domain: spi phys

text:
Giuseppa Donelli
email: giuseppa.donelli@dlr.de

João M G D Mello
email: joao.mello@embraer.com.br

São José

Dos Campos

Felipe I K Odaguil
email: felipe.odaguil@gmail.com

Ton Van Der Laan
email: ton.vanderlaan@fokker.com.

Thierry Lefebvre
email: thierry.lefebvre@onera.fr

Nathalie Bartoli
email: nathalie.bartoli@onera.fr

Luca Boggero
email: luca.boggero@dlr.de

Nagel Björn
email: nagel.bjoern@dlr.de

Value-driven Systems Engineering Approach addressing Manufacturing, Supply-chain and Aircraft Design in the Decision-Making Process

In the last decades, some studies have highlighted that the integration of the product design and supply chain management leads to an increase of the profitability and efficiency of companies. However, considering manufacturing, supply chain and overall aircraft design variables in the early design phase increases the size of the solutions tradespace and thus the complexity in performing the decision-making process. This paper, follow-up of previous research activities addressed within the European project AGILE4.0, demonstrates how to leverage the value-model theory to simplify the decision-process when multiple criteria are accounted in the design phase.

Introduction

To meet the heterogeneous societal needs, even more complex, innovative, sustainable and circular aeronautical systems are required nowadays. The objective of the sustainable and circular aviation is to reduce the environmental impact in terms of fuel consumption, waste and emissions associated with all the aeronautical system activities and operations (Flightpath2050, 2011. Hence, the necessity to extend the branches of the aeronautical research to the entire aircraft life-cycle, from the design to the production, to the disposal after the end of the system activity. This surely enlarges the design space, having to consider even more variables related to different stages of the aircraft development in the design phase. However, it provides great possibility to aeronautical industries to win the nowadays global and competitive market [START_REF] Wu | A concurrent engineering approach to design for assembly[END_REF]. In this frame, the DLR Institute of System Architecture in Aeronautics aims at developing methodologies enabling the concurrent coupling of multiple domains (e.g. design, manufacturing) in the early stages of aircraft design to achieve solutions optimizing the entire aircraft life-cycle. The first step of this ambitious goal has been addressed within the European funded H2020 project AGILE 4.0 (INEA & Consortium, 2019), followup of the AGILE project (AGILE Website, 2022), also led by the DLR. By leveraging Multidisciplinary Design Optimization (MDO) and Model Based Systems Engineering (MBSE) technologies, this project aims at creating a digital representation of the system under design throughout the entire lifecycle (Ciampa & Nagel, 2021). Particularly, one of the challenges is to include all the main pillars of the aeronautical supply-chain during the early stages of aircraft design, with the objective to address innovative trade-off studies never performed before.

With this purpose, a value-driven methodology concurrently coupling manufacturing (MfG), overall aircraft design (OAD) and supply chain (SC) has been developed within one of the AGILE4.0 application cases. A schematic representation of this methodology, applied to the horizontal tail plane (HTP) of a regional 90-pax aircraft (AGILE Consortium, 2016), is reported in Figure 1. The manufacturing domain, dealing with the choice of materials, manufacturing and assembly processes for the HTP, is linked with the supply chain domain through the production quantity (PQ) that each enterprise has to perform. Based on this production quantity and on the characteristics of each individual enterprise, in the supply chain domain, production cost, time, quality and risk for each supply chain architecture, combination of multiple enterprises, are estimated. On the other side, a technology factor (TF) links the manufacturing domain with the overall aircraft design domain. In the OAD domain, the aircraft performance and specifically the aircraft fuel mass consumption (in cruise) is estimated based on this TF, which quantifies the impact that the HTP manufacturing choices have on the HTP mass and drag. The methodology then leverages the value-model theory to allow the concurrent coupling of these domains. Therefore, the production risk, time and quality characterizing the SC domain and the fuel mass consumption characterizing the OAD domain, are aggregated in the value by assigning to each one a weight and a single utility attribute (SAU) function. For more information about the methodology itself, reader can refer to [START_REF] Donelli | A Model-Based Approach to Trade-Space Evaluation Coupling Design-Manufacturing-Supply Chain in the Early Stages of Aircraft Development[END_REF]. In this study, however, the value-model theory has been leveraged only as weighted-sum function. This means that the same weights and linear utility functions have been assigned to all the attributes before being aggregated in the value. As consequence, the value-driven tradespace has not been influenced by stakeholders 'expectations. Therefore, the solution with the highest value, in this case, might not be the best solution for stakeholders. In fact, stakeholders 'expectations are modelled through the utility functions. By using linear utility functions on linearized data, the stakeholders´ expectations with respect to the criteria aggregated in the value have not been considered. The focus of this paper, instead, is on the use of the value theory to address and simplify the decisionmaking process when multiple criteria, also called attributes in the value-model theory, have to be taken in account. In fact, including manufacturing and supply chain decisions in the early development stage of aircraft design increases product competitiveness, giving to aeronautical industries the chance to win the nowadays global market. However, it also complicates the decision-making process having stakeholders consider even more criteria while taking a decision. In this paper, the challenge is to identify the best solution for stakeholders considering not only aircraft performance, but also production aspects being the supply chain and manufacturing domains included in the early design stage. For this purpose, the DLR internal tool, called VALORISE, is used to support stakeholders in modelling their expectations with respect to these criteria. Thus, first stakeholders identify the attributes, meaning the criteria playing a key-role when they have to take a decision. Then, an analysis (e.g. an MDO problem) is executed to numerically estimate these attributes. Finally, stakeholders model their expectations with respect to the defined attributes through the utility functions. Once defined the utility functions and the weights to attributes, the value-driven tradespace is generated by leveraging the value-model theory. In this case, the solution with highest value is the best solution for stakeholders.

In this paper, details on the value-model theory are provided in Section 2. In the same section, the AGILE4.0 MBSE-MDO framework is introduced to clearly describe the role of VALORISE, the interactive dashboard developed at DLR, based on the value-model theory and supporting the multicriteria decision-making process. An application case addressing the design, manufacturing a supply chain of the HTP is described in Section 3. However, before addressing the decision-making process, the attributes and the MDO problem providing the solutions of interest for the decision-making process are introduced and described in the same section. Finally, conclusions are provided in Section 4.

Value-driven Decision-making Process Definition

Including manufacturing and supply chain decisions in the early development stage of aircraft design complicates even more the decision-making process. In fact, identifying the best solution while multicriteria have to be considered is a great challenge for the decision maker. He/she might prefer an alternative because of one of the criteria, but not others. The objective of this research activity is to leverage the interactive dashboard VALORISE, implemented at DLR, to simplify the multiple criteria decision-making process and easily identify the best solution based on stakeholder´s expectations.

But, what does best solution means? In this study, it is assumed as best solution the alternative on the value-driven pareto-front that perfectly matches stakeholder´s expectations with respect to all the selected decision criteria (or attributes). The theory supporting this activity and also implemented in VALORISE is the value-model theory. For this reason, first details on the value-model theory are provided in Section 2.1. Then, the process explaining how VALORISE is leveraged to identify the best solution on the value-driven pareto-front based on stakeholder´s expectations is explained in Section 2.2.

Value Model Theory

The value-model theory is used in this research activity to simplify the multi-criteria decision-making process. Among all the available techniques presented in literature, the Multi Attribute Utility (MAU) theory is here adopted [START_REF] Keeney | Decisions with Multiple Objectives, Preferences and Value Tradeoffs[END_REF]. It is a good practice to use MAU when, at least, three criteria (also called attributes) are considered for the decision-making process [START_REF] Ross | Value-Driven Tradespace Exploration ofr System Design[END_REF]. The MAU formula is the following:

𝑈(𝑋) = ∑ 𝜆 𝑖 𝑈(𝑋 𝑖 ) 𝑁 𝑖=1
In which:

-N is the number of attributes; -U (𝑋), U (𝑋 𝑖 ) are the multi-attribute and single attribute utility function respectively; -𝜆 𝑖 is the weight associate with attributes 𝑋 𝑖 :

0 < 𝜆 𝑖 < 1 ∶ ∑ 𝜆 𝑖 = 1 𝑁 𝑖=1
For the value estimation, it is important to define:

-Attributes; -Single Attribute Utility (SAU) Functions; -Attributes Weights.

Attributes are the criteria (or the expectations) selected by stakeholders for identifying the best solution. Once defined the attributes, the SAU functions are used to quantify stakeholder´s expectations with respect to each attribute. They represent the way stakeholders would select a solution by only considering this attribute. For this reason, they are used to translate the qualitative stakeholder´s preferences in analytical curves. For the SAU functions elicitation, the function trend and the boundaries of each attribute have to be defined. An example of linear and not-linear SAU functions is reported in Figure 2. Utility (y-axis) usually ranges from 0 to 1 and expresses the stakeholder´s satisfaction with respect to the attribute content. The attribute content (x-axis) is the numerical value of the attribute. It is indicated in the entire study as attribute content instead of attribute value for sake of clarity (value is the dimensionless measure aggregating all the criteeria). Given an attribute content, higher is its utility, higher is the stakeholder´s wiliness in accepting a solution with this specific attribute content. As consequence, chancing the utility functions trend, it is possible to represent stakeholder´s preferences. In the example of Figure 2, for a specific attribute content (grey arrow), the stakeholder´s wiliness in accepting a solution with this attribute content drastically reduced by switching from the linear case (red point, higher utility) to the not-linear curve (blue point, lower utility). The elicitation of the utility functions is the most challenging step for the value estimation. Several methods are available in literature for the attribute utility function design, from questionnaire to advanced tools. However, interactive tools in which the decision maker can directly design the utility functions, might support decision makers in well-represent the qualitative preferences [START_REF] Ross | Interactive value model trading for resilient systems decisions[END_REF]. For this purpose, the DLR internal tool called VALORISE is used in this research activity as later described. Once assessed the SAU function trend (e.g. linear, exponential), the x-axis boundaries (attribute content) can be also defined to create scenarios of interest. In fact, solutions with an attribute content lower than the lower boundary and/or higher than the upper boundary are excluded from the value-driven tradespace. Finally, the weight combination has to be defined. Weights are associated to attributes to express the relative importance of each other. Therefore, also in this case, several strategic scenarios can be analyzed by prioritizing attributes according to stakeholder´s needs.

VALORISE supporting the Value-driven Decision-Making Process

As already mentioned in the previous sections, the DLR internal tool called VALORISE is used in this study to support the multi-criteria decision-making process. However, since this tool is part of the AGILE4.0 MBSE-MDO framework, shown in Figure 3, a brief overview of this framework is first provided for reader´s complete comprehension.

The AGILE4.0 MBSE-MDO framework, web-based and therefore accessible from multiple partners, combines several blocks ranging from the system identification, thus identification of stakeholders, needs and requirements to the system exploration, thus the optimization of the system under design.

In previous research activities, all the steps of the framework have been already addressed [START_REF] Donelli | A Value-driven Quantitative Framework coupling Aircraft Design, Manufacturing and Supply Chain by leveraging the AGILE4.0 MBSE-MDO Framework[END_REF]. Briefly, once selected all the stakeholders, needs and requirements, functional requirements have been collected to generate the system architectures in ADORE (Bussemaker, Ciampa, & Nagel, 2020) [START_REF] Donelli | A Value-driven Quantitative Framework coupling Aircraft Design, Manufacturing and Supply Chain by leveraging the AGILE4.0 MBSE-MDO Framework[END_REF]. To check the consistency between the architectures and the MDO problems, the DLR internal tool called MULTILINQ has been used [START_REF] Bussemaker | From System Architecting to System Design and Optimization: A Link between MBSE and MDAO[END_REF]. VALORISE, which stands for Value-driven trAdespace visuaLizatiOn, exploRatIon and asSEssment is an interactive dashboard, developed by DLR, leveraging the value-model theory to simplify the multi-criteria decision-making process. Within the AGILE4.0 MBSE-MDO framework, as shown in Figure 3, it is used to link stakeholder´s expectations with the MDO exploration to identify the best solution on the value-driven pareto-front. Thus, the best solution is the one on the pareto-front matching stakeholders' expectations with respect to multiple criteria. Before performing the decision-making process two steps have to be addressed: the definition of the attributes (first block of the AGILE4.0 framework) and the numerical estimation of the attributes through the MDO problem execution (last block of the AGILE4.0 framework). Thus, from one side stakeholders have to define attributes, that are the criteria playing a key role when they have to take a decision. On the other side, an MDO problem is set-up and executed to estimate attributes and achieve the pareto-front. Particularly, the solver SEGOMOE based on Bayesian optimizer has been adopted to solve multi-objective optimization problems (Bartoli, et al., 2019) (Grapin, et al., 2022). In this way, the best solution, matching stakeholders' expectations with respect to all the attributes, is identified on the value-driven tradespace as the one with the highest value. The modelling of this process, shown in Figure 4, is supported by the AGILE4.0 framework technologies. The attributes defined by stakeholders are collected in a CPACS file -called baseline in Figure 4 -then used to run the MDO workflow which provides the attributes contents and pareto-front. All the CPACS files, generated for each solution of this MDO problem, are collected in a .zip file and given as input to VALORISE. In fact, real-time scenarios can be investigated in VALORISE since changes on the attributes weights and/or on utility functions (e.g. on the boundaries of contents) are directly visualized on the dashboard. Finally, VALORISE can be also used as stand-alone tool and be integrated in a toolchain with other tools, for instance in RCE, to perform specific analysis. However, this is out of scope since in this research activity, VALORISE has been used by stakeholders as interactive dashboard.

Clarified the process, technologies and information needed to address the decision-making, in the next section an application case is presented. First attributes are defined, then the MDO problem is executed to estimate attributes and finally the best solution is identified on the value-driven paretofront as the one with the highest value.

Value-driven Decision-Making Application

As explained in the previous section, to identify the best solution it is necessary to identify the key decision-criteria (attributes) for stakeholders from one side and run an MDO workflow to estimate the attributes and obtain the pareto-front on the other. In this section, an application case addressing the identification of the best solution when multiple criteria have to be considered at the same time is described. In particular, in the first sub-section the attributes selected by stakeholders are introduced; in the second one the optimization problem executed to obtain the pareto-front is presented and finally the decision-making process allowing the identification of the best solution on the value-driven pareto-front is addressed.

Attributes Definition

The first step to find the best solution consist in the identification of the key criteria playing a key role when stakeholders have to take a decision. These criteria, also called attributes, are selected by stakeholders (first step of AGILE4.0 framework). In this activity, several attributes, reported in Table 1, are identified by stakeholders. Particularly, the production time, risk, quality and cost, characterizing the SC domain, are selected as attributes by stakeholders because of their significant impact on the production management (Guide, 2001). The production risk, time and quality are therefore the key criteria considered by stakeholders in finding the best solution. These criteria, as reported in Section 2.1, are then aggregated in a single dimensionless measure which is the value, once a weight and a utility function is assigned to each of them. In addition to these, also the production cost plays a key role for stakeholders in the decisionmaking process. However, it is used in this research activity, as other variable for the value-cost tradespace generation. In this way, stakeholders can perform trade-off studies of interest deciding whenever to go for a solution with lower value and cost or spend more to achieve an even more "valuable" solution.

Once defined the attributes, the next step is to numerically estimate the attributes. An MDO problem is here executed to evaluate attributes and identify the pareto-front.

Attribute contents estimation: MDO problem execution

Once defined the attributes, the next step for the best solution identification relays in the numerical estimation of these attributes. The contents of attributes are estimated by running an MDO problem, addressed in this section.

The MDO problem aims at maximizing the production performance while changing the production quantity assigned to enterprises. In this case, the manufacturing properties are fixed and all the combinations of the enterprises able to perform the selected materials and processes are analyzed, as reported in Table 2. Among all the 9•10 6 possible combinations, 19 solutions are identified on the 4objective pareto-front. Reader can refer to (Merola, et al.) to know more about this MDO problem since this out of scope for this research activity. Once estimated the attributes, the next step is to aggregate these attributes in the value, defining the utility functions and weights, to identify then the best solution on the value-driven tradespace.

Thus, for the decision-making process the 4-objective pareto-front is "translated" in a value-driven pareto-front. Two strategies could be addressed here as shown in Figure 6: Strategy I in which attributes are estimated through an optimization and then aggregated in the value through a weight and a utility function; Strategy II in which attributes are aggregated in the value through a weight and a utility function and then the value and cost are optimized. However, in case of strategy II, the optimization has to be executed anytime weights and utility functions are changed, as needed to address the decision-making process. In addition, it has been demonstrated that the 2-objective pareto-front is included in the 4-objective pareto-front [START_REF] Donelli | A Value-driven Quantitative Framework coupling Aircraft Design, Manufacturing and Supply Chain by leveraging the AGILE4.0 MBSE-MDO Framework[END_REF]. Therefore, to reduce the computational time, it has been decided to follow Strategy I in this research activity. Hence, first the optimization is executed and then the value-driven tradespace is generated by changing weights and utility functions. Therefore, in the next section, the 4-objective pareto-front is "translated" in a value-driven tradespace by aggregating the optimized attributes of each solution of the pareto-front in the value.

Changing the utility functions or the weight combination, it is shown in next section how the value of these solutions changes.

Figure 6 -Strategy I (in red) adopted to perform the decision-making process: first a 4-objective optimization is executed, then strategic scenarios are addressed by changing the value settings

Value-driven Decision-Making Process Analysis

Once defined attributes and run the MDO workflow, the best solution can be identified on the valuedriven pareto-front. From the execution of the MDO problem previously introduced, 19 points have been identified on the 4-objective pareto-front. In this section, the value is first used as weighted-sum function to simplify the visualization of the 4-objective pareto-front and then to address and simplify the multi-criteria decision-making process.

To use value as weighted-sum function, the hypothesis of linear utility functions and same weights for all the attributes is mandatory. These assumptions are summarized in Table 3, while the utility functions are plotted in Figure 9a. To all the attributes the same boundaries have been assigned (x-axis ranging from 0 to 100) to include all the solutions in the value-driven tradespace. Attributes contents are already normalized (0 to 100) due to industrial partners´intelletual properties. In this case, utility functions do not represent decision maker´s preferences. They can be read as analytical curves used to translate different parameters having different scale of measures in the same dimensionless one to be correctly compared. In addition, curves follow the logic flow of higher the worst for risk and time, higher the better for quality.

In this case, the value-driven tradespace is therefore not influenced by decision-maker´s preference.

Identified as "reference" value-driven tradespace, it is reported in Figure 7. Higher is the value, better is the solution. This value-driven tradespace highlights as solution with highest value the alternative 1, red circle in the plot. However, it is worth to underline that this solution might not be the best solution for stakeholders since decision-maker´s preferences have not been considered yet. The next step instead is to introduce stakeholders' expectations to identify in the value-driven tradespace the best solution as the one with the highest value. In Figure 8, the qualitative stakehold-ers´ preferences with respect to the selected attributes are represented. As expected, stakeholders prefer solutions with lower production cost, time and risk but higher quality. To account for stakeholders' expectations, it is necessary to quantify such preferences with respect to all the attributes. The utility functions are used to quantify such expectations. Particularly, stakeholders used VALORISE to draw the utility functions representing their preferences respect to these attributes. The utility functions defined by two stakeholders, industrial partner supporting this research activity, are respectively reported in Figure 9b) and Figure 9c). Each stakeholder (identified as Stakeholder A and Stakeholder B) define own curves to reflect their preferences with respect to these criteria. The qualitative preferences are the same for both stakeholders (Figure 8), instead the quantification of those expectations change with respect to the each stakeholder (Figure 9a,b). In fact, a different utility is associated by each stakeholder to each attribute and attribute content. For example, referring to the quality utility functions, stakeholder A wiliness in accepting solutions with low quality is higher than stakeholder B, who instead assigned a low utility to solutions with low quality. These curves defining stakeholders' expectations impact the value-driven tradespace. With the assumption of same weight for all the attributes, the value-driven tradespace implementing stakeholders' expectations are represented in Figure 10. In this case, the solution with the highest value is the best solution, meaning that this solution matches stakeholders' expectations with respect to all the attributes. In both plots of Figure 10, the best solution is highlighted by a red circle: the solution 1 for the valuedriven tradespace implementing stakeholder A utility functions (Figure 10a), by alternative solution 10 in the value-driven tradespace implementing stakeholder B utility functions (Figure 10b). The best solution is different because of the different stakeholders 'expectations with respect to all the attributes. These two solutions are related to supply chain, involving enterprises performing the selected HTP materials and manufacturing processes. In details, both solutions involve the same enterprises, but the processes they perform are different as shown in Figure 11. Stakeholders will therefore choose a different option in case they decide to go for the best solution, thus the one with highest value. However, a trade-off is always possible. For instance, stakeholder A might decide to go for solution 10 instead of solution 1 being the reduction in value very small with respect to the reduction in cost. Going in a solution with lower value implies, of course, to lose something in terms production time, risk and quality being these criteria aggregated in the value. In this way, the decision-making process is simplified. In fact, stakeholders can perform the trade-off studies considering multiple criteria (aggregating in the value) but only considering the value vs. cost pareto-front. In fact, the way stakeholders would select solutions with respect to each of those criteria are implemented in the value through the single utility functions. Thus, the challenge in well-design utility functions that perfectly match stakeholders 'expectations.

In addition, in both value-driven tradespaces of Figure 10, solutions 5 and 13 are crossed. These two solutions are in fact not part of both pareto-front but instead they are included in the "reference" pareto-front. Similarly, solution 12 is included in the pareto-front of stakeholder A, but not in the "reference" and "Stakeholder B" pareto-front. The motivation always relays in the implementation of the stakeholders' expectations and thus on the utility curves defined by them. The utility of each attribute associated to each solution for the three different cases are summarized in Table 4. The value-driven tradespaces reported in Figure 10 have been obtained by assuming the same weights to all the attributes. This assumption implies is no prioritization among attributes and, as consequence, the value-driven tradespace is influenced only by the implementation of the utility functions representing stakeholders' expectations. However, once defined the utility functions, more strategic scenarios can be analyzed by prioritizing attributes in different ways. This can be easily done in VALORISE since it gives the possibility to check real-time the value-driven tradespace changes. As example, a value-driven tradespace is represented in Figure 12 when time is prioritized with respect to the other attributes. The weight combination analyzed is reported in Table 5 while the utility functions are the same plotted in Figure 9a,b.

Table 5 - From the mathematical point of view, the shift of solutions (up and down) is related to the utility functions chosen by stakeholders. In this case, in fact, utilities reported in Table 4 for some of the solutions are still valid, being the utility functions always the same. The value of each solution now changes because of the weight assigned to each attribute. From Table 4, it appears that solutions for the stakeholder A always have a utility for time lower than the one of risk and quality. Thus, the value of these solutions goes down when prioritizing the attribute with lowest utility, that is time. Similarly, for the stakeholder B, in which the trend is opposite, time has utility higher than others. For this reason, the value of these solutions goes up when prioritizing time. In any case, what is worth to underline here is that with this analysis stakeholders can easily identify the best solution in any scenarios of interest. For some stakeholders, in fact, quality might be even more important than time and risk or risk than quality and time, and so on. In VALORISE, these scenarios can be easily set and visualized since any change in weights and utility functions is real-time plotted in the dashboard.

Finally, as already mentioned, other scenarios can be investigated by changing the boundaries of the x-axis of the utility functions, thus the boundaries of the attributes contents. This allows stakeholders to visualize on the value-driven tradespace only solutions of interest. In this application case, the boundaries have been not changed since solutions are characterized by attributes contents really close to each other. That´s because the attributes contents have been estimated in an MDO problem.

Conclusions

The research activity aims at leveraging the value-model theory to simplify the multi-criteria decision-making process. The DLR internal tool called VALORISE is used to support decision-maker in identifying the best solution, thus the alternative on the tradespace perfectly matching the decision-maker´s preferences. First attributes, that are the criteria playing a key role when stakeholders have to take a decision are identified. Then, an MDO problem is set-up and executed to achieve the numerical quantification of the attributes and the pareto-front. At this point, attributes are aggregated in the value, through a weight and a utility function. In the application case, first the value is used as weighted-sum function to obtain the value-driven tradespace not influenced by stakeholders 'preferences. In this case the solution with the highest value might be not the best solution for stakeholders. Then stakeholders ´expectations have been implemented through utility functions. Therefore, two value-driven tradespaces have been shown in Figure 10, one implementing utility function of each stakeholder. The solution with highest value in the tradespace is the best one for stakeholder A, but not for stakeholder B which is instead represented by solution 10. Thus, implementing stakeholders 'expectations lead to identify different best solution for each stakeholder. However, in this case, the same weight has been assigned to all the attributes meaning that stakeholders are not prioritizing any of them while taking a decision. To show how the value-driven tradespace change when an attribute is prioritized, an example related to the time prioritization has been addressed in Section 3.3. This analysis allows stakeholders to easily identified the best solution when one of the criteria has to be prioritized. The multi-criteria decision-making is so simplified. In fact, criteria are aggregated in the value and stakeholders can perform trade-off studies accounting for multiple criteria while only referring at value or cost. In fact, the way stakeholders select solutions based on each criterion are represented by the single utility functions. Making a single utility function for time means to represent how stakeholders expect to select a solution if only looking at this attribute. From here, it appears clear how important is to well-design the utility functions. That´s why VALORISE has been implemented. By using VALORISE stakeholders can directly draw their own curves for each attribute thinking how they would take solutions based on that attribute. Unfortunately, it was not possible to instead analyze scenarios in which the boundaries of the utility functions were limited to exclude solutions from the pareto-front. That´s because solutions analyzed in the decision-making process are output of an optimization problem. It can be interesting in future, to address decisionmaking after the system architecting (third block of the AGILE4.0 framework). In this case the best solution is identified on the tradespace and it might be not an optimal solution. However, once defined for the best solution, thus the best architecture, an MDO problem might be performed. In addition, further studies might deeply analyze the relation between the two optimization strategies presented in Section 3.2 and understand which solutions always relay on the pareto-front. Furthermore, it is already in progress activities including the fuel consumption, characterizing the OAD domain, in the decision-making process. From the decision-making perspective, the process is the same, the only difference is in the number of criteria aggregated in the value. In fact, by running the MDO problem addressed in this research activity, the fuel consumption mass of the aircraft doesn´t change. As consequence, it doesn't make sense to consider the fuel consumption in the decision-making process since there is no variation of stakeholders' expectations with respect to this attribute. Last, but not least, uncertainty can be included in the value estimation, especially on the design on the utility functions and weight set. In any case, in a society in which even more criteria will be considered in the early aircraft design stage because of the circular and sustainable requirements to address, the valuemodel theory can represent a powerful mean for a first decision-making analysis.

Figure 1 -

 1 Figure 1 -Value-driven methodology concurrently coupling manufacturing, supply chain and overall aircraft design domain

Figure 2 -

 2 Figure 2 -Single Attribute Utility (SAU) functions: linear (red) and not-linear (blue) trend

  The MDO problems have been then setting-up by using MDAx (Page Risueño, Bussemaker, Ciampa, & Nagel, 2020) and executed within RCE (RCE Website, 2022). The MDO problem set-up and execution has been automatized through CPACS (CPACS Webpage, 2022).

Figure 3 -

 3 Figure 3 -AGILE4.0 MBSE-MDO Framework including Decision-Making, adapted from(Ciampa & Nagel, 2021)

Figure 4 -

 4 Figure 4 -Value-driven Decision-Making Process and Technologies

Figure 5 -

 5 Figure 5 -CPACS input file for VALORISE toolBased on these inputs, VALORISE is then used to simplify the multi-criteria decision-making process. All the attributes are aggregated in one single dimensionless measure, which is the value, by assigning a weight and a utility function to each attribute. In this way, the value-driven tradespace is generated. In other words, the pareto-front of the optimization problem is translated in the valuedriven tradespace in which the best solution is identified as the one with the highest value. The settings of the value model, weights and utility functions, can be easily defined by stakeholders directly in VALORISE. Single utility functions, as already mentioned in the previous section, represent the way stakeholders would select solutions based on this attribute. In VALORISE, stakeholders can interactively draw utility functions to represent in the best way possible their expectations with respect to each selected attribute. Instead weight represents the relative importance of each attribute. In VALORISE stakeholders can set several weight combinations to analyze the scenario of interest. In fact, real-time scenarios can be investigated in VALORISE since changes on the attributes weights and/or on utility functions (e.g. on the boundaries of contents) are directly visualized on the dashboard. Finally, VALORISE can be also used as stand-alone tool and be integrated in a toolchain with other tools, for instance in RCE, to perform specific analysis. However, this is out of scope since in this research activity, VALORISE has been used by stakeholders as interactive dashboard.

Figure 7 -

 7 Figure 7 -"Reference" Value-driven Tradespace -same weight and the same linear utility function for all the attributes (production risk, time and quality)

Figure 8 -

 8 Figure 8 -Stakeholders Qualitative Preferences

  Figure 9 -Utility Functions

  Figure 10 -Value-driven Tradespace influenced by Statekholders' preferences

  Figure 11 -Value-driven Tradespace influenced by Statekholders´preferences

  Figure 12 -Value-driven Tradespaces prioritizing Time

Table 1

 1

		-Attributes Definition
	Parameter	Domain	Role
	Production Time	Supply Chain	Attribute
	Production Risk	Supply Chain	Attribute
	Production Quality	Supply Chain	Attribute
	Production Cost	Supply Chain	Attribute

Table 2

 2

			-HTP Configuration Description
	HTP	Number of	Materials &	Number of	Number of
	Components	Components	Processes	Production Site	Assembly Sites
	Skins Stringers	2 30	Sheet Metal Stretch Formed Metal by Z-Extrusion	13 13	4
	Spars	2	Machined Aluminium	14
	Ribs	20	Machined Aluminium, Sheet Metal Stretch Formed	10	9

Table 3

 3

		-"Reference" Value-driven Tradespace Characterization
	"Reference"	Attributes	Trend	Utility Function Boundaries	Weight Combination
	Value-driven	Time	Linear	[100,0] [0,1]	0.33
	Tradespace	Quality	Linear	[0,100] [0,1]	0.33
		Risk	Linear	[100,0] [0,1]	0.33

Table 4 -

 4 SAU and Value for specific solutions of the value-driven tradespace

				SAU (-)
	ID	Linear		Stakeholder A		Stakeholder B
		Time Risk Quality Time Risk Quality Time Risk Quality
	1 0.86 0.84	0.89	0.47 0.78	1	1	0.67	0.99
	5 0.85 0.84	0.87	0.44 0.77	1	1	0.64	0.99
	10 0.86 0.84	0.89	0.47 0.78	1	1	0.67	1
	12 0.86 0.84	0.85	0.47 0.78	1	1	0.66	0.99
	13 0.84 0.83	0.88	0.44 0.75	1	1	0.60	0.99

Biography


Anton Tkachuk

Tim Krake

Jan Gade

Malte Von Scheven

Alexander Popp

Review of "Efficient computation of redundancy matrices for moderately redundant truss and frame structures"

Reviewer 1 (Anonymous) Reviewer In the manuscript, a new approach is presented to compute the redundancy matrix of structures in an efficient way. The approach is well explained and applied to three academic examples. However, it would be interesting to also see an application on how the redundancy matrix information is used in structural design. I suggest to accept the manuscript after minor revision.

Reviewer Please check the scale of Figure 1. It doesn't look like a cylinder with 1 m radius and 10 m height.

Authors A remark about different scaling along different axes is added to captions of Figure 1 instead of the plain text describing the figure. An additional view for each version of the structure is added either.

Reviewer In the caption of Figure 1 it is written, that n = 8 but in the text above it is written n = 10. Please clarify.

Authors Yes, there was an inconsistency. The value n=8 is taken in both cases now.

Reviewer In all examples, the cross section and the modulus of elasticity is constant for all elements. Is this a requirement of the presented approach?

Authors On page 8 a comment is added "For all examples, even a random distribution of the stiffnesses of the bars does not affect the increase in efficiency. "

Reviewer Also if the focus of the paper is on the computational efficiency of the redundancy matrix evaluation, some more motivation on how this information is used in structural design would be interesting. In the introduction, it is written: "Information about the spatial distribution of the degree of statical indeterminacy, as well as information about different load-carrying mechanisms, can be obtained from the redundancy matrix. This information shows the variety of load paths in the structure, which can be later used in the analysis and design of the structure. " Maybe you can add to one of the examples a short application to show how the structural design can be improved by the introduced approach.

Authors Adding such an example would diffuse the focus of the current paper. Therefore, an additional comment is added on page 1. "A detailed discussion how the redundancy matrix can be used in the design and analysis of structures is beyond the scope of this paper and can be found in the references given above".

Reviewer 2 (Anonymous)

Reviewer This paper presents a new approach for efficient computation for redundancy matrices for truss and frame structures. The paper is interesting and well written. It first gives a nice introduction to the theory of redundancy matrices and then proposes a new approach for computing these redundancy matrices based on a single value decomposition, which is presented clearly and in much detail. Finally, the method is applied to a series of truss and frame structures, which demonstrates the applicability and the advantage of the method. The numerical examples confirm that the new method is more efficient than the standard one in computing redundancy matrices if the degree of statical indeterminacy of the structure is low or moderate. I have only some minor comment on the presentation of the structures in the examples section. For the more complex structures, the illustrations could be improved, it is sometimes hard to understand them. I guess that views from different perspectives may help. Moreover, the geometry description of the examples could be provided (in the appendix), which might also help understanding the geometries without the need to retrieve all the data from the repository.

Reviewer Important: The provided DaRUS link (Tkachuk 2023) does not work for me (DOI not found)!

Authors A provisionary link to the implementation and examples for review process was added in the cover letter of the submission. The DaRUS link will be activated as soon as the paper is accepted.

Reviewer Detailed comments on the illustrations:

The third one in Figure 1: I cannot really make up the geometry from this figure, but I think that there are additional diagonals which connect non-neighboring nodes, right? This could probably be seen better in an additional top view.

Authors The view is added for each version of the structure.

Reviewer Similar for the Mero rood in Figure 4. The figure already shows 2 different perspectives, but I find it very hard to make sense of them.

Authors An additional view is added for a better description of the structure. Furthermore, additional explanation for Figure 5 is added to explain the obtained results.

Reviewer Regarding Figure 7: it probably would be useful to distinguish between hinges and rigid joints in the figures.

Authors A consistent change in all figures is made. Empty circles denote hinged connection and the rigid joints are at intersections of the members.

Editor's assessment (Alexander Popp)

The present contribution focuses on efficient computations of redundancy matrices for truss and frame structures that are characterized by a moderate degree of redundancy. The proposed methods are derived following the concept of singular value decomposition and yield closed-form expressions of redundancy contributions that are computationally efficient for large, statically indeterminate systems. The initial manuscript underwent a rigorous review process, with two expert referees offering their assessments. Both review reports have been very positive and only requested minor modifications. The authors provided such a minor revision and the manuscript was accepted for publication accordingly.

Open Access This review is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the authors-the copyright holder. To view a copy of this license, visit creativecommons.org/licenses/by/4.0.


text:
Nathan Letheule

Flora Weissgerber

Sylvain Lobry

Elise Colin

AUTOMATIC SIMULATION OF SAR IMAGES: COMPARING A DEEP-LEARNING BASED METHOD TO A HYBRID METHOD

Keywords: Simulation, Radar, Deep Learning, Remote sensing, Semantic segmentation

This study compares two approaches for simulating synthetic aperture radar (SAR) images. The first approach uses a conditional Generative Adversarial Network (cGAN) to learn statistical image distributions from optical images. In a second approach, we generate SAR images using a electromagnetic simulator taking into input material maps obtained by segmenting optical images. We propose two metrics to evaluate the quality of the simulation. We evaluate the methods on existing Sentinel-1 SAR images of France using the DREAM database. The results suggest that the physical simulator with automatically created material maps is better suited for generating realistic SAR images compared to the cGAN approach, even if a lot of work remains to be done on the complexity of the description of the scene.

INTRODUCTION

Radar images have diverse applications in civil, industrial, and defense sectors, such as land surveillance and mapping. Simulating radar images is crucial for generating large quantities of SAR images for deep learning, evaluating algorithmic performance, and meeting specific acquisition conditions.

Simulation tools face significant challenges in balancing computational complexity, fidelity to physics, and image size. Several tools exist for solving Maxwell's equations, such as RaySAR [START_REF] Auer | RaySAR -3D SAR simulator: Now open source[END_REF] which uses ray tracing to account for topography and interactions, CohRaS [START_REF] Hammer | Coherent simulation of SAR images[END_REF] which utilizes coherence simulation, and EMPRISE (www.emprise-em.fr) developed in Onera [START_REF] Lebarbu | Complete radar simulation chain: Application to maritime patrol surveillance using SAR/ISAR modes[END_REF]. EMPRISE is capable of generating high-resolution images over large areas and in various operational scenarios. This capability is achieved through a combination of two modules. The first module accurately describes the Radar Cross Section (RCS) of objects by utilizing their detailed 3D mesh and material properties. The second module generates speckle zones by employing curves that describe the variation of σ 0 with respect to incidence angle and polarization. However, this simulation tool faces two significant challenges: ensuring the electromagnetic model is accurately represented in the resulting image, and properly describing the input data for the simulator. In the case of the EMPRISE simulator, the input is typically a manual annotation of the scene from an optical image. This study aims to replace this process with an automatic algorithm and evaluate the quality of the resulting simulation product.

We propose two approaches to automatically simulate SAR images, depicted in Figure 1. The first one uses a conditional generative adversarial network (cGAN) architecture to simulate a SAR image from its optical equivalent, without taking into account the physics. The second approach involves automatic segmentation of an optical image to generate a scene description, which can be used within our physical simulation tool EMPRISE.

In section 2, we describe the methods used for the simulation and the performance evaluation, including the implementation of the metrics. In section 3, we present the main results of our approaches before concluding.

Fully supervised simulation

To generate a SAR image from an optical image, we train a cGAN using a Pix2Pix architecture [START_REF] Dietrich-Sussner | Synthetic glacier SAR image generation from arbitrary masks using pix2pix algorithm[END_REF]. The cGAN incorporates an additional condition, such as a reference image or la-bel, to guide the generation process. Specifically, we use the Pix2Pix architecture [START_REF] Dietrich-Sussner | Synthetic glacier SAR image generation from arbitrary masks using pix2pix algorithm[END_REF] to learn the mapping between optical and SAR domains, resulting in realistic SAR images from corresponding optical images.

EMPRISE-based simulation

Our second approach, based on hybrid simulation, utilizes the EMPRISE simulator which requires a material map as input for scene simulation. To generate the material map, we propose assigning a label to each pixel in an image through semantic segmentation. In recent years, deep learning, particularly convolutional neural networks (CNNs), has emerged as the dominant paradigm for image semantic segmentation [START_REF] Kotaridis | Remote sensing image segmentation advances: A meta-analysis[END_REF]. U-Net [START_REF] Ronneberger | U-net: Convolutional networks for biomedical image segmentation[END_REF], a CNN specifically designed for biomedical image segmentation, has become a popular choice for this task. We train a U-Net-based neural network on both the optical image and OpenStreetMap (OSM) data rasterized in the DREAM database and used as classification labels. Once the segmentation map S is obtained, it is used as input for automatically simulating SAR images with the EMPRISE simulator. Each label of S is assigned a backscatter curve for each class type, which is parameterized by the local incidence angle and acquisition frequency. allows us to generate simulated SAR images with accurate backscatter properties for each labeled class in the material map.

Evaluation Frameworks

To evaluate the quality of simulated results, we propose dedicated metrics considering the specific statistical properties of SAR. Previous literature commonly uses metrics like the structural similarity index measure (SSIM) [START_REF] Dietrich-Sussner | Synthetic glacier SAR image generation from arbitrary masks using pix2pix algorithm[END_REF] and mean square error (MSE) for image assessment. However, it is important to note that speckle can cause MSE variations even when comparing different real images of the same area. Similarly, the SSIM may not accurately distinguish between texture and statistical differences in images. In this study, we propose two new metrics designed for SAR images to address limitations in previous approaches. These metrics offer a comprehensive evaluation of SAR simulations.

Speckle statistics

The first metric focuses on analyzing the statistical properties of speckle. When speckle satisfies Goodman's model assumptions [START_REF] Goodman | Some fundamental properties of speckle[END_REF], the second and third order log-cumulants become independent of the first moment of the speckle distribution. To compare the micro-texture properties of two speckle realizations, regardless of their mean values, we propose to compute the Bhattacharyya distance between the bivariate distribution of the log-cumulants κ 2 and κ 3 [START_REF] Nicolas | Statistical models for sar amplitude data: a unified vision through mellin transform and meijer functions[END_REF] and we call this metrics BDLC. The distribution is estimated through the joint histogram of these log-cumulants. We recommend estimating the log-cumulant of the intensity distributions over 5x5 pixel estimation windows. To ensure that the estimated Bhattacharyya distance is as independent as possible from content of the image, we propose to estimate the histogram on the support of [0, 5σ κ2,L ] and [µ κ3,L -9σ κ3,L , µ κ3,L +9σ κ3,L ] where µ κi,L and σ κi are respectively the theoretical mean and theoretical standard deviation for κ i (i = 2, 3) of a gamma distribution. The number of bins required for histogram discretization is chosen to be equal to the square root of the number of pixels in the considered area. We found that an adequate image area size for calculating this criterion is approximately 90,000 pixels. With these parameters, the Bhattacharyya distance established on these two statistical parameters can effectively distinguish different distributions such as gamma distributions with different equivalent number of looks L or distinguish gamma distribution with log-normal and inverse gamma.

Bright scatterer detection and matching

We introduce our second evaluation process, which aims to identify bright scatterers in the scene. These scatterers are characterized by their high reflectivity, typically resulting from the material or geometric structure of objects. Our approach involves comparing the presence of these bright scatterers in both real and simulated images. To achieve this, we propose a multiscale detection method inspired by the work of Armand and Lopes [START_REF] Lopes | Optimal target detection using one channel sar complex imagery: application to ship detection[END_REF]. This method entails computing intensity averages over central and peripheral neighborhoods. By employing various neighborhood sizes, we are able to detect bright scatterers at different scales. For each detection involving a set of related pixels, we retain only the pixel with the highest value, resulting in a refined final detection.

We introduce three metrics for comparing the sets of points extracted from both the real and simulated images. These metrics include the Chamfer distance [START_REF] Wu | Density-aware chamfer distance as a comprehensive metric for point cloud completion[END_REF], the recall comparing at a fixed matching distance of N pixels with the true image as the reference, called RBMP for Recall-Based Matching Performance, and the minimum matching distance required to attain a recall score of 0.9 with the simulated image as the reference called 90RD.

The Chamfer distance can be interpreted as the average distance between two point clouds. When the points in the simulated image closely match those in the real image, the Chamfer distance will be small. Conversely, if no bright points are detected in the simulated image, the distance will be infinite, indicating a lack of correspondence between the two point clouds.

The recall measures the ability of our simulator to accurately locate and match real points in the simulated image. It calculates the percentage of real points that have a corresponding point in the simulated image within a certain distance threshold, fixed here to N=5 pixels. A high recall indicates accurate localization and matching of the simulated image. The distance to a fixed recall score, assesses the abil-ity of our simulator to reproduce the points present in the real image. It returns the maximum distance threshold required to match 90% of the bright scatterers in the simulated image to the points in the real image. A smaller distance indicates a closer resemblance between the simulated points and the real image. If the density of bright scatterers in the real image deviates significantly from that of the simulated images, the first and third metrics will reflect this discrepancy by increasing their respective values.

RESULTS

Our global Approach

We evaluate the performance of our two simulation approaches on Sentinel-1 images, using a Sentinel-2 optical image as input. However, comparing the results is not straightforward due to differences in simulation geometries. Specifically, for the cGAN approach, we simulate images in the terrain-corrected Ground Range Detected (GRD) geometry, which allows for a direct transfer from the optical images. On the other hand, EMPRISE generates images in the antenna reference frame, resulting in Single Look Complex (SLC) products. Despite these differences, we analyze the simulation results using our proposed metrics, as shown in Figure 3.

In both of our approaches, we rely on the DREAM database [START_REF] Colin | The DREAM database: a multimode database including optics, radar, DSM (SRTM) and OSM labels for deep machine learning purposes[END_REF]. DREAM is a multimodal database containing optical images from Sentinel-2, material map composed of 7 different labels extracted from OSM, and SAR images from Sentinel-1 acquired within less than 10 days apart from the optical images. To validate the hybrid simulations, we also download Single Look Complex (SLC) images corresponding to the same products than the initial GRD images.

Figure 2 consists of a real Sentinel-1 image on the left, a cGAN-simulated image on the right, and an EMPRISEsimulated image in the center after automated segmentation and rescaled to the same squared grid for consistency. In a qualitative assessment, the overall results appear to be more realistic with cGAN. However, our metrics will provide a more accurate and detailed evaluation of these simulations.

Accuracy of statistics

Taking into account the entire image footprint, both the cGAN and EMPRISE approaches closely approximate the speckle distributions observed in real images. However, at the local level, this is no longer the case for cGAN.

Specifically, in the case of EMPRISE, the log-cumulant diagrams exhibit remarkable similarity between the real and simulated data. Conversely, cGAN fails to adequately capture the microtexture. In Figure 3, the example diagram clearly illustrates that the points in the log-cumulants plane for cGAN are considerably less dispersed around the theoretical value, indicating a notable deviation from the desired representation. Furthermore, a complementary parametric analysis of EM-PRISE indicates that the fidelity of the simulation depends much more on the number of materials used, and especially on the attribution of their electromagnetic properties, than on the performance of the U-Net segmentation.

Accuracy of bright points

Both types of simulators struggle to retrieve bright scatterers. cGAN fails to generate any bright point. For this simulation scenario, the Chamfer distance thus is infinite. This issue may be attributed to the convolutional filters used in conjunction with the loss functions, which tend to smooth the results at finer scales, hindering the representation of locally higher intensities. The EMPRISE approach performs slightly better with a Chamfer distance of 21 pixels. It produces an excess of bright points, due to the selection of a material for urban areas with inadapted directivity. The attribution of electromagnetic behaviors to labels plays a critical role in achieving an adequate number of bright points. More accurate modeling would require either the use of more materials or the use of object with RCS computed from canonical mechanisms.

The table 1 gathers all the criteria computed on the relative test dataset for the U-net and cGAN learning methods. It should be noted that it is not possible to compare the BDLC criterion in this situation, as it depends on the equivalent number of looks, which differs in our two cases (L=4.4 for the GRD data of cGAN, and L=1 for the SLC data of EMPRISE).

CONCLUSION

In conclusion, two approaches are proposed and tested to simulate SAR images from optical data. The first approach uses a cGAN while the second approach uses a physical simulator based on electromagnetic simulation of segmented areas and the 3D geometry of the terrain. We propose metrics to evaluate the simulation results, showing that EMPRISE is better at reproducing local statistics, while both approaches struggle to retrieve bright scatterers. Future progress should focus on assigning materials to segmented areas through supervision and adding specific 3D object modeling, rather than improving segmentation accuracy. Diffusion networks could be used to guide physical simulation by distributing these specific objects, such as vehicles or urban details, with realistic distributions.

Fig. 1 .

 1 Fig. 1. Block diagram of the two SAR image simulation methods. Below: in GRD geometry with cGan; Up: in SLC geometry with EMPRISE

Fig. 2 .Fig. 3 .

 23 Fig. 2. Left: initial real Sentinel-1 image. Middle: EMPRISE-simulation approach. Right: cGAN approach

Table 1 .

 1 Evaluation metrics for simulation results

	Simu	Metrics BDLC Chamfer 90RD RBMP
	EMPRISE	0.013	16.86	43.57 24.27%
		cGAN	1.81	inf	inf	0%


domain: spi math phys

text:
Thomas Di Martino

Régis Guinvarc'h

Laetitia Thirion-Lefevre

Elise Colin

TOWARDS THE UNDERSTANDING OF THE C-BAND TEMPORAL SIGNATURE OF BOREAL FOREST THROUGH PHYSIOLOGY PARAMETERS RETRIEVAL FROM SENTINEL-1 TIME SERIES AND MACHINE LEARNING

Keywords: Deep Learning, Inversion, SAR, Time Series, Explainable AI

The C-Band radiometric signature of boreal forests is highly seasonal, with apparent correlations to temperature changes. Within these seasonal components, we assume that information related to tree height can be extracted. We apply a onedimensional Convolutional Neural Network to assess this assumption, intending to retrieve tree height measured by Airborne Laser Scanning from C-Band Sentinel-1 time series. A study site in the Parc National des Grands Jardins, in Québec, Canada, was selected for this analysis. Prediction-wise, we reach an R2 score of 0.45 and an RMSE of 1.84m, following a 4-fold cross-validation, which exhibits a non-negligible influence of the tree height parameter on boreal forest radiometric response in C-Band Synthetic Aperture Radar, despite the presumed fast saturation of this wavelength, when observing forested environments. In addition to performance metrics, we use a gradient-based explainability tool to diagnose the most contributing periods of the input time series to predict tree height to better correlate the seasonal conditions of this parameter's influence on the forests' radiometry.

INTRODUCTION

The extraction of forest parameters relies on long wavelength configurations such as P-band or L-band SAR [START_REF] Dobson | Dependence of radar backscatter on coniferous forest biomass[END_REF], as they offer better penetration of the canopy layer. However, access to P-band and L-band SAR data is limited. The current SAR satellite offering the best spatial and temporal coverage is ESA's Sentinel-1, which on-boards a C-band sensor. The understanding of the interaction of the electromagnetic wave of C-Band Synthetic Aperture Radar (SAR) imagery with complex environments such as boreal forests is still partial because of its more limited penetration capacities. The few studies analyzing the interaction of C-band wave with a forested environment alleviated the limitations mentioned above by relying on temporal approaches [START_REF] Cohen | A modeling-based approach for soil frost detection in the northern boreal forest region with C-band SAR[END_REF][START_REF] Santoro | Retrieval of growing stock volume in boreal forest using hypertemporal series of envisat asar scansar backscatter measurements[END_REF]. An example of such a study can be found in [START_REF] Rignot | Monitoring of environmental conditions in Taiga forests using ERS-1 SAR[END_REF], where the seasonal variations of the radiometric response of the forest are correlated with the soil state, with higher radiometry during summer and lower radiometry during winter under freezing conditions. The amplitude of summer-to-winter radiometric changes appears to be a direct function of the physiology of the tree [START_REF] Pulliainen | Multitemporal behavior of l-and c-band sar observations of boreal forests[END_REF]. Thus, with the problem of inversion in mind, while a single C-Band SAR acquisition may be insufficient, a shift towards multitemporal SAR is expected to lead to better results [START_REF] Bruggisser | Potential of Sentinel-1 C-band time series to derive structural parameters of temperate deciduous forests[END_REF]. In addition, with the launch of the new generation of Sentinel-1 C-Band satellites, unparalleled worldwide open-source coverage is available, with a short revisit time ranging from 12 to 6 days for some regions of the globe. This unprecedented amount of temporal information has the potential to ease the understanding of the relationship between the physiology of a forested area and its C-Band temporal profile. This paper uses a deep learning approach to assess the retrievability of a commonly used forest parameter, tree height, from Sentinel-1 time series. We first present the study area and illustrate the seasonalities of the C-Band radiometry of forests by introducing Sentinel-1 and airborne laser scanning data. We then detail the model used, particularly a 1D Convolutional Neural Network. In the recent literature, 1D convolutions were shown to be excellent feature extractors when processing Sentinel-1 time series, even in unsupervised conditions [START_REF] Di | Beets or cotton? blind extraction of fine agricultural classes using a convolutional autoencoder applied to temporal sar signatures[END_REF]. The presentation of the application performance of the network to the retrieval of tree height from Sentinel-1 time series is accompanied by a study on its explainability, using the Grad-CAM algorithm, to study date contribution to the prediction.

MATERIALS AND METHODS

Study area: Parc National des Grands Jardins, Québec, Canada

Located in the Québec province of Canada, the Parc National des Grands Jardins is a unique ecosystem with a boreal forest similar to the taiga while remaining accessible for experimentation. Thus, considering the need to monitor boreal forests at scale, this park is prone to experimentation and measurements. In this paper, the ground truth forest measurements used consist of a rasterized 30m resolution tree height data product, originating from airborne LiDAR acquisitions, and illustrated in Fig. 1.

The tree height metric estimates the height of the top of the canopy, from the ground, in meters. The point measure is then averaged over resolution cells of 30m in length. The Parc appears highly diverse in tree height: the western side has lower tree heights, represented by light color. This may relate to more sparse vegetation and younger vegetation. Oppositely, the eastern side of the Parc contains many areas with up to 10m of height and more. In addition, in the center of the image, a vast area appears unwooded, deforested by a forest fire from the late 90s. In addition to LiDAR data, Sentinel-1 multitemporal imagery was also retrieved.

Sentinel-1 multitemporal data

To exploit the seasonality of the radiometric signature of boreal forest, we use two years' worth of ground range detected Sentinel-1 data, which include the years 2018 and 2019. 56 acquisitions, with an average revisit time of 12 days, constitute the multitemporal stack of Sentinel-1 data.

Each image consists of a decibel-scaled intensity value of both Vertical-Vertical and Vertical-Horizontal polarizations.

A snippet of the average temporal image of the area layered over the LiDAR acquisition is displayed in Fig. 2. While While we expect these temporal patterns to be connected to the physiological information of forests so that their inversion becomes possible to some degree, its demonstration requires a powerful modeling tool. In this paper, we rely on a 1D convolutional neural network that we now introduce.

The model, the Convolutional Neural Network

Architecture

A convolutional neural network (CNN) is a specific kind of artificial neural network using learned convolution layers to extract patterns in data. A familiar context for the application of CNNs is image processing, which relies on 2D convolutions to extract spatial features from images. In our case, the input data are time series, which are 1-dimensional sequences. They thus rather induce the use of 1D convolutions. The complete detail of the used architecture can be found in Tab. 1.

In particular, temporal feature extraction relies on three stacks of convolutions and pooling, all 1D-parameterized.

The activation function in all layers is the Exponential Linear Unit (ELU) [START_REF] Clevert | Fast and accurate deep network learning by exponential linear units (elus)[END_REF]. Four separate fully-connected layers then process the features before making a tree height prediction.

Explainability, and Grad-CAM

To extend the outcomes of applying a CNN to this regression task, we rely on a gradient-based explainability tool called Grad-CAM [START_REF] Ramprasaath | Grad-CAM: Visual explanations from deep networks via gradient-based localization[END_REF]. Initially developed for image processing and classification, we adapt its functioning and equations to consider the sequential nature of our input data and the regression task at hand. With this tool, we can extract date contribution to the regression prediction and better understand important dates where height information radiometrically manifests itself.

Training setup

The proposed network's training involves using various set of hyper-parameters, which include a learning rate set to 1e -3, a batch size to 1024, and an epoch count to 20. In addition, the network is trained using the Adam optimizer. The choice of parameters was empirically optimized. In addition, a 4fold cross-validation scheme was used for validation, with a handmade slice of the dataset in 4 separate spatially coherent areas. Using spatially coherent splitting prevents any data leakage between sets coming from the correlation of neighboring time series, which could be caused by Sentinel-1 preprocessing steps such as multi-looking or despeckling.

RESULTS

Following the previously mentioned 4-fold cross-validation scheme, we obtain an R2 score between the ground truth and the predictions of 0.45 and an RMSE of 1.84m. A first visualization of the proposed CNN network's prediction results is illustrated in Fig. 4, and overall, the predictions appear accurate. In particular, unwooded areas are well retrieved: the forest fire in the center of the image is retrieved, and areas of lower/higher height appear distinguish- difficulties encountered by the network: high environments are also high in biomass density. Aside from underestimating high trees, this scatter plot also sheds light on the overestimation of the smallest. This underestimation showcases a similar difficulty in dealing with extremums which are, in this case, highly sparse vegetated environments. Thus, a combination of Sentinel-1 time series and a 1D CNN is not operational for inversion purposes. However, it still successfully extracts groups of vegetation types with similar heights.

ANALYSIS AND EXPLAINABILITY

Now that we have answered the question "does it work ?", we orient our analysis towards "How does it work?". We plot the Grad-CAM-based date importance using a color-coded average Sentinel-1 profile over the Parc. To assess particular radiometric events which play a significant role in the differentiation between tree heights, we illustrate 20 S1 colorcoded time series, averaged iteratively over the groups of data points, gathered in buckets of 5% of heights' percentiles. Their respective Grad-CAM coloring is thus calculated separately. These time series are displayed in Fig. 6. The main observation regarding a significant difference in temporal radiometry between the various tree height groups lies in three distinct signal drops. Their intensity appears inversely proportional to tree height cover, and these drops almost entirely disappear in high tree scenarios. These three drops are the most important dates for tree height regression, particularly for smaller trees. In addition to helping to understand the difference between various tree height radiometries, Fig. 6 also sheds light on the confusion between groups of higher trees: their average signature appears highly similar in C-Band, which disturbs their regression.

CONCLUSION

This paper assessed the retrievability of tree height from C-Band SAR time series. Acknowledging the task's difficulty by the theorized saturation of C-band wavelength to biomass levels, we relied on a state-of-the-art prediction algorithm using a 1D Convolutional Neural Network. We achieve performances of 0.45 as an R2 score which, while not strong enough for pure inversion, exhibits the potential of multitemporal C-Band SAR to be used for inversion of tree height alongside other imagery, such as optical imagery or different wavelengths of SAR data. In particular, thanks to its high temporal resolution, Sentinel-1 has the unique capacity among open data earth observation sensors to monitor forest seasonality finely.

Fig. 1 :

 1 Fig. 1: Tree Height product illustration

Fig. 2 :Fig. 3 :

 23 Fig. 2: False color Sentinel-1 image (R: VV, G: VH, B: VV-VH) with the respective value intervals {[-15, -5], [-20, -10], [0, 15]}

Fig. 4 :

 4 Fig. 4: Tree Height prediction map

Fig. 5 :

 5 Fig. 5: Tree Height density-colored prediction scatter plot

Fig. 6 :

 6 Fig. 6: Grad-CAM colored average radiometric time series, split into groups of tree height

Table 1 :

 1 Architecture of the 1D Convolutional Neural Network used in this study

	Layer	(Count, size, stride, padding)	Output size
	Input time series	-	58 × 2
	1D Convolution	(64, 7, 1, 1)	54 × 64
	Max Pooling 1D	(-, 2, 2, -)	27 × 64
	1D Convolution	(128, 5, 1, 0)	23 × 128
	Max Pooling 1D	(-, 2, 2, -)	11 × 128
	1D Convolution	(256, 3, 1, 0)	11 × 256
	Max Pooling 1D	(-, 2, 2, -)	5 × 25
	Flatten	-	1280
	Fully Connected	-	128
	Fully Connected	-	64
	Fully Connected	-	32
	Fully Connected	-	1