerable to sea level rise from global warming. Based on sea level rise predictions for the year 2100 given in the 6th Assessment Report by the Intergovernmental Panel on Climate Change (IPCC), we examine the effect of uncertainty in sea level rise on maximum tidal elevation statistics at several locations along the Bangladesh coastline using a discretized derived distribution approach. For five IPCC scenarios, the standard deviation in maximum tidal elevation including sea level rise is predicted to increase by between 3% and 61% at three different locations for a 41% increase in the standard deviation of mean sea level rise. By excluding the linear effect of sea level rise, the increase in the standard deviation of maximum tidal elevation is found to vary spatially from 2% to 68% with a 41% increase in the standard deviation of sea level rise.


| INTRODUCTION
Bangladesh is one of the countries most threatened by climate change impact.It is conjectured that oceanic water will penetrate the river system of Bangladesh as the mean sea level rises, and the resulting salt intrusion will adversely affect freshwater resources and food production (Kusche et al., 2016).The rising sea level is also expected to affect the flow-sediment regime, which in turn will alter flood inundation patterns and local sediment balances (Akter et al., 2016).The risen sea level will increase drainage blocking and cause more frequent storm surge inundation (Karim & Mimura, 2008).Sea level rise could even threaten the sustainability of mangrove forests, coral reefs, and salt marshes (Lovelock et al., 2015).It has been suggested that the Sundarbans mangrove forest may be unable to provide a suitable habitat for Bengal tigers after 2070 (Mukul et al., 2019).More than 2 million people residing in the southern half of Bangladesh are vulnerable to displacement due to sea level rise (Davies et al., 2018).The risks of flooding, salinization, and erosion are expected to increase by the year 2100 in low-lying countries if proper measures are not adopted (Oppenheimer et al., 2019).The Bangladesh coast is subsiding, and taken together with sea level rise, the magnitude and frequency of flood events are projected to increase (Becker et al., 2020;Ranasinghe et al., 2021).Consequently, the whole of South Asia is under threat of severe shoreline retreat (Ranasinghe et al., 2021).
Without protective measures, flood inundation risk is likely to increase for densely populated coastal cities in South Asia under the projected sea level rise by 2100, and the impacts of frequent flood events will be severely exacerbated by sea level rise (Yin et al., 2020).Low-lying coastal regions, subject to frequent tropical cyclones, are already under threat of flooding even under lowprobability flood events (Yin et al., 2021).Recent research advances in coastal flood risk assessment and prediction include the application of data analytics such as machine learning to estimate actual risk (Park & Lee, 2020;Pollard et al., 2018), coastal hydromorphodynamic process-based models to predict flood inundation (Roelvink & Reniers, 2011), dynamic reduced complexity models (Purvis et al., 2008;Ramirez et al., 2016;Vousdoukas et al., 2016), and composite vulnerability indices to indicate relative risk (Pantusa et al., 2018;Sahana & Sajjad, 2019).Ongoing work is addressing future coastal risk in the context of climate change by considering sea level rise, extreme precipitation, and typhoon events using probabilistic approaches such as multivariate elliptical copulas (Wahl et al., 2016) and Monte Carlo simulations (Vousdoukas et al., 2018), and integrated climatologicalhydrodynamic methods (Yin et al., 2020).A recent study by Xu et al. (2022) used an index-based framework to assess the flood risk posed by future sea level rise scenarios in the coastal zone of China.Taken overall, the present research trend is toward the inclusion of climate change impacts in the spatial assessment of coastal flood risk, utilizing developments in data analytics and statistical analysis.The present paper complements this trend by proposing a fast method for estimating the propagation of uncertainty in the context of coastal flood risk assessment.
The coastal zone of Bangladesh is densely populated.Its coastline is 710 km long, with coastal and island communities comprising about 38.5 million people, nearly one-third of the total population of Bangladesh (Bangladesh Planning Commission, 2018).The coastal population of Bangladesh is very vulnerable to natural disasters.Seven major risks have been reported for the coastal zone in a baseline study conducted by the Bangladesh Planning Commission for the Delta Plan 2100-coastal floods, cyclone and storm surges, riverbank erosion and vulnerability of islands, sea level rise, salinity intrusion, waterlogging, and coastal erosion (Bangladesh Planning Commission, 2018).Furthermore, land scarcity exposes the coastal community to a much higher level of risk, and so policymakers are striving to develop an efficient coastal zone management plan.Formulation of the Bangladesh Delta Plan 2100 commenced in 2014 to establish a long-term holistic and integrated strategy for the delta.The Bangladesh Government's policy is to integrate climate change adaptation with a knowledge-based approach in order to make efficient use of the limited natural resources in Bangladesh.Our research finds that there is considerable uncertainty in maximum tidal elevations due to uncertainty in mean sea level rise.This is particularly important for flood risk assessment and could inform flood mitigation planning in Bangladesh.
Over the recent past, the global mean sea level (GMSL) rise has accelerated for anthropogenic reasons (Becker et al., 2020;Oppenheimer et al., 2019).Analysis of data from the northern Indian Ocean obtained over a period of more than 40 years up to 2004 has shown the local rate of sea level rise off the Indian subcontinent during that period was 1.06-1.75mm/year (Brammer, 2014).In its 5th Assessment Report (AR5), the IPCC examined glaciers, ice sheet surface mass balance, ice sheet dynamics, groundwater and reservoirs, gravity and solid Earth effects, thermal expansion, density, and circulation changes, waves and storm surges to forecast sea level change scenarios (Church et al., 2013).The IPCC considered four Representative Concentration Pathway (RCP) scenarios, namely RCP2.6,RCP4.5, RCP6.0, and RCP8.5, based on combinations of contributory factors that had different value ranges.The IPCC predicted 0.44 m, 0.53 m, 0.56 m, and 0.73 m mean rises in global sea level and estimated likely 5-95 percentile ranges for these scenarios (Church et al., 2013).
In the IPCC Special Report on the Ocean and Cryosphere in a Changing Climate (SROCC), the Antarctic contribution to global mean sea level rise was updated due to the availability of new studies since the AR5 report (Oppenheimer et al., 2019).The RCP scenarios were rederived with updated mean values and likely ranges.The IPCC's SROCC report predicted sea level rises of 0.43 m (0.29-0.59 m), 0.55 m (0.39-0.72 m) and 0.84 m (0.61-1.10 m) for the RCP2.6,RCP 4.5 and RCP8.5 scenarios.The Antarctic contribution produced the largest uncertainty in the RCP8.5 scenario.The components underpinning probabilistic sea level rise projections led to different uncertainty ranges, because of variations in methodology, interpretation of findings and definition of uncertainty range among the studies on which SROCC was based.Although the uncertainties projected in the SROCC report are slightly higher than in AR5, the results are generally consistent (Oppenheimer et al., 2019).
Both the AR5 and SROCC reports utilized a baseline period of 1986-2005, whereas the latest 6th Assessment Report (AR6) by IPCC considered an updated baseline period of 1995-2014(Fox-Kemper et al., 2021).The AR6 report analyzed five carbon emission scenarios, called Shared Socioeconomic Pathways (SSPs), to predict sea level rise due to anthropogenic drivers of climate change.The SSP5-8.5, SSP3-7.0, and SSP2-4.5 scenarios represent very high, high, moderate and current levels until midcentury of greenhouse gas (GHG) emissions.The SSP1-1.9 and SSP1-2.6 scenarios represent very low and low GHG emissions which decline to net zero around 2050 followed by net negative emissions (Fox-Kemper et al., 2021).Table 1 lists the median values and 5-95 percentile ranges for these five scenarios.
In this paper, we consider all five SSP scenarios in the IPCC 6th Assessment Report to analyze how uncertainty in sea level rise influences uncertainty in maximum tidal elevation at Tiger Point, Char Purulia, and Sandwip, in the Meghna estuary, Bangladesh.It should be noted that none of the IPCC reports considered land subsidence in projections of GMSL because of the high dependence on local conditions (Church et al., 2013;Fox-Kemper et al., 2021;Oppenheimer et al., 2019).
Under the aegis of the Bangladesh Water Development Board (https://www.bwdb.gov.bd/project),extensive embankment construction has been undertaken along the Bangladesh coastline (including the three locations of interest).These embankments have been designed to protect people and properties in nearby cities and villages from flood inundation and land erosion.Given an accurate estimate of the uncertainty in maximum tidal elevation arising from uncertainty in mean sea level rise, then the resulting effect on risk can be assessed properly, and safety factors modified if necessary during the embankment design phase.Furthermore, flood mitigation planning measures and warning system operations can be executed more effectively by taking such uncertainty into consideration.
Bangladesh is extremely poor in terms of field data, and very often it becomes difficult to establish a validated hydrodynamic model.Uncertainty analysis, therefore, offers a suitable approach for parameter studies in this region.Noting that the IPCC prediction itself contains uncertainty in mean sea level rise, the method presented herein is very efficient at estimating the propagation of uncertainty from mean sea level rise to maximum tidal elevation.Sea level rise is likely to promote many natural and societal disasters, such as salinity intrusion, storm surge inundation, land loss, displacement and migration of people, reduced food and water security, lower biodiversity, and ecological damage.The research approach presented herein is particularly appropriate to the assessment of multiple coastal risks.
For about 50 years, an analytic version of the derived distribution approach has been applied to problems involving parameter uncertainty, where the functional dependence of the output parameter on the input parameter is known.Ang and Tang (1975) demonstrated that the approach directly mapped the probability density function of an uncertain input parameter onto that of an output parameter.By relating uncertainty to the variation of the input parameter about its mean value, it is then possible to estimate the propagated uncertainty from the variation of the output parameter.The method extends to all higher moment statistics, such as skewness, and kurtosis, and so provides a very useful way of examining the overall effect of parameter uncertainty on the system.The analytical version of the derived distribution method has been widely applied by hydrologists and water resources engineers, for example, for flood frequency estimation (Loukas, 2002), flood prediction in a poorly gauged basin in Italy (Brocchiola & Rosso, 2009), stormwater quality modeling (Chen & Adams, 2007), and variability in annual precipitation (Meier et al., 2016).A discrete approach was adopted by Kreitmair et al. (2019) to estimate uncertainty propagation in tidal stream power assessment.Sarwar and Borthwick (2023) adopted the discrete approach to estimate uncertainty in the deposition rate of suspended cohesive sediment.To the authors' knowledge, our paper is the first to report on the application of the numerical analogue of the derived distribution approach to maximum tidal elevations in the context of an actual large-scale bay subject to sea level rise.A basic hydrodynamic model has been established for this study using the freely available open access modeling tool Delft3D.With further development of this model, the same method can be applied to risk assessments related to flood inundation, sea level rise, land erosion and accretion, salinity intrusion, cyclonic surges, and ocean wave propagation, etc., contributing in the future to the planning and management of the coastal zone of Bangladesh.
T A B L E 1 Median values and 5-95 percentile ranges for global mean sea level rise scenarios by IPCC 6th Assessment Report (Fox-Kemper et al., 2021), and the standard deviation for each scenario obtained from the 90% range.

Scenarios
Median value (m) 5-95 percentile range (m)  It should be noted that a high-accuracy map for global terrain elevation called Multi-Error-Removed Improved-Terrain DEM (MERIT DEM) has recently become available (http://hydro.iis.u-tokyo.ac.jp/ $yamadai/MERIT_DEM/). MERIT DEM separates out certain major error components in the SRTM DEM and AW3D DEM including speckle noise, stripe noise, absolute bias, and tree height bias (Yamazaki et al., 2017).Speckle (or multiplicative) noise refers to the random error generated through variation in surface reflectance over a few pixels.Stripe noise is a regular, 500 m to 1 km, variation in elevation caused by residual mast motion error for SRTM data and artificial tilt in elevation for AW3D DEM (Yamazaki et al., 2017).Absolute bias is a shift in average elevation over large areas, whose linear dimension usually exceeds 20 km.Tree height bias arises from instrumentation errors in measuring ground elevation beneath the tree canopy.However, such errors are predominantly terrestrial and so have only a minimal effect on the hydrodynamic model used herein because the model mostly comprises sea and river areas where bathymetric field data were utilized.Land elevation data were required for a few islands near the coast to facilitate wetting and drying in the hydrodynamic simulation.Apart from the coastline, most of the land areas of these islands remained dry throughout all the simulations, and so the land elevation data did not influence the coastal flow patterns considered herein.Moreover, Bhola, the largest island, has an area of 1441 km 2 , length 130 km, width 25 km, and highest elevation 1.8 m above mean sea level.Other islands are even smaller.Thus, the islands do not cover a significantly large area compared to the whole domain which occupies several hundred km 2 .
The depth-averaged version of Delft3D-FLOW software (Deltares, 2600 MH Delft, the Netherlands, https:// oss.deltares.nl/web/delft3d)was used to simulate the flow hydrodynamics in the Meghna estuary and hence to predict the maximum tidal elevation (resulting from river flow and tide).The cell dimension of the curvilinear grid varied from 100 m in the estuary to 3400 m in the bay.Smaller cell size was provided near the location of interest to reproduce the bathymetry and channel alignment correctly, and hence minimize computational error.Larger cell size was provided in the bay far from the estuary, where the bathymetry is less complex.The grid followed the riverbank and coastline as closely as possible, but the fractal geometry along the coast comprising multiple tributaries was obviously simplified.The computational grid covered the entire bay, including islands and tidal flats to facilitate unconstrained wetting and drying.
The Delft3D model contained two upstream open boundaries carrying river discharge and one downstream open boundary with tidal constituents.For simplification purposes, constant river discharge values were prescribed at both upstream boundaries.Flow inputs for the Padma river (located beyond the confluence between the Ganges and Brahmaputra) and the Meghna river were set to 120,000 m 3 /s and 20,000 m 3 /s respectively (Figure 3).These river discharges represented the annual peak flood in the year 2007.The bathymetry used in the model corresponded to the year 2007-2008 (Figure 2), and so river discharges for the same time period were input to the model.Both sets of river data were obtained with permission from the MorphoFlood project by IHE Delft, the Netherlands.At the downstream boundary, the flow input comprised M 2 and S 2 tidal components, which are dominant in the Bay of Bengal.Table 2 lists the tidal components at the east and west ends of the offshore boundary.It also lists the representative amplitudes and phases, obtained from the TPXO 8.0 database using the DelftDashboard tool (https://publicwiki.deltares.nl/display/DDB/Delft+Dashboard) from Deltares, the Netherlands.Further details of the model are given by Sarwar (2021).
Bed friction was incorporated using the Manning's roughness parameter, n.In absence of field data concerning bed roughness, we adopted the approach proposed by F I G U R E 2 Model bathymetry.River bed levels (data obtained from MorphoFlood project, IHE Delft, the Netherlands) and island land elevations (downloaded from https://earthexplorer.usgs.gov/) in the Delft3D domain for the Meghna estuary and Bay of Bengal.Soulsby (1997) and Whitehouse et al. (2000) to determine the spatial distribution of Manning's n as follows.Using values of depth-averaged velocity u and water depth h obtained from the Delft3D model, the log-law was invoked to determine the bed friction velocity iteratively from where κ is the von K arm an constant ($0.4), h is water depth, and z 0 is the bed roughness length given by in which k s ¼ 2:5d 50 is the Nikuradse roughness, d 50 is the median particle diameter, and ν is the kinematic viscosity of water.Assuming that the dominant contribution was from skin friction, the bed roughness drag coefficient C D was then estimated from The Manning bed roughness coefficient was finally determined (assuming the ocean was analogous to a very wide channel from the perspective of hydraulic radius) from: where g is acceleration due to gravity. Figure 3   lies roughly in the range 0.01-0.03s/m 1/3 for the Meghna estuary under conditions of zero sea level rise.The Delft3D model was then run for different values of sea level rise varying from 0.1 m to 1.2 m.After a spin-up of 5 days, the model was run for a further 2 spring-neap cycles, and predicted maximum tidal elevations were recorded at the three locations of interest.Functional relationships were then established between sea level rise and maximum tidal elevation at the three locations of interest (marked on Figure 1).Truncated normal distributions of sea level rise were fitted to 2100 data for IPCC global scenarios SSP1-1.9,SSP1-2.6,SSP2-4.5, SSP3-7.0, and SSP5-8.5.Convergence tests were carried out to decide the number of bins required for accurate discretization of the probability distribution of sea level rise (see Section 1 of Supporting Information).The probability distribution for sea level rise was obtained using the mean value and the 90% confidence limit listed by IPCC for each scenario (Fox-Kemper et al., 2021).
Then the numerical derived distribution approach (originally proposed for uncertain tidal stream power assessment by Kreitmair et al., 2019) was adopted to derive the probability distribution of maximum sea level rise.In the numerical derived distribution method, probability distributions of independent and dependent variables are considered in discrete, rather than continuous, form.The derived distribution approach is summarized as follows.First, an analytical expression is specified for the probability density function of the input parameter, with the variance representing uncertainty.Then, the cumulative distribution function of the input parameter is determined by integration, and the cumulative distribution function of the output parameter is then evaluated through conservation of probability noting the functional relationship between the input and output.Being direct, the derived distribution method offers fast assessment of uncertainty propagation provided the functional relationship between input and output is monotonic.The derived distribution approach is based on the concept of conservation of probability (Benjamin & Cornell, 1970).A full description is given by Sarwar (2021).In short, after changing the variable of integration, the resulting cumulative distribution function of the output variable is differentiated to provide its probability density function.
Let the Manning roughness coefficient n be a random variable represented by a Gaussian distribution with mean μ n and standard deviation σ n .Then, the probability that the value of n lies between n A and n B , where n A < n B , may be written as For finely spaced discrete values of n, the likelihood of a particular value n i is given by Pr From the definition of the derived probability distribution, Equation ( 6) relates to the probability of maximum tidal elevation Z, given that Z ¼ f n ð Þ.Hence, the expected value and variance of Z may be determined as and with (higher) m th order statistical moments given by Here m ¼ 3 denotes skewness (i.e., asymmetry) and m ¼ 4 denotes kurtosis (i.e., tailed-ness or peakedness).
The detailed procedure is as follows: v. Divide the PDF and CDF into a prescribed number i max of bins, ensuring that the probabilities are always positive-valued.vi.For each bin from i ¼ 1, 2, …,i max determine the middle value x i .Use the relationship between y and x to determine corresponding y i .vii.Determine CDF values c x ð Þ iÀ1=2 and c x ð Þ iþ1=2 at either end of the bin, and hence determine the probability Pr and plot p y ð Þ i against y i for i ¼ 1, 2,…, i max .xiv.Hence determine CDF of y by numerical integration, c y ð Þ i ¼ P i max i¼1 p y ð Þ i Δy i , and plot c y ð Þ i against y i for i ¼ 1, 2…,i max .Whereas the bin widths Δx i may each have the same value, the bin widths Δy i will be different to each other if there is a nonlinear relationship between y and x.

| UNCERTAINTY ANALYSIS
Figure 5 shows the functional relationship between sea level rise and maximum tidal elevation (including the rise in sea level from datum), probability distribution (PDF) of maximum tidal elevation and cumulative probability distribution (CDF) of maximum tidal elevation derived from the PDF of sea level rise using the numerical derived distribution approach at Sandwip, Tiger Point, and Char Purulia along the coastline of Bangladesh.Table 3 lists the expected values and higher order moments for maximum tidal elevation obtained for the five scenarios of sea level rise.As the mean sea level rise increases from 0.38 m to 0.77 m (103% increase), the expected maximum tidal elevation at Sandwip increases by about 15%.The standard deviation of maximum tidal elevation at Sandwip increases by 3% for a 41% increase in standard deviation in sea level rise.The nondimensional skewness remains about zero, indicating the PDFs to be truncated normal distributions for maximum tidal elevation.The magnitude of nondimensional kurtosis remains very close to 3 in each case, in keeping with a truncated normal distribution.As the mean value of sea level rise increases, the peak in the PDF of maximum tidal elevation at Sandwip progressively moves to higher values of maximum tidal elevation, but the magnitude of the peak decreases.Similarly, the standard deviation of maximum tidal elevation increases as the mean value of sea level rise increases.
At Tiger Point, both the river discharge and tide are dominant forcing parameters.Increasing sea level rise causes a corresponding increase in tidal elevation.As at Sandwip, the peak in the PDF of maximum tidal elevation is reached at progressively higher values of maximum tidal elevation as the sea level rise increases.Again, the magnitude of the peak of the PDF of maximum tidal elevation decreases with increasing sea level rise.The expected value of maximum tidal elevation increases by about 20% when the mean sea level rise alters from 0.38 m to 0.77 m (103% increase), and the standard deviation in maximum tidal elevation increases by about 38% as the standard deviation of sea level rise increases by 41% (Table 3).The nondimensional skewness is almost zero and the nondimensional kurtosis is about 3, indicating that the five distributions of maximum tidal elevation can be effectively represented by truncated normal distributions.
The third location at which the propagation of uncertainty from sea level rise to maximum tidal elevation is considered is at Char Purulia.It can be seen in Table 3 that the expected value of maximum tidal elevation at Char Purulia increases by about 8% with a 103% increase in the mean value of sea level rise.The standard deviation in maximum tidal elevation increases by about 61% for a 41% increase in the standard deviation in sea level rise.The magnitude of nondimensional skewness for all the PDFs is invariably very small, indicating that the probability density function of maximum tidal elevation is close to a normal distribution.The nondimensional kurtosis has values near 3 which again indicate the PDF is very well represented by a normal distribution.
At all three locations, the nondimensional kurtosis remained at a value of about 3, indicating that the probability distributions are truncated normal distributions.There is no specific increasing or decreasing trend in the nondimensional kurtosis (Table 3), but instead there is a decrease in the trend in peakedness of the probability distribution (Figure 5).By definition, nondimensional kurtosis expresses how far the extreme values of the distribution are dispersed from the mean (Equation 9, where m ¼ 4).The shape of the probability distribution of maximum tidal elevation depends on the probability distribution of sea level rise, and the relationship between maximum tidal elevation and sea level rise.The PDFs of sea level rise have more dispersed extreme values when the sea level rise increases.Since maximum tidal elevation is proportional to sea level rise (Figure 5, row 1), the probability distributions of maximum tidal elevation also contain more dispersed extreme values with a lower peak.
The foregoing results quantify how uncertainty differs from place to place in the vicinity of the Bangladesh coast.It is expected that the maximum tidal elevation will increase with the rise of sea level; however, for the same global mean sea level rise (same uncertainty and same PDF), it is found that the values of uncertainty in maximum tidal elevations at different locations differ from each other.The geometry of the bottom surface plays a vital role causing tidal asymmetry.Char Purulia experiences additional effects from river discharge and channel curvature on its flow regime.So, the relationships between sea level rise and maximum tidal elevation vary, and that causes the PDFs and uncertainty to change across the locations considered.
Next, the rise in sea level is subtracted from the model output of maximum tidal elevation (denoted by s from here on); this enables assessment as to how uncertainty in sea level rise affects uncertainty in maximum tidal elevation when the linear effect of sea level rise is removed.Figure 6 shows the cubic spline relationship between s at Sandwip, Char Purlia, and Tiger Point, as well as the derived probability distribution of s.In this case, F I G U R E 5 Relationships between maximum water level and sea level rise at (a) Sandwip, (b) Tiger Point, and (c) Char Purulia are depicted in the top row of the figure.In the middle row, (d), (e), and (f) show the derived PDFs of maximum tidal elevation for SSP1-1.9,SSP1-2.6,SSP2-4.5, SSP3-7.0, and SSP5-8.5 from IPCC 6th Assessment Report.The bottom row shows the derived CDFs of maximum tidal elevation at (g) Sandwip, (h) Tiger Point and (i) Char Purulia.
T A B L E 3 Expected value and other statistical moments of maximum water level, maximum water level without the linear effect of sea level rise and excess maximum water level at Sandwip, Char Purulia, and Tiger Point for five different IPCC sea level rise scenarios.the functional relationships are no longer linear, although the functions are either monotonically increasing or decreasing.Table 3 summarizes the expected values and statistical moments of s at the three selected locations.As the mean of sea level rise increases 103% at Sandwip, the expected s increases by only 2%, but the standard deviation increases by 68%.The nondimensional skewness confirms that the distribution related to SSP1-19 sea level rise is left-skewed (nondimensional skewness is negative).The probability density distribution is almost a normal distribution for SSP1-1.26(near-zero nondimensional skewness).The PDF becomes right-skewed for the SSP2-4.5 scenario (positive nondimensional skewness).Then the PDFs become left-skewed for SSP3.-7.0 and SSP5-8.5 scenarios (negative nondimensional skewness).All the respective CDFs reach 1 despite their various shapes.The functional relationship between s and sea level rise is monotonic, but nonlinear where the slope changes multiple times.Thus the derived probability distributions of s exhibit starkly different shapes.The nondimensional kurtosis for SSP1-2.6 is well above 3, which indicates that the shape of distribution is narrower than a standard normal distribution, whereas for SSP3-7.0 the distribution is lower than 3, indicating that the probability distribution is flatter than a normal distribution.For the other three distributions, the nondimensional kurtosis is slightly above 3, suggesting that these are slightly narrower than a normal distribution.
At Tiger Point, as the sea level rise increases, there is a monotonic decreasing trend in s and the function is not linear.From Table 3, all the PDFs of s have nondimensional kurtosis of about 3, which indicates that the steepness of these PDFs is very close to that of the standard normal distribution.The nondimensional skewness values for SSP1-1.9 and SSP1-2.6 are positive, so the distributions are right-skewed.The other three PDFs have negative nondimensional skewness, and so are leftskewed.The mean of s hardly varies (with 1% reduction) indicating it is almost insensitive to sea level rise.However, the standard deviation increases by about 54%, indicating that there is much higher propagation of uncertainty.
Derived probability distributions for s at Char Purulia are shown in the third column of panels in Figure 6, where it can be seen that the tallest peak in the PDF occurs for SSP5-8.5.The PDFs and CDFs are similar across the scenarios in that they all exhibit near-normal distributions.shows that the three probability distributions are slightly skewed to the right of the mean of the distribution (positive values), and the nondimensional kurtosis (all values near 3) suggests that the derived PDFs are very close in form to a standard normal distribution.When the mean of the sea level rise increases by 103%, the mean of s decreases by about 3% only.An increase of 41% in standard deviation in sea level rise leads to an almost invariable standard deviation (2% variation) in s.
The functional relationship between sea level rise and s indicates that the effect of global mean sea level rise on the regional sea level rise near the coast of Bangladesh is not a straightforward phenomenon.When a tide enters shallow water, it is deformed.The regional sea level rise varies spatially, and may not be equal to the amount of global sea level rise.Thus, when the global sea level rise is deducted from the local maximum tidal elevation, the remaining water level can exhibit either increasing or decreasing trends.The expected value of s does not significantly vary in space, unlike the standard deviation which varies across the three locations (from 2% variation at Char Purulia to 68% at Sandwip).The primary reason for this large span of standard deviation concerns the local bed topography, and the resulting hydrodynamics.More specifically, the fractal geometry of the land, the narrow gorges containing river channels, shallow mudflats at the mouth of the estuary, amplification of the flood tide as it crosses the continental shelf when entering the estuary and passing over mudflats and through islands, all cause a striking variation in flow pattern at different locations.In future work, it is hoped that a clearer picture will emerge by considering more locations along the coast and creating locally fine maps of key statistics such as s.
Lastly, in order to examine the effect of mean sea level rise on excess maximum tidal elevation, further analysis is carried out concerning the statistical relationship between excess maximum tidal elevation and sea level rise.Here, the excess maximum tidal elevation is defined as the maximum tidal elevation minus both the mean sea level rise and the reference maximum tidal elevation obtained for no sea level rise.Figure 7 shows the cubic spline fit to data on excess maximum tidal elevation at Sandwip, Tiger Point, and Char Purulia plotted against sea level rise.The relationship at Sandwip is nonlinear, even though there is a progressive increase in excess maximum tidal elevation from about 0.02 m to 0.12 m for a sea level rise from 0.1 to 1.2 m.All the PDFs and functional relationships shown in Figure 7 show the same trend in PDFs and functional relationship as in Figure 6 F I G U R E 7 Relationships between excess tidal elevation (i.e., maximum tidal elevation À sea level rise À reference maximum tidal elevation) and sea level rise at (a) Sandwip, (b) Tiger Point, and (c) Char Purulia depicted in the top row of the figure.In the middle row, (d), (e), and (f) show the PDFs of excess tidal elevation for SSP1-1.9,SSP1-2.6,SSP2-4.5, SSP3-7.0, and SSP5-8.5 from IPCC 6th Assessment Assessment Report.The bottom row shows the derived CDFs of excess tidal elevation at (g) Sandwip, (h) Tiger Point, and (i) Char Purulia.
(i.e., cases of maximum tidal elevation minus sea level rise).Table 3 lists the expected value of excess maximum tidal elevation and other statistical moments.When the mean of the sea level rise increases by 103%, the expected value of excess maximum tidal elevation at Sandwip increases by 85%, at Tiger Point decreases by 92%, and at Char Purulia decreases by 76%.With an increase in standard deviation in sea level rise of 41%, the standard deviation in excess maximum tidal elevation at Sandwip increases by 68%, at Tiger Point decreases by 54%, and at Char Purulia decreases by 2%.
A more detailed description of the uncertainty analysis is provided by Sarwar (2021), and its extension to sediment transport and bed morphological change by Sarwar and Borthwick (2023).

| CONCLUSIONS
This article has investigated the effect of sea level rise on maximum tidal elevation at three locations along the Bangladesh coastline, based on fits of IPCC 6th Assessment scenario forecasts (Fox-Kemper et al., 2021) of sea level rise by 2100 to a truncated normal probability density distribution.For the five different IPCC scenarios considered, the standard deviation (i.e., uncertainty) in maximum tidal elevation including sea level rise increased by between 3% at Sandwip and 61% at Char Purulia for a 41% increase in standard deviation of mean sea level rise.By excluding the linear effect of sea level rise, the increase in standard deviation in maximum tidal elevation was found to vary remarkably from 2% at Char Purulia to 68% at Sandwip for the 41% increase in standard deviation of sea level rise.For the cases considered, a novel numerical analogy of the derived distribution method proved effective at estimating output uncertainty in maximum tidal elevation using results from very few model simulations.
Mean and extreme sea level rise promote coastal hazards that include permanent submergence of land, more frequent intense flooding, severe coastal erosion, drainage congestion, salinization of soil, groundwater and surface water, and ecosystem degradation (Oppenheimer et al., 2019).Bangladesh is a low-lying country, and its coastal communities are under substantial risk of such coastal hazards.The IPCC SROCC report (Oppenheimer et al., 2019) suggested that century-scale adaptation and protection strategy is needed for low-lying countries, given that their communities may experience the upper end of the high to very-high risk scenarios.Despite the large uncertainty, adaptation planning and implementation for 2100 can be beneficial.Decision-analysis methods, particularly for situations with large uncertainty are available nowadays and can be used to assist decisions covering timescales from decades to centuries.Thus, the present study of uncertainty in maximum tidal elevation is of high importance for coastal regions prone to inundation, and specifically relevant to coastal management strategies in Bangladesh.
Although only three locations were chosen for quantification of output uncertainty arising from a single input parameter (i.e., sea level rise), the present analysis is straightforward to extend to the estimation of output uncertainty for multiple input parameters at multiple locations along the coastline of Bangladesh.Such an approach, enhanced by bias correction, would provide a holistic picture of uncertainty propagation throughout the whole coastal region of Bangladesh, contributing to the planning and management of the coastal zone of Bangladesh for many years to come.

Figure 1
Figure1shows the study area, computational domain, and three locations of interest.The input bathymetry (Figure2) comprised measured bed elevations of rivers (data obtained from MorphoFlood project, IHE Delft, the Netherlands) and land elevations of islands (downloaded from the Earthexplorer website of the United States Geological Survey (USGS); https://earthexplorer.usgs.gov/).The computational model followed the local projected coordinate system Bangladesh Transverse Mercator (BTM).This is because Bangladesh falls under two UTM zones: 45 N and 46 N, and so the local coordinate system BTM is used in Bangladesh to avoid the complication of having two UTM zones.Land elevation data (SRTM data) were downloaded from Earthexplorer webpage (https:// earthexplorer.usgs.gov/)belonging to the United States Geological Survey (USGS).The resulting data were available in WGS 1984 geographic coordinate system in longitude and latitude form and were converted to BTM before incorporation in the model.All the maps in this paper were produced in geographic coordinate system WGS 1984 (Figures 1, 2 and 3) for universal communication.

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I G U R E 1 Map showing land elevation of Bangladesh (data downloaded from https://earthexplorer.usgs.gov/),domain of the Delft3D model used for this paper shown in red, and three locations of study-Sandwip, Tiger Point, and Char Purulia.The geographic boundary of Bangladesh is downloaded from www.geodash.gov.bd.
displays the resulting contour map of Manning's n whose value F I G U R E 3 Spatial map of Manning's n calculated from Soulsby (1997) and Whitehouse et al. (2000) for the Meghna estuary and the Bay of Bengal domain (including major river tributaries).The open boundaries are also indicated.T A B L E 2 Tidal constituents at the offshore boundary.

Figure 4
Figure4presents a graphical representation of the probability transfer process.
Probability density transfer from a PDF of x to a PDF of y through a function y ¼ gðxÞ.

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I G U R E 6 Relationships between maximum water level without the linear effect of sea level rise (i.e., "s") and sea level rise at (a) Sandwip, (b) Tiger Point, and (c) Char Purulia depicted in the top row of the figure.In the middle row, (d), (e), and (f) show the PDFs of s for SSP1-1.9,SSP1-2.6,SSP2-4.5, SSP3-7.0, and SSP5-8.5 from IPCC 6th Assessment Report.The bottom row shows the derived CDFs of s at (g) Sandwip, (h) Tiger Point, and (i) Char Purulia.
Δy i ¼ y iþ1=2 À y iÀ1=2 .xiii.Determine PDF for y as p y ð Þ i ¼ Pr i Table 3 lists the expected values and other statistical parameters.The nondimensional skewness