Innovative Vulnerability and Risk Assessment of Urban Areas against Flood Events: Prognosis of Structural Damage with a New Approach Considering Flow Velocity

Abstract

The floods in 2002 and 2013, as well as the recent flood of 2021, caused billions Euros worth of property damage in Germany. The aim of the project Innovative Vulnerability and Risk Assessment of Urban Areas against Flood Events (INNOVARU) involved the development of a practicable flood damage model that enables realistic damage statements for the residential building stock. In addition to the determination of local flood risks, it also takes into account the vulnerability of individual buildings and allows for the prognosis of structural damage. In this paper, we discuss an improved method for the prognosis of structural damage due to flood impact. Detailed correlations between inundation level and flow velocities depending on the vulnerability of the building types, as well as the number of storeys, are considered. Because reliable damage data from events with high flow velocities were not available, an innovative approach was adopted to cover a wide range of flow velocities. The proposed approach combines comprehensive damage data collected after the 2002 flood in Germany with damage data of the 2011 Tohoku earthquake tsunami in Japan. The application of the developed methods enables a reliable reinterpretation of the structural damage caused by the August flood of 2002 in six study areas in the Free State of Saxony.

1. Introduction

All Federal States in Germany are legally obliged to prepare and regularly update flood risk assessments and action programs for their rivers in accordance with the EU Flood Directive (2007/60/EC). However, there is a lack of uniform methods for analysis of the vulnerability of potentially affected buildings and to determine their potential damage for risk and cost-benefit considerations. Currently adopted area-related statistical damage values, particularly in urban areas, significantly underestimate the actual costs of radiation of damage caused by flooding.

The frequent sequence of devastating floods in Germany (2002, 2011, 2013 and 2021) highlights the relevance of studying extreme flood scenarios, considering that the probability of occurrence is not as low as previously assumed and that damage could be catastrophic for the affected areas and overdemanding for the responsible authorities and decision makers. In particular, the 2002 flood, along with flash flood events, such as the 2016 flood in Braunsbach [1,2] and the flood of 2021 in Rhineland-Palatinate and North Rhine-Westphalia [3], have shown that in addition to moisture penetration and water impact, the severity of structural damage identified in buildings can vary. Conventional flood loss models (e.g., overview in [4]) are usually limited to the prognosis of loss, considering the inundation level, and cannot adequately address the structural damage caused by dynamic flood processes.

Previous studies of structural damage to buildings have examined the correlation between flood actions (i.e., flow velocity and inundation level) and the collapse of structures [5,6,7]. Additionally, criteria for the delimitation of partial failure were set in [8]. However, these studies lack a refined methodology according to which various damage patterns of structural damage can be differentiated and transformed into tangible loss statements.

The flood damage model developed by the Earthquake Damage Analysis Center–(EDAC) [9,10] can predict and classify such structural damage in the form of damage grades. The vulnerability functions developed for the prognosis take into account the vulnerability of individual building types, the inundation level and the flow velocity.

The particular characteristics of flash flood events (cf. [1,3]), with their high flow velocities, are currently not sufficiently supported by damage data in this damage model.

Therefore, one objective of the INNOVARU research project funded by the German Federal Ministry of Education and Research (BMBF) was the development of an improved application-ready model for the monetary assessment of expected flood damage to buildings as an essential basis for planning flood risk management measures in the Free State of Saxony, Germany. Application target groups are primarily the Dam Authority of the Free State of Saxony (“Landestalsperrenverwaltung Sachsen”; LTV) and engineering offices, which can use the damage model for cost-benefit analyses in preliminary studies on flood protection measures, as well as in the selection of measures for future flood risk management plans.

The objective of the investigations presented in this paper is to provide an improved model for the prognosis of structural damage caused by flooding at the microscale level (individual buildings) depending on the flow velocity, inundation level and the vulnerability of the building (building type, state of maintenance and number of storeys). This ultimately serves as the basis for conversion into detailed synthetic loss statements.

The examined new damage prognosis tools will be applied to different investigation areas in Saxony (Germany) on a microscale level. The results are compared and validated based on the observed damage grades in the corresponding investigation areas.

Some sections and graphs presented this paper have already been published in [11] to discuss the interim results of the INNOVARU project at the FLOODrisk 2020 conference. The present paper completes these investigations, addressing the necessary coefficients for the application of the proposed damage model and providing more detailed explanations. Based on the reinterpretation of the damage caused by the 2002 flood in Saxony and the associated error analysis, the most suitable calculation variants are identified. The influence of flow velocity and the number of storeys on the scenarios is highlighted.

The accompanying paper [12] will link the subsequently presented methodology with newly derived synthetic damage functions covering the entire range of residential building stock in a more differentiated manner. An overview of efforts to reduce procedures using improved open geodata on a microscale level is provided in [13].

2. Basic Elements of the Procedure

2.1. Damage Data

2.1.1. EDAC Flood Damage Database

The damage caused by the 2002 flood in Saxony exceeded the financial resources of most homeowners in Saxony. The recovery of residential buildings was supported by grants from the Saxonian Relief Bank (SAB). For recovery costs of more than EUR 30,000, damage reports had to be prepared and submitted. The SAB funded more than 22,000 applications for financial support to recovery [14]. Detailed damage reports were available for approximately 8000 buildings covering nearly all districts in Saxony affected by the 2002 flood.

These damage reports were processed and analyzed by EDAC on behalf of the Dam Authority of the Free State of Saxony. Reliable damage functions for residential buildings were derived from the first evaluations, which were required in the context of cost-benefit calculations for planned flood protection measures in the Free State of Saxony, Germany [15].

The relevant building parameters, the observed damage patterns, the impact parameters (inundation level) and the actual recovery costs were documented. Approximately 5000 cases of damage were evaluated, which currently form the core of the EDAC flood damage database. Additional damage data from previous research projects [16,17] are available but were not considered in this study because of possible data overlaps.

For approximately 1200 damage cases, flow velocities were assigned based on hydraulic calculations (see Section 2.3) in four of the selected investigation areas in Saxony [18]. The value for the assigned flow velocities was v_max ≈ 2.5 m/s, which corresponds to moderate water movement typical of river floods. Data on damage cases caused by higher flow velocities (such as those that occur with flash floods) were not available. Therefore, an unconventional but innovative approach had to be implemented in the INNOVARU project. The damage data of the 2011 Tohoku earthquake tsunami were included, assuming that tsunamis represent an extreme form of (cyclic) flooding.

2.1.2. Tsunami Damage Data

In general, inherent differences can be assumed between the impact of a flood and that of a tsunami, including the difference in quality of time-dependent characteristics. Changes in the direction of flow with incoming and outgoing waves or the surge forces of the wave front are more pronounced in the event of a tsunami.

Tsunamis generate large amounts of debris, the impact of which can significantly increase structural damage to buildings. This is usually not the case for river floods with low flow velocities. However, the flood of 2021, in particular, showed that in flash flood events, the impact of debris has a significant effect on structural damage [3]. For higher flow velocities, the application of tsunami damage data seems justified.

The damage data in the EDAC flood damage database also include damage cases in which the impact of debris contributed to structural damage. Thus, the impact of debris is included in the two damage databases, although not specifically extracted. More detailed investigations are to be carried out in the future, including damage data on the 2021 flood [3].

In this paper, comparable forces of hydrostatic and hydrodynamic pressure are assumed, as well as the impact of floating debris and buoyancy, which act on the affected buildings, making it possible to enrich the dataset for higher flow velocities.

Data on building damage sustained during the 2011 Tohoku earthquake tsunami were collected in a comprehensive damage database (n ≈ 252.000) and are available in an aggregated form [19]. So-called “fragility functions” were derived in [20,21] from this damage data depending on the inundation level (h) and building type for damage prognosis due to tsunami impact. Based on the concept of damage grades for earthquakes according to the European Macroseismic Scale (1998-EMS-98) [22], a six-stage damage scale for tsunamis was introduced for these investigations.

The additional damage grade D6 was assigned to buildings that were completely washed away or completely overturned (see [21]), whereas the lowest damage grade assigned is D1, as buildings not affected by the tsunami (i.e., D0) are not included in the evaluations (see also [23]).

The unified damage scales for the main natural hazards—earthquake, flood and wind—in the sense of a multihazard approach in [24] consider the damage classification described in [20,21]. The MLIT database [19] was re-evaluated in [25] to derive a mathematically based vulnerability table for tsunami impact (analogous to EMS-98). The various building types in the database were classified into vulnerability classes and their ranges of scatter.

Using the flow velocities estimated from video recordings in [26,27], Froude numbers were derived for the coastal characteristics (“plain coast” and “ria coast”). This enables an estimation of the flow velocities associated with the water levels in the MLIT database (see [25]).

The classification of vulnerability classes ensures the comparability of the behavior (i.e., the expected damage) of the considered individual building types in the two datasets (see also Section 3.2).

The application of the unified damage scales in [24] contributes to the compatibility and comparability between the damage grades of the individual natural hazards and, therefore, also between the available damage data.

Information on the damaged buildings in the EDAC flood damage database, for which the flow velocities were determined, as well as the tsunami damage data, were used to derive the vulnerability functions described in Section 4.

2.2. Investigation Areas

For the validation of the developed model on a microscale level, the building data from various flood-affected investigation areas in Saxony had to be collected. These data form the basis for the vulnerability assessment (see Section 3.2) and the application of the corresponding vulnerability functions to predict the structural damage in Section 5.

To this end, towns in the Free State of Saxony were taken as case studies, and different flooding parameters (i.e., inundation level and flow velocities) describe the events that occurred. Figure 1 displays the location of the investigated areas, and

Table 1. Overview of the investigation areas (for interim state cf. [11]).

Investigation Area	Buildings Inspected (Affected) 1		Damage Cases: SAB 2 (EDAC) 3	Year(s) of Survey
	Residential	Total
Pirna	1209 (938)	1405 (1067)	1148 (366)	2008
Grimma	773 (690)	1280 (1186)	616 (306)	2009, 2017
Freital	1048 (946)	2096 (1842)	865 (277)	2019
Eilenburg	1041 (1028)	2184 (2149)	961 (551)	2003, 2004
Döbeln	832 (788)	1429 (1348)	681 (276)	2004
Flöha	734 (721)	1872 (1828)	582 (154)	2009

¹ Acc. to flood scenarios (see Section 2.3). ² Reported to Saxonian Relief Bank (SAB) [14]. ³ Included in EDAC flood damage database.

provides an overview with respect to the inspected buildings per study area. Existing buildings in the Saxonian towns of Döbeln, Eilenburg, Grimma and Flöha affected by the 2002 flood were systematically investigated in previous research projects and formed the basis for the validation of the previous EDAC flood damage model with respect to real observed damage [9,10,18,28]. In addition to the relevant building parameters, the existing flood marks were also documented during building inspections. The towns of Pirna, Grimma and Freital were selected as study areas for joint investigations with INNOVARU project partners.

The building stock data of the investigation area of Grimma were updated in 2017 as part of a previous research project and are also available as part of INNOVARU.

Pirna is located at the confluence of the tributary rivers Gottleuba and Seidewitz with the Elbe River. The town experienced severe flooding in recent years, particularly in 2002 and 2013. Therefore, both state and local authorities developed various flood protection concepts that had to be assessed and prioritized. In this context, potential flood damage to residential buildings was previously analyzed in [29,30] for selected scenarios in Pirna.

The affected buildings on the northeastern bank of the Elbe remain unconsidered. Therefore, the number of SAB damage cases for Pirna apparently exceeds the number of residential buildings in Table 1.

The building stock of the city of Freital was documented on a microscale level using a coordinated parameter list, which was implemented in an updated version of the EQUIP building survey tool developed by EDAC (cf. [31,32]). The EQUIP (elaboration, qualification and interpretation) tool supports the documentation of building characteristics for subsequent detailed vulnerability analyses and damage modelling.

In the course of data collection, building plans (e.g., the building outlines of the official real estate cadaster information system, ALKIS^®) were integrated into EQUIP and linked to the internal database of the program (Figure 2). The database fields can be activated in the program by simply selecting of the relevant building plan. Predefined selection fields and the option of entering free text enable exceedingly efficient data input. The various background maps that can be activated (satellite, street map or hybrid view), in combination with the display of the present location (on GPS-capable tablet PCs), simplify orientation in the investigation area. For the identification and classification of flood-prone buildings in the investigation areas, and an established building typology approach (cf. [12,30]) is used.

This approach was implemented in the applied version of the EQUIP tool as a quick selection matrix (Figure 3), providing windows with lists of predefined inspection parameters, which are adjustable to fit the real building properties. Therefore, the tool considerably accelerates data collection. Photographs of the structures were taken for plausibility checks and data supplementation. A further increase in efficiency can be expected as a result of future integration of a building typology for non-residential buildings.

Following the local building surveys, the data records of the individual members of the survey team were merged, GIS-specifically prepared and checked. A plausibility check and supplementation of partially non-visible parameters (especially in the roof areas of the buildings) was performed with high-resolution 3D models of Freital and Pirna, which are available in Google Earth^®.

2.3. Flood Scenarios

The proposed approaches were validated by evaluating the damage prognosis in comparison to the damage that was actually observed after the 2002 flood. Hydraulic simulations of the 2002 flood event in the study areas are required to determine the impact parameters (inundation levels and flow velocities) on the affected buildings.

Due to the influence of flow velocity on damage, detailed hydraulic models are necessary. Velocity can change by a factor of ten within a few meters of buildings that are closely spaced.

Such accurate hydraulic models were provided for the INNOVARU study areas (Freital, Grimma, Pirna) and also for the town of Döbeln. For the investigation areas of Eilenburg and Flöha, hydraulic calculations were available, in which the existing building stock is taken into account with a mean surface roughness.

Table 2. Overview of the 2002 flood scenarios (for interim state cf. [11]).

Investigation Area	2D Model Approach	Grid Size (m × m) 1	Inundation Level hgl (m) 2	Flow Velocity vfl (m/s) 2
Pirna	detailed	variable	0–4.1	0–5.3
Grimma	detailed	5 × 5 (hgl) 1 × 1 (vfl)	0–5.0	0–2.7
Freital	detailed	2 × 2	0–3.5	0–4.5
Eilenburg	mean roughness	25 × 25	0–3.5	0–1.9
Döbeln	detailed	variable	0–4.7	0–2.4
Flöha	mean roughness	5 × 5	0–2.8	0–2.3

¹ Output from hydraulic calculation. ² At building location.

provides an overview of the applied 2D scenarios.

2D hydraulic calculations were compiled for the study areas of Grimma, Flöha, Döbeln and Eilenburg. In this project, the flow velocities for Eilenburg were derived from the calculations initiated and presented by the RIMAX project MEDIS (cf. [17]). The hydraulic models for Grimma and Flöha were provided by the Dam Authority of the Free State of Saxony (Landestalsperrenverwaltung; LTV). The LTV also later submitted a refined hydraulic model for Döbeln [33] for the investigations in [18]. Figure 4a shows the spatial distribution of inundation levels and flow velocities in Grimma.

The INNOVARU project comprises the reinterpretation of the 2002 flood event in the investigation area of Freital (Figure 4b). Two-dimensional (2D) unsteady-flow modeling was performed. Flow hydrographs of the tributaries “Rote Weisseritz” and “Wilde Weisseritz” were analyzed by rainfall–runoff models. Manning’s values were calibrated considering the observed flood line of the 2002 flood. Interestingly the maximum inundation level and maximum flow velocity are not simultaneous as a result of the unsteady flow. Discharge conditions at the maximum inundation level were selected for further analysis of flood damage.

A hydraulic calculation for the river “Elbe” for the 2002 flood was provided by the Dam Authority of Saxony for the investigation area of Pirna. The “HQ100” scenario was used for the “Gottleuba” and “Seidewitz” tributaries, demonstrating a satisfactory match with the observed flood areas of 2002 (Figure 4c).

Hydraulic calculations for the reinterpretation of extreme flood events, such as the 2002 flood, are usually subject to considerably uncertainties, as the hydraulic boundary conditions change due to erosion and rearrangement processes during the event. Therefore, the documented flood marks were used in the investigation areas of Eilenburg, Flöha and Döbeln to derive a simplified but realistic inundation level model based on the digital elevation model (DEM) with a resolution of 2 m × 2 m [28].

3. Flood Damage and Vulnerability of Buildings

3.1. Damage Scale for Flooding

One of the elements of the EDAC flood damage model [9] is a five-stage damage scale, which was developed in [23] based on the observed damage from the 2002 flood.

Research on the unification of the description of building damage and the assessment of vulnerability for different natural hazards in terms of a multihazard approach [31,32] also led to further development of the damage scale for floods in [24], which is illustrated in

Table 3. Extended flood damage scale with examples of the 2002 flood in Saxony [24] and the flood of 2021 [3].

Damage Grade	Damage		Description	Drawing	Example 1
	Structural	Non-Structural
D1	none	light	moisture damage, dirt
D2	light	moderate	slight cracking of loadbearing walls doors/windows pushed in washing out of foundations contamination replacement of finshings necessary
D3	moderate	heavy	larger cracking in loadbearing walls and slabs settlements collapse of non-loadbearing walls replacement of non-loadbearing building elements necessary
D4	heavy	very heavy	collapse of loadbearing walls, slabs replacement of loadbearing walls, slabs
D5	very heavy	very heavy	collapse of larger parts of building
D6	very heavy	very heavy	dislocation: building completely washed away, toppled or displaced from foundation

¹ Photos of D1 to D6 damage taken by EDAC (D1 to D5: flood in 2002; D6: flood in 2021, cf. [3]).

. The examples for damage grades D1–D5 show damage cases of the 2002 flood in Saxony documented by EDAC in the days after the flood.

The damage cases of the 2021 flood in Germany show that damage patterns are possible in which buildings are completely washed away [3]. Such damage patterns have rarely been observed during previous floods but can occur during “flash flood” events.

Furthermore, considering the experience of tsunami damage [34], damage grade D6 (cf. Table 3) was introduced in [24] in order to differentiate these extreme damage cases from the grade of common collapse (D5).

The introduction of damage grade D6 in the flood damage scale enables comprehensive identification of damage patterns. Although damage grades D5 and D6 both represent total loss, there are differences that still need to be investigated.

With damage grade D5, additional demolition and disposal costs must be contemplated, which does not apply to buildings that have been completely washed away (i.e., damage grade D6).

The example of the washed away building for damage grade D6 in Table 3 was recently documented during a damage field survey immediately after the flood in Germany in 2021 in the Ahr Valley (Federal State of Rhineland Palatinate, cf. [3]). In principle, damage grade D6 can also be assigned to damage cases in which the building has been moved from its foundation or the building has tilted as a whole due to the scour of the foundation, which can be characterized as a disproportionate collapse.

3.2. Flood Vulnerability Classes

The concept of vulnerability classes was originally established by the European Macroseismic Scale (1998–EMS 98) [22] to define the intensity of an earthquake (and its shaking) based on observed effects, including, for higher intensities, the quality and quantity of damage to buildings as the most relevant indicator. One of the EMS 98 achievements is a vulnerability table, which enables a simple assessment of the vulnerability of various types of structures (cf. [25]).

The building type and the structural system of the buildings are taken into account. Additional factors, such as the quality of construction, state of disrepair, irregularities in shape, floor plan and design “defects”, can be considered according to the specified ranges of scatter (most likely, still probable and exceptional cases). The concept was successfully introduced for flood damage prognosis in [23] and further developed in [31]. Vulnerability classes categorize building types with a comparable vulnerability. A similar damage expectation (in the form of damage grades) is assumed for the same impact level from a natural hazard.

The introduced vulnerability tables for earthquakes [22], floods and wind [31] provide four vulnerability classes (A to D) for the consideration of the typical building stock and two classes (E and F) for buildings specially designed for the corresponding natural hazard [35].

Buildings of class HW-E (i.e., HW, referring to German “Hochwasser”, can be read as “High Water”) usually consist of reinforced concrete or masonry in a flood-resistant design and are characterized by a separation of vulnerable building parts from the flood water level, for instance, by raising the ground floor onto storey-high columns of steel or reinforced concrete [10]. Vulnerability class HW-F (newly introduced in [31]) is related to constructions such as floating homes (e.g., [36]), which represent a construction method specially adapted to floods.

Built on steel or concrete pontoons, these buildings avoid flooding by floating when the water level rises. Because only the pontoon is exposed to the flood water, the construction of the actual building is of minor importance with respect to the vulnerability. No damage data are currently available for the HW-E and HW-F vulnerability classes; therefore these classes were excluded from the investigations presented in this paper.

An essential criterion for determining flood vulnerability classes is structural damage (described by the mean damage grades, D_m) of similar impact levels.

The flood vulnerability classes were determined according to the method described in [10]. Evaluation of the damage cases in the EDAC flood damage database reveals significant differences in the mean damage grades (D_m) for the various main building types depending on the impact (inundation) level. Owing to these differences, damageability levels could be defined that are assumed to be typical for the vulnerability classes HW-A to HW-D. New or previously unclassified construction methods are classified based on their mean damage grades (D_m) for the corresponding impact level [10].

The most likely classes were determined by the engineering evaluation of various damage cases and confirmed using procedure presented in [10,25]. The range of scatter is initially determined on the basis of empirical values. In the vulnerability table for floods (cf.

Table 4. Classification of building types in vulnerability classes and identification of ranges of scatter [31].

Building Type	Vulnerability Class HW-
	A	B	C	D	E	F
Clay
Prefabricated timber frame
Timber frame with masonry or clay infill
Masonry
Reinforced concrete
Flood-resistant design
Flood-evasive design

Most likely vulnerability class. Probable range. Range of less probable, exceptional cases.

), the symbols of the EMS-98 are used to identify the most likely vulnerability class and the range of scatter. Exact classification is then performed for the corresponding building type, inspecting the condition and structural design of the building.

The final determination requires a competent, engineering-based assignment of the vulnerability class. For example, in the investigation areas, the quality of the materials used, which is related to the age of the building, and any previous damage (settlement cracks, plaster detachment, etc.) was evaluated as part of the vulnerability assessment. With a more detailed level of knowledge about the internal building construction, changes to buildings could also be taken into account in the vulnerability assessment, such as the removal of walls, which can weaken the building and degrade the structural performance.

3.3. Prognosis of Structural Damage

The vulnerability-based approach of the EDAC flood damage model [9,10,23] considers the inundation level and the flow velocity in the form of the specific energy height (H) (see Equation (1)) to predict structural damage considering the building type.

I.It could be demonstrated that this approach provides the best correlations with structural damage and losses in residential buildings. However, due to comparatively moderate flow velocities, the correlations are not clearly visible in the existing datasets [10].

$$H=h_{gl}+\frac{v_{fl}^2}{2g}$$

(1)

The basis for the derivation of the specific vulnerability function (SVF) type 2b in Figure 5 [9,10,18] were 1200 damage cases from the 2002 flood, for which flow velocities were assigned based on hydraulic calculations (see Section 2.1.1).

A hyperbolic tangent function was selected as a mathematical approach according to Equations (2) and (3) to determine the mean damage grades (D_m) in the original interval (1 to 5; D1 to D5) depending on the vulnerability class.

$$D_m=2·\tanh \left(f\left(h_{gl},v_{fl}\right)\right)+3$$

(2)

$$D_m=2·\tanh \left(A·\left(H-2\right)+B\right)+3$$

(3)

3.4. Loss Prediction

In the existing EDAC flood damage model, the vulnerability-relevant parameters (building type or vulnerability classes) are also considered in the loss prognosis [9,10,23,28,37].

One type of specific damage functions takes into account the building type (SDF Type 1a) or the vulnerability class (SDF type 1b), depending on the inundation level. The second type of specific damage functions (SDF type 2) converts the calculated damage grades (D_m) into relative losses. For all of the developed damage functions, the number of storeys and the presence of a cellar is considered [9]. An exponential approach is selected as a mathematical formulation of the various specific damage functions [9,10,23].

The main application is the general residential building stock. An application for other uses should also be possible for similar building constructions [9,10].

The results of the specific damage functions are expressed as relative losses in relation to the recovery value. The specific damage function calculates the losses (L) as a relative fraction of the replacement value. In Germany, so-called normal construction costs (Normalherstellungskosten-NHK 2000 [38] provide an efficient means of determining the replacement costs for the affected buildings. These still have to be scaled to the corresponding reference year using the building price index of the Federal Statistical Office [39].

The reliability and prognosis quality of the specific damage functions of the EDAC flood damage model were verified on the basis of the reported losses caused by the 2002 flood to residential buildings in the cities Eilenburg, Döbeln, Grimma and Flöha [10,18,28]. Additionally, the results for the losses based on the improved methods for the prognosis of structural damage presented in this paper are in agreement with the observed losses of the 2002 flood in all of the considered investigation areas [11].

However, due to the special boundary conditions in the loss compensation after the extreme flood in 2002 and the existence of an affected building stock with considerable renovation backlog, overcompensation likely occurred. Thus, application in other areas would require an adjustment of the damage functions and provision of further damage data.

Nevertheless, this would mitigate the general limitations of empirical damage models (i.e., large scatter of damage data, and insufficient damage data for some individual groups of buildings).

Therefore, the empirically supported prognosis of damage grades—as a measure of structural damage—and the adapted damage functions of a synthetic damage model (cf. [29,30]) are combined in the outcome of INNOVARU project. The procedure and the results of the loss calculations are presented in detail in the accompanying paper [12].

4. Improved Prognosis of Structural Damages

4.1. Consideration of Inundation Level and Flow Velocity

Validations of the existing model (see Section 3.3) showed good agreement with the actually observed damage (cf. [9,10,28]) However, when using the mathematical formulation of the specific energy height (H) (cf. Equation (1)), the contribution of the flow velocity is relatively small. In addition, the introduction of damage grade D6 requires an adjustment of the vulnerability functions.

The unified damage scales proposed in [24] and the vulnerability table for flood and tsunami impact [31] enable the combination of the two damage datasets described in Section 2.1. This combined, well-adjusted damage database can be analyzed for the different vulnerability classes. Because the evaluations are carried out for a flood damage model, tsunami damage cases with an inundation level of up to 6 m are taken into account. However, an extension to higher water heights, such as those that occurred in the Ahr Valley (cf. [3]), is also possible.

The combination of the two datasets based on the mean damage grades (D_m) of the clustered damage data for vulnerability classes HW-B and HW-C is shown in Figure 6a,b, respectively. Due to the simplified relationship between inundation level and flow velocity, the tsunami damage data follow a clearly defined area with respect to the impact level. However, assuming a plausible mathematical regression model, a realistic damage model can be derived. For this purpose, the previous approach of a hyperbolic tangent function for prognosis of the mean damage grades (D_m) is extended to intervals 1 to 6 (D1 to D6) according to Equation (4).

Five variants acc. to Equations (5) to (9) were initially investigated in [11]:

• Variant V1 only converts the existing approach from [18] to the six-stage damage scale;
• Variant V2 uses the inundation level (h_gl) and flood intensity (I_fl = h_gl × v_fl) from the so-called “Swiss model”, representing a combination of inundation level and flow velocity but without an extended physical background;
• Variant V3 includes the inundation level (h_gl) and the momentum flux (h_gl × v_fl²) (which is related to the hydrodynamic forces);
• Variant V4 considers only the momentum flux as a physically based input parameter;
• Variant V5 is similar to Variant 3 but weights the influence of the inundation level in a differentiated way.

The terms in Equations (6), (7) and (9) are not true to the unit, but they consider that, according to low or unavailable flow velocities, the inundation level alone has an influence on structural damage.

$$D_m=2.5·\tanh \left(f\left(h_{gl},v_{fl}\right)\right)+3.5$$

(4)

Variant V1: specific energy height (H):

$$D_m=2.5·\tanh \left(C_1·H+C_2\right)+3.5$$

(5)

Variant V2: inundation level (h_gl) + flood intensity (h_gl × v_fl):

$$D_m=2.5·\tanh \left(C_1·h_{gl}+C_2·h_{gl}·v_{fl}+C_3\right)+3.5$$

(6)

Variant V3: inundation level (h_gl) + momentum flux (h_gl × v_fl²):

$$D_m=2.5·\tanh \left(C_1·h_{gl}+C_2·h_{gl}·v_{fl}^2+C_3\right)+3.5$$

(7)

Variant V4: momentum flux (h_gl × v_fl²):

$$D_m=2.5·\tanh \left(C_1·h_{gl}·v_{fl}^2+C_2\right)+3.5$$

(8)

Variant V5: Inundation level (h_gl) + momentum flux (h_gl × v_fl²):

$$D_m=2.5·\tanh \left(C_1·\sqrt{h_{gl}}+C_2·h_{gl}·v_{fl}^2+C_3\right)+3.5$$

(9)

C₁, C₂, >C₃–Regression parameters (coefficients)

The regression parameters of the individual calculation variants can are presented in

Table 5. Coefficients of the vulnerability functions D_m = f(h_gl,v_fl).

Variant	VC	Coefficients			Coefficient of Determination (R2)
		C1	C2	C3
V1	HW-A 1	0.351	−0.730	-	-
	HW-B	0.292	−0.853	-	0.72
	HW-C	0.238	−0.914	-	0.84
	HW-D 2	0.189	−0.920	-	0.76
V2	HW-A 1	0.255	0.066	−0.572	-
	HW-B	0.135	0.053	−0.621	0.78
	HW-C	0.062	0.042	−0.647	0.88
	HW-D 2	0.035	0.030	−0.650	0.84
V3	HW-A 1	0.230	0.017	−0.496	-
	HW-B	0.143	0.011	−0.571	0.79
	HW-C	0.090	0.007	−0.623	0.85
	HW-D 2	0.071	0.004	−0.650	0.79
V4	HW-A 1	0.017	0.105	-	-
	HW-B	0.013	−0.250	-	0.74
	HW-C	0.009	−0.456	-	0.83
	HW-D	0.005	−0.512	-	0.78
V5	HW-A 1,2	0.578	0.017	−0.800	-
	HW-B 2	0.381	0.011	−0.800	0.79
	HW-C 2	0.264	0.007	−0.800	0.86
	HW-D 2	0.227	0.004	−0.800	0.80

¹ For HW-A, the coefficients were extrapolated from vulnerability classes HW-B, HW-C and HW-D. ² Coefficients slightly modified.

. For vulnerability classes HW-B to HW-D, these parameters were determined by a non-linear regression procedure. Due to the lack of damage data, the coefficients for vulnerability class HW-A were determined using an extrapolation procedure. In order to avoid overlapping or an intersection of the functions, the coefficients of some functions were slightly modified.

Improvements to the regression model analogous to the investigations in [40,41] have to be discussed in the future. However, assignment of the impact parameter, building type and other vulnerability-related parameters to the damage data is associated with considerable uncertainties. An improvement of the model is therefore unlikely at this point of the research [11].

Figure 7 displays the developed mathematical relationships in a 3D parameter surface. The mean damage grade (D_m) starts with values >D1, even at the 0 m inundation level above ground level, as groundwater inundation could also cause structural damage. Figure 7 graphically demonstrates that there are (practically) implausible relationships in some variants. Here, the physically consistent variants prove to be problematic with respect to their qualitative course (cf. [11]):

Variant V1 implies increasing structural damage at an inundation level of h_gl = 0 m with increasing flow velocities (which cannot occur here).

Variant V4 does not take into account the increase in structural damage with increasing inundation levels when the flow velocity is still v_fl = 0 m/s.

Although variants 2, 3 and 5 represent meaningful correlations from an engineering point of view, variants 1 and 4 are considered further.

4.2. Consideration of Inundation Level, Flow Velocity and the Number of Storeys

Maiwald [37] demonstrated (with only a small (limited) dataset for masonry buildings) that with a similar inundation level above the ground level (h_gl) the mean damage grade (D_m) tends to decrease with an increasing number of storeys (n_st) due to the higher static requirements of multi-storey buildings.

In the INNOVARU project, the flow velocity had to be taken into account, in addition to the influence of the number of storeys, depending on the inundation level (cf. [11]).

In the MLIT database, no differentiation is made according to the number of floors. Therefore, the influence of the number of storeys on structural damage cannot be investigated in the combined dataset. In addition, the damage data of the 2002 flood, for which flow velocities are available, only allow for insufficient further differentiation according to the influence of the number of storeys. Therefore, a systematic transfer approach is necessary to obtain improved vulnerability functions that additionally consider the number of storeys.

The information supplement from the entire EDAC flood damage database allows for derivation of vulnerability functions according to the number of storeys and vulnerability classes depending on the inundation level above ground level. Figure 8 displays the trend in the clustered damage data. Due to the small amount of damage data in the clusters at higher inundation levels (h_gl ≥ 3.5 m), some outliers are visible.

The evaluations show that buildings with fewer storeys are correlated with an increase in mean damage grade (D_m), which is considered and weighted via the natural logarithm in the corresponding term in the simplified vulnerability functions acc. to Equation (10).

Figure 8 shows that the generalized vulnerability function without differentiation according to the number of storeys is similar to the function for two-storey building. This is also true for the other vulnerability classes. For the sake of simplicity, in this study, it is assumed that the obtained coefficients for the vulnerability functions are valid for v_fl = 0 and that the vulnerability functions presented in Section 4.1 are also valid for two-storey buildings.

In a second step, the determined mathematical relationships and the obtained coefficients were used to extend the vulnerability functions presented in Section 4.1, resulting in the coefficients listed in

Table 6. Derived coefficients of the vulnerability functions, D_m = f(h_gl, v_fl, n_st).

Variant	VC	Coefficients
		C1	C2	C3	C4
V1	HW-A	0.351	−0.155	−0.622	-
	HW-B	0.292	−0.064	−0.809	-
	HW-C	0.238	−0.127	−0.829	-
	HW-D	0.189	−0.074	−0.869	-
V2	HW-A	0.255	0.066	−0.155	−0.465
	HW-B	0.135	0.053	−0.064	−0.577
	HW-C	0.062	0.042	−0.127	−0.559
	HW-D	0.035	0.030	−0.074	−0.599
V3	HW-A	0.230	0.017	−0.155	−0.388
	HW-B	0.143	0.011	−0.064	−0.527
	HW-C	0.090	0.007	−0.127	−0.535
	HW-D	0.071	0.004	−0.074	−0.599
V4	HW-A	0.017	−0.155	0.212	-
	HW-B	0.013	−0.064	−0.206	-
	HW-C	0.009	−0.127	−0.368	-
	HW-D	0.005	−0.074	−0.460	-
V5	HW-A	0.578	0.017	−0.155	−0.693
	HW-B	0.381	0.011	−0.064	−0.756
	HW-C	0.264	0.007	−0.127	−0.712
	HW-D	0.227	0.004	−0.074	−0.749

, which are applicable to Equations (11) to (15). Because the coefficients of these vulnerability functions were not created by a data regression but by merging the functions presented in Section 4.1 with the simplified functions, no coefficients of determination are included in Table 6.

The 3D surface diagrams for variant V2 for vulnerability classes HW-A to HW-D can be derived from the graphs shown in Figure 9. Variant 3 is shown in [11]. The functions are specified up to the number of storeys that were recorded in the damage data and in the investigation areas. For HW-C (typical for masonry constructions) and HW-D (typical for reinforced concrete buildings), number of storeys n_st > 5 are also possible.

$$D_m=2.5·\tanh \left(C_1·h_{gl}+C_2·\ln \left(n_{st}\right)+C_3\right)+3.5$$

(10)

Variant V1: specific energy height (H):

$$D_m=2.5·\tanh \left(C_1·H+C_2·\ln \left(n_{st}\right)+C_3\right)+3.5$$

(11)

Variant V2: inundation level (h_gl) + flood intensity (h_gl × v_fl):

$$D_m=2.5·\tanh (C_1·h_{gl}+C_2·h_{gl}·v_{fl}+C_3·\ln (n_{st})+C_4)+3.5$$

(12)

Variant V3: inundation level (h_gl) + momentum flux (h_gl × v_fl²):

$$D_m=2.5·\tanh (C_1·h_{gl}+C_2·h_{gl}·v_{fl}^2+C_3·\ln (n_{st})+C_4)+3.5$$

(13)

Variant V4: momentum flux (h_gl × v_fl²):

$$D_m=2.5·\tanh [C_1·h_{gl}·v_{fl}^2+C_2·\ln (n_{st})+C_3]+3.5$$

(14)

Variant V5: inundation level (h_gl) + momentum flux (h_gl × v_fl²):

$$D_m=2.5·\tanh (C_1·\sqrt{h_{gl}}+C_2·h_{gl}·v_{fl}^2+C_3·\ln (n_{st})+C_4)+3.5$$

(15)

C₁, C₂, C₃, C₄– Regression parameters (coefficients).

5. Validation of the Improved Model

The developed innovative procedure was validated using the observed damage of the 2002 flood in the six investigation areas in Saxony. The validation process takes place in two stages:

Level I: the model approaches according to Section 4.1 (Equations (5) to (9), coefficients according to Table 5) are taken into account, but the flow velocity is neglected (v_fl = 0 m/s).

Level II: the model approaches according to Section 4.2 (Equations (11) to (15), coefficients according to Table 6) are taken into account. The flow velocity is applied according to hydraulic models (Section 2.3). The influence of the number of storeys on the vulnerability is considered.

For all variants of the newly derived vulnerability functions, the mean damage grades (D_m) for the 2002 flood (see Section 2.3) were calculated for the individual affected buildings in the investigation areas. Because a comparison with the observed damage only makes sense for a larger number of buildings, the calculation results are aggregated at the level of land-use units. Therefore, comparison of the calculated and observed mean damage grades (MD_m) in the land-use areas (which are composed by the calculated mean damage grades (D_m) or the observed damage grades (D_i) for the individual buildings) is based on the land-use areas according to the “Official topographic-cartographic information system” (Amtliches Topographisch-Kartographisches Informationssystem-ATKIS^®) for Germany (cf. procedure in [10]).

The ATKIS^® land-use areas suitable for the investigation areas of Döbeln, Eilenburg, Flöha and Grimma are available from previous research projects with state of the year 2009.

Nowadays, these ATKIS^® land-use areas are much more coarsely or overlapping subdivided and therefore not suitable for aggregation of the calculation results at the land-use level. In contrast, the usable areas that are currently contained in the datasets of the “Official real estate cadaster information system-ALKIS^®” (Amtliches Liegenschaftskataster_Informationssystem) are much more finely divided. Therefore, these are also considered unsuitable for validation purposes. Alternatively, the built-up areas were subdivided independently into area units that provide a sufficient number of buildings/damage cases for the Pirna and Freital investigation areas.

Figure 10 shows the calculated mean damage grades (MD_m,calc) in comparison with the observed mean damage grades (MD_m,obs) for the investigated variants for Grimma for Level II.

For residential buildings, the aggregated deviations can be derived between reinterpretation and observation in all of the investigation areas. To avoid the outbalancing of the deviations in the individual areas of use, the mean absolute error (MAE) and the root mean square error (RMSE) were calculated according to Equations (16) and (17), where N is the total number of land-use areas in the corresponding investigation area. These statistical benchmarks are well-known and widely used (e.g., [42])

Mean absolute error (MAE):

$$MAE=\frac{1}{N}{\displaystyle \sum _{i=1}^N\left|M{D}_{m,obs,i}-M{D}_{m,calc,i}\right|}$$

(16)

Root mean square error (RMSE):

$$RMSE=\sqrt{\frac{{\displaystyle \sum _{i=1}^N(M{D}_{m,calc,i}-M{D}_{m,calc,i}}{)}^2}{N}}$$

(17)

Figure 11 displays the procedure for the numerical validation of the models.

The outcome of the error analysis s is displayed for Level I in

Table 7. Error analysis for Level I (excluding flow velocity).

Investigation Area	MAE					RMSE
	V1	V2	V3	V4	V5	V1	V2	V3	V4	V5
Döbeln	0.38	0.34	0.41	0.44	0.41	0.48	0.48	0.51	0.54	0.51
Eilenburg	0.48	0.41	0.49	0.51	0.50	0.75	0.68	0.70	0.68	0.71
Flöha	0.23	0.26	0.34	0.44	0.33	0.32	0.34	0.40	0.50	0.40
Freital	0.28	0.24	0.28	0.43	0.26	0.42	0.38	0.41	0.52	0.40
Grimma	0.40	0.39	0.33	0.37	0.34	0.51	0.52	0.46	0.51	0.47
Pirna	0.36	0.21	0.32	0.34	0.33	0.40	0.23	0.35	0.37	0.37
Total	0.36	0.31	0.36	0.42	0.36	0.48	0.44	0.47	0.52	0.48

and for Level II in

Table 8. Error analysis for Level II (including flow velocity).

Investigation Area	MAE					RMSE
	V1	V2	V3	V4	V5	V1	V2	V3	V4	V5
Döbeln	0.35	0.34	0.39	0.42	0.39	0.46	0.47	0.49	0.52	0.49
Eilenburg	0.49	0.43	0.49	0.52	0.50	0.65	0.60	0.62	0.62	0.63
Flöha	0.25	0.30	0.34	0.45	0.34	0.31	0.36	0.40	0.50	0.40
Freital	0.27	0.24	0.26	0.41	0.25	0.42	0.39	0.40	0.50	0.39
Grimma	0.41	0.35	0.34	0.37	0.34	0.51	0.48	0.46	0.50	0.47
Pirna	0.33	0.23	0.28	0.30	0.29	0.37	0.25	0.31	0.33	0.33
Total	0.35	0.32	0.35	0.41	0.35	0.45	0.43	0.44	0.50	0.45

. In a first step, the comparison of MAE and RMSE of both model levels in Table 7 and Table 8 shows only minor differences for the individual study areas. However, the deviations across all test areas (see row “Total” in both Table 7 and Table 8) are slightly reduced for most variants by considering the flow velocity, as well as the number of storeys (Level II).

In the second step, the best-suited must be determined. With regard to the calculation effort, as well as provision of the required hydraulic data (inundation level and flow velocity) and microscale building data, the variants should be assessed as equivalent.

For Level II and with an MAE of 0.32–0.41, all variants have relatively low mean absolute deviation. Assuming the range of the damage grades from D1 to D6 as 100%, this corresponds to 6.3–8.2% relative deviation. Evaluation of the RMSE indicates a comparable trend, as can be determined in the previous investigations.

Variants V1 and V4 are subject to particularities in terms of quality that make them appear less suitable for practical use. At an inundation level of h_gl = 0 m (e.g., with penetrating groundwater), variant V1 shows an increase in the mean damage grade (D_m) as the flow velocity increases (see Figure 7a). Because practically no increase in the flow velocity can occur at h_gl = 0 m, no increase in D_m should be forecast here either. In contrast, variant V4 predicts a constant mean damage grade (D_m) with standing water (v_fl = 0 m/s), even with an increase in the inundation level (see Figure 7d).

The lowest deviations for MAE and RMSE are associated with variant V2. This variant, which is currently regarded as somewhat more realistic than variants V3 or V5, is used as the basis for the loss calculations in an accompanying paper [12].

In the final report of the INNOVARU project [43], variants V2 and V3 are recommended for damage prognosis based on the current evaluation status.

Variant V2 shows the smallest deviations with respect to actually observed damage. In contrast, variant V3 is more physically justified by including the momentum flux (h_gl × v_fl²).

Figure 12, Figure 13, Figure 14 and Figure 15 show the corresponding distribution of the calculated mean damage grades (MD_m,calc) for variant V2 at Levels I and II and the observed mean damage grades (MD_m,obs) for the investigation areas of Döbeln, Eilenburg, Freital and Pirna. A slight increase in the calculated mean damage grades (MD_m,calc) caused by the flow velocity at Level II is visible.

Apart from slight deviations in individual areas of use, the graphic representation also shows a realistic damage prognosis. The deviations can be traced back to the low density of damage data in some areas under consideration.

The small deviations of the forecasts compared to the real observed structural damage demonstrate the realism of the chosen approach on a detailed micro-/mesoscale level. This successful validation stands out from previous validations of other damage models, in which mostly aggregated losses for larger towns or river catchment areas were compared with observations or with the results of other models (see e.g., [44]).

With the developed approach, a realistic prognosis of the expected structural damages is also possible in flood areas in which the flow velocities can no longer be neglected.

6. Conclusions

Based on the concept of the EDAC flood damage model, new approaches for vulnerability functions to predict structural flood damages are presented in this paper. The specific building vulnerability, the inundation level and the flow velocity are taken into account in the refined damage model. Realistic reinterpretations of the real observed damage from the 2002 flood in Saxony are presented for different investigation areas with moderate flow velocities, which are typical for river floods.

The calculation variant with the best agreement between the predicted and the reported damage grades is achieved with variant V2. The calculation results for this variant are the basis for the loss estimations via the application of new synthetic damage functions that consider the concept of the damage grades in the accompanying paper [12].

The new damage model initially intended for use in the Free State of Saxony can also be used in other Federal States and is therefore ultimately suitable for general application in Germany. If the regionally predominant building types are classified in vulnerability classes, the described part of the model would also be applicable internationally.

In the future, climate change will not only cause changes on the impact side but also require adaptation of the prevailing building types. Possible changes in the vulnerability of the existing structures to natural hazards represent a challenge for damage prognoses. The concept of vulnerability classes in the EDAC damage model allows for a flexible response to such changes. Newly designed or adapted conventional building types would have to be classified according to experience with the corresponding vulnerability classes. The vulnerability functions presented in this paper would still be applicable, although requiring slight modifications due to an extended damage database.

7. Outlook

The derived new vulnerability functions, in principle, also enable damage prognosis for events with high flow velocities, which can be expected in associated with flash floods. The model would have to be validated again for such extreme hydraulic conditions. In this context, the damage caused in the Ahr Valley (Germany) in 2021 might serve as a reference event. Initial attempts at engineering analysis of damage cases from this event in [3] indicate that the impact of debris and foundation erosion are important intensifying factors with respect to structural damage.

Evaluations of the damage in the Ahr Valley also confirm the influence of exposure (location) of buildings, which was previously emphasized to explain the variety of failure mechanisms and unexpected damage patterns in [10,18]. The basic concept is presented in

Table 9. Classification scheme with respect to the location of buildings and direction of inflow [10,18].

No.	Type/ Location	Description	Flow Direction	Scheme
1	Stand-alone		Direct
2a	Front house	Beginning of a row of houses	Direct/flow around
2b	End house	End of a row of houses	Flow around/circulation
2c	Front/end house	Beginning/end of a row of houses	Orthogonal/circulation
3a	Central house	In the middle of a row of houses	Tangential
3b	Central house	In the middle of a row of houses	Direct/orthogonal
4	Corner house	Cross situation	Flow around/circulation

. All these effects would have to be examined and integrated into the damage model in the future. Interesting options are represented by random forest techniques, including consideration of other factors that are not directly related to the impact and loading conditions responsible for structural damage [42].

For the application of the developed prognosis model for the structural damage, a detailed building inspection is necessary, which is not always feasible due to cost and effort limitations. An effort to reduce procedures using improved geodata on a microscale level is described in [13]. In connection with another type of new damage functions, realistic loss statements can also be obtained. As noted in the outlook of a previous study [10], in, such cases, reasonable assumptions about the distribution of building types or vulnerability classes in the investigation areas must be derived and inserted into the scenarios.

However, the developed vulnerability functions only indicate expected values with the mean damage grades (D_m), and a large scatter of flood damage cannot be considered.

So-called fragility functions were presented in [35], which indicate the probability of exceeding a certain damage grade depending on the inundation level and flow velocity. In principle, this makes it possible to characterize the spread of structural damage and the associated losses. According to simulative damage prognosis using the Monte Carlo method, the scatter is quite small. In further investigations, the complete chain of uncertainty in the flood damage prognosis should be highlighted.

Concerning the need of further data harvesting, remote sensing data and damage interpretation solutions for the areal images might represent key tools for development. Initial studies indicate the success of such applications, in particular for the “delta consideration” of the situation before and after the event [45]. Accepting the multihazard exposure of built environment and the cascading effects of the majority of natural disasters, we recommend evaluation of floods in the sense of dynamic processes with time histories of action parameters and loading conditions [46].