3D-Modelling of Charlemagne’s Summit Canal (Southern Germany)—Merging Remote Sensing and Geoarchaeological Subsurface Data

The Early Medieval Fossa Carolina is the first hydro-engineering construction that bridges the Central European Watershed. The canal was built in 792/793 AD on order of Charlemagne and should connect the drainage systems of the Rhine-Main catchment and the Danube catchment. In this study, we show for the first time, the integration of Airborne LiDAR (Light Detection and Ranging) and geoarchaeological subsurface datasets with the aim to create a 3D-model of Charlemagne’s summit canal. We used a purged Digital Terrain Model that reflects the pre-modern topography. The geometries of buried canal cross-sections are derived from three archaeological excavations and four high-resolution direct push sensing transects. By means of extensive core data, we interpolate the trench bottom and adjacent edges along the entire canal course. As a result, we are able to create a 3D-model that reflects the maximum construction depth of the Carolingian canal and calculate an excavation volume of approx. 297,000 m3. Additionally, we compute the volume of the present dam remnants by Airborne LiDAR data. Surprisingly, the volume of the dam remnants reveals only 120,000 m3 and is much smaller than the computed Carolingian excavation volume. The difference reflects the erosion and anthropogenic overprint since the 8th century AD.

. Depth accuracy, scale and stratigraphical resolution of trench geometry data; scale classification according to Zielhofer et al. [35].

Technique
Number Name (Label in Figure 2) References Depth Accuracy

Resolution of Stratigraphy
Pace Excavation 3 trenches "2013" (c) "2016-S1" (d) "2016-S2" (e) Werther  As a first step, we present standard cross-section reference geometries derived from archaeological excavations and direct push sensing transects. Subsequently, we conduct our 3D-modelling approach by the transfer of the cross-section reference geometries to extensive vibra-coring positions along the entire canal course. Further, we integrate the ground truth data within the pre-modern topography to establish the 3D-model. Finally, we calculate the volume of the present dam remnants and compare both volume results in the context of the construction and decay of the Fossa Carolina.
The main objectives of our study are: 1. Integrating different geoarchaeological datasets and creating a standard routine for high precision 3D-modelling of the Carolingian excavation depth and volume.
2. Calculation of the earth excavation volume based on the Fossa Carolina 3D-model and the pre-modern DTM.
3. Calculation of the dam volume based on the present LiDAR DTM and the pre-modern DTM. 4. Comparison of both volumes from the canal trench and dams, describing the differences and their implications for the Carolingian and post-Carolingian history of the canal.
The canal course can be divided into five sections in relation to surface structures and trench bottom depths (see Figure 2).
(a) The Altmühl floodplain consists of late Pleistocene fluvial deposits, mainly gravels and sands. Holocene alluvial sediments accumulated subsequently with an intercalated mid-Holocene soil indicating neglectable meander migration during the late Holocene [40]. Further, Kirchner et al. [40] deduced that the Fossa Carolina was never built in this section and, therefore, the canal has never been finished.
(b) The area of the watershed (Central Section) is sedimentary characterised by Pleistocene valley fills of sandy grain sizes [3]. The sediments are of reddish to greyish colours, due to different redox conditions [27,32,38]. Here, large lateral dams are still present in this section. These reach altitudes up to 13 m above present pond level.
(c) The West-East Section marks the transition from the watershed to the Swabian Rezat floodplain. The sandy parent material is similar to the Central Section [27,38]. The lateral dams are smaller compared to the Central Section, but still prominent with altitudes up to 5 m above the inner trench levels.
(d) The Northern Section represents the South-North canal course parallel to the Rezat fen in the East. Sandy to loamy fluvial sediments dominate and close to the Rezat fen and a half-bog soil is developed [32]. Here, the dams are almost not visible in the field, but noticeably identifiable in the LiDAR DTM.
(e) The North-East Section is similar to the Northern Section, with clastic valley fills but the influence the organic sediments in the northernmost part increases. The Swabian Rezat floodplain The canal course can be divided into five sections in relation to surface structures and trench bottom depths (see Figure 2).
(a) The Altmühl floodplain consists of late Pleistocene fluvial deposits, mainly gravels and sands. Holocene alluvial sediments accumulated subsequently with an intercalated mid-Holocene soil indicating neglectable meander migration during the late Holocene [40]. Further, Kirchner et al. [40] deduced that the Fossa Carolina was never built in this section and, therefore, the canal has never been finished.
(b) The area of the watershed (Central Section) is sedimentary characterised by Pleistocene valley fills of sandy grain sizes [3]. The sediments are of reddish to greyish colours, due to different redox conditions [27,32,38]. Here, large lateral dams are still present in this section. These reach altitudes up to 13 m above present pond level.
(c) The West-East Section marks the transition from the watershed to the Swabian Rezat floodplain. The sandy parent material is similar to the Central Section [27,38]. The lateral dams are smaller compared to the Central Section, but still prominent with altitudes up to 5 m above the inner trench levels.
(d) The Northern Section represents the South-North canal course parallel to the Rezat fen in the East. Sandy to loamy fluvial sediments dominate and close to the Rezat fen and a half-bog soil is developed [32]. Here, the dams are almost not visible in the field, but noticeably identifiable in the LiDAR DTM.
(e) The North-East Section is similar to the Northern Section, with clastic valley fills but the influence the organic sediments in the northernmost part increases. The Swabian Rezat floodplain (Rezat fen) is characterised by organic sediments that reach up to 3 m thickness [42]. Similar to the Northern Section, the dams are almost not visible in the field but can be clearly detected in the DTM. (Rezat fen) is characterised by organic sediments that reach up to 3 m thickness [42]. Similar to the Northern Section, the dams are almost not visible in the field but can be clearly detected in the DTM.

Figure 2:
Local setting and course of the Fossa Carolina and its subdivision in I) Central Section, II) West-East Section, III) Northern Section, IV) North-Eastern Section. All input data for the subsequent modelling are shown, including drillings, modelled trench bottom transects and additive transects. Cross-section reference geometries a) "West-East Section", b) "The Anomaly" based on direct push sensing data. Cross-section reference geometries c) "2013", d) "2016 -S1", e) "2016 -S2" based on archaeological excavations. LiDAR data have been provided by the Bavarian Land Surveying Office.
The Fossa Carolina has a length of approx. 2.9 km [32] and proof of the Early Medieval summit canal was found by vibra-coring [3,32,39,40], direct push sensing [27,38] and archaeological excavations [33,36]. The canal course starts on the southern slope of the valley watershed and runs with a noticeable s-shape in the northern direction ( Figure 2). The s-shape is a result of the impressive knowledge of the Carolingian constructors to make the best alignment of the canal course in relation to a minimal excavation workload [34].
According to Werther and Feiner [33] and Völlmer et al. [38] the general stratigraphy of the canal fills can be summarised as follows: the pre-Carolingian parent material consists of sandy to loamy fluvial sediments with almost no organic remains. The timber (oaks recovered during archaeological excavations, which were used to stabilise the embankments of the Carolingian canal West-East Section, III) Northern Section, IV) North-Eastern Section. All input data for the subsequent modelling are shown, including drillings, modelled trench bottom transects and additive transects. Cross-section reference geometries (a) "West-East Section", (b) "The Anomaly" based on direct push sensing data. Cross-section reference geometries (c) "2013", (d) "2016-S1", (e) "2016-S2" based on archaeological excavations. LiDAR data have been provided by the Bavarian Land Surveying Office.
The Fossa Carolina has a length of approx. 2.9 km [32] and proof of the Early Medieval summit canal was found by vibra-coring [3,32,39,40], direct push sensing [27,38] and archaeological excavations [33,36]. The canal course starts on the southern slope of the valley watershed and runs with a noticeable s-shape in the northern direction ( Figure 2). The s-shape is a result of the impressive knowledge of the Carolingian constructors to make the best alignment of the canal course in relation to a minimal excavation workload [34].
According to Werther and Feiner [33] and Völlmer et al. [38] the general stratigraphy of the canal fills can be summarised as follows: the pre-Carolingian parent material consists of sandy to loamy fluvial sediments with almost no organic remains. The timber (oaks recovered during archaeological excavations, which were used to stabilise the embankments of the Carolingian canal trench) documents the construction time. The initial trench fills feature abruptly redeposited sediments with less organic material. Subsequently, thick organic sediments cover the initial, sandy trench fills. These organic sediments consist of peat and sapropel layers, representing open water bodies and former ponds. The youngest fills feature clastic sediments from mainly modern times. They indicate the ongoing erosion of the dams and an intensive land use with intentional levelling of the northern canal sections. Today, parts of the canal structures are fully eroded, refilled and under agricultural use. However, in the Central and West-East Section, massive dam remnants and the course of the canal are still visible ( Figure 3).
Remote Sens. 2019, 11, x FOR PEER REVIEW 6 of 22 trench) documents the construction time. The initial trench fills feature abruptly redeposited sediments with less organic material. Subsequently, thick organic sediments cover the initial, sandy trench fills. These organic sediments consist of peat and sapropel layers, representing open water bodies and former ponds. The youngest fills feature clastic sediments from mainly modern times. They indicate the ongoing erosion of the dams and an intensive land use with intentional levelling of the northern canal sections. Today, parts of the canal structures are fully eroded, refilled and under agricultural use. However, in the Central and West-East Section, massive dam remnants and the course of the canal are still visible ( Figure 3).

LiDAR Digital Terrain Model
High-resolution airborne laser scanning data were provided by the Bavarian Land Surveying Office (provided 6 November 2012 and 8 August 2013) [44,45]. We used the derived 1 × 1 m spatial resolved DTM for the localisation of the canal course and the determination of the midway position between both dam ridges. Furthermore, we use the present LiDAR DTM for the modelling and volume calculation of the present remnants of the dams.

Pre-Modern Digital Terrain Model
For the modelling of the canal geometry and its integration in the former landscape, the availability of a high-resolution pre-modern DTM is important. In this study, we use the pre-modern DTM with a spatial resolution of 1 × 1 m from Schmidt et al. [34]. This model is based on the LiDAR DTM mentioned above. The DTM is purged of all modern anthropogenic structures, by removing all grid cells which are affected by anthropogenic disturbance. Finally, the remaining cells were interpolated to create a smoothed pre-modern terrain.

Magnetic Survey
We prospected the large areas of the Northern and North-Eastern Section magnetically with a fast, motorised measurement system developed for geoarchaeological issues by Linzen et al. [46]. The system is based on a set of SQUIDs (Superconducting Quantum Interference Device) which provided a very high resolution (local centimetre resolution; sensor sampling rate of 1 kHz) and a maximum of magnetic information [47,48]. Thus, we detected buried canal remains and localised their course precisely over a distance of more than 1.2 km [49].
Within the West-East Section of the canal, we conducted a manually operated Fluxgate magnetic survey to detect the canal course between the present dams. We used a Bartington Grad601 Fluxgate magnetometer and analysed and visualised the data with the software Geoplot 3.0 [32]. We mapped the precise canal course with the aid of the georeferenced magnetic maps.

Vibra-Coring
The basic data for the 3D-modelling approach represent sedimentary stratigraphic data recovered from vibra-coring, direct push sensing and archaeological excavations (Table 1). For vibra-coring, we used an Atlas Copco Cobra Pro hammer and 60 mm open corer. In total, we drilled 39 cores within the trench fills with coring depths between 200 and 800 cm. Mostly, we could identify the trench bottom macroscopically but we verified it by geochemical analysis. The main contrast of the natural sediments and the first anthropogenic backfills is the organic carbon content, because the Pleistocene sediments are sterile and backfills are TOC (Total organic carbon) enriched [32].

Direct Push Sensing
Direct push sensing is a fast, minimally invasive and depth accurate tool for in-situ characterisation of sediment stratigraphies [50,51]. Steel rods with a small diameter (38 mm) and different probes were pushed into the unconsolidated sediments. For the data acquisition we used the colour logging tool (SCOST™, Dakota Technologies, Fargo, USA) to describe different sediment layers and their colour-dependent properties like organic content or redox characteristics [27,52]. An appropriate pace of measuring (2 cm/s) and an integration time of 300 ms resulted in three values per 2 cm. This high vertical resolution was accompanied by horizontal spacing of a minimum of 12.5 cm. Usually, we used 50 cm spacing. Additionally, an electrical conductivity probe (SC-500, Keijr Engineering Inc.-Geoprobe Systems, USA) provided evidence for grain size changes [53,54]. A Geoprobe 6610DT caterpillar drove the system. For this study, we used 105 direct push colour logs divided in two  (Table 1). The sensing depths ranged between 400 and 800 cm, depending on the specific depth of the canal.

Archaeological Excavations
Based on our detailed geoarchaeological and geophysical survey, we conducted three archaeological excavations in 2013 and 2016, cutting the canal rectangular to the embankments [33,36,37]. The stratigraphy has been documented in detail in the field and validated afterwards by sedimentological analysis (grain size, organic carbon content) as well as archaeobotanical samples [36]. We measured all points of the local network with a Topcon HiPER II D-GPS. We took the absolute position official measuring points. Additionally, we proved the reference heights with an analogue levelling tool. The edges of the Carolingian trench bottom at both banks could be identified precisely with cm-accuracy, because the well-preserved timber revetments show clear signs of decay in the upper part, which was exposed to the water [36]. The trench bottom between both banks has been identified based on initial infills such as sapropel and re-located sandy material with higher organic content compared to the Pleistocene parent material. In 2013, a decimetre-uncertainty of the depth in some parts of the cross-section has been inevitable due to excavation conditions [36]. In 2016, depth accuracy is on a cm-scale in all parts of the cross-section.

Modelling Routine
The main challenge of our study is the combination of geoarchaeological datasets (vibra-coring data, direct push sensing data, archaeological excavation data) and the subsequent integration of the combined data within the pre-modern topography. High-resolution cross-sections of the archaeological excavations and direct push sensing transects are not equally distributed along the canal course. This is mainly due to the high effort of archaeological excavations [33] and the impassibility for the direct push caterpillar in the Central Section. Furthermore, the vibra-coring positions are also not equally distributed throughout the canal. Thus, we developed a modelling approach which suits this challenge, with four steps to create a 3D-model (Figure 4a a) First of all, we created cross-section reference geometries that are based on high-resolution transects. Here, archaeological excavations provided the most precise data in terms of identifying the trench bottom geometry ( Table 1). The spatial and vertical resolution of the direct push transects were also very high. The horizontal spacing was usually 50 cm but could be as small as 12.5 cm. The vertical resolution was at centimetre scale. Hence direct push transects provided significant data for cross-section reference geometries (Figure 4a). In total, we compiled cross-section reference geometries for five cross-sections (a-e in Figure 2). These geometries are also representative for their adjacent sections. b) In the second step, we transferred the cross-section reference geometries to the respective vibra-coring positions (Table 2). Here, we used trench bottom levels from recovered core stratigraphies. The specific reference geometry was adjusted to the level of the trench bottom, inferred from individual cores (Figure 4b). If the trench bottom was deeper than the cross-section reference geometry, we extended the embankments with its specific slope angle until they reached the pre-modern DTM surface. If the trench bottom was above the depth of the cross-section reference geometry, we cut the supernatant embankments at the pre-modern DTM surface.  (a) First of all, we created cross-section reference geometries that are based on high-resolution transects. Here, archaeological excavations provided the most precise data in terms of identifying the trench bottom geometry ( Table 1). The spatial and vertical resolution of the direct push transects were also very high. The horizontal spacing was usually 50 cm but could be as small as 12.5 cm. The vertical resolution was at centimetre scale. Hence direct push transects provided significant data for cross-section reference geometries (Figure 4a). In total, we compiled cross-section reference geometries for five cross-sections (a-e in Figure 2). These geometries are also representative for their adjacent sections.
(b) In the second step, we transferred the cross-section reference geometries to the respective vibra-coring positions (Table 2). Here, we used trench bottom levels from recovered core stratigraphies. The specific reference geometry was adjusted to the level of the trench bottom, inferred from individual cores (Figure 4b). If the trench bottom was deeper than the cross-section reference geometry, we extended the embankments with its specific slope angle until they reached the pre-modern DTM surface. If the trench bottom was above the depth of the cross-section reference geometry, we cut the supernatant embankments at the pre-modern DTM surface. 198 "2016-S1" archaeological excavation 2 North-Eastern Section III (N) 120 "2016-S2" archaeological excavation 4 (c) Subsequently, we created additional transects, which were not based on vibra-coring. These additional transects were important for the interpolation of all transects in the next step. The larger the distance between two transects, the bigger the potential interpolation errors and disturbances [55]. For this reason, we added additional transects equidistantly (~50 m spacing) (Figure 2). The depths were transferred from neighbouring vibra-coring trench bottom levels, direct push sensing transects or archaeological excavations (Figure 4c).
(d) Once all transects had been calculated, we spatially interpolated neighbouring transects via triangulation with an output raster of 0.5 m resolution (Figure 4d). Altogether, we interpolated 73 single segments. Finally, we merged all single segments into one raster dataset and used a low-pass filter (5m radius) to slightly smooth the raster data. The result was a 3D digital terrain model of the Fossa Carolina at its maximum construction depth in a 0.5 x 0.5 m resolution.
(e) and (f) To answer the question of the volumes, we integrated the 3D-model in the pre-modern landscape (Figure 4e) provided by Schmidt et al. [34]. We determined the excavation volume of the canal trench by calculating the difference between the 3D-model and the pre-modern DTM (Figure 4f). The volume of the Fossa Carolina dam remnants was computed as the difference between the present LiDAR DTM and the pre-modern DTM.

Canal Course
We localised the canal course with different prospection methods (SQUID magnetic and Fluxgate magnetic prospection as well as DTM analyses), depending on the data availability. In the Central Section, no geophysical data were available due to wet and barley passable ground conditions. Therefore, we used the middle of the dam ridges as alignment of the canal centre (Figure 5b). In the West-East Section, we used a combination of dam ridge positions and a Fluxgate magnetic map to explore the canal course (Figure 5c). Low relief changes and nearly no visible dams characterise the Northern and North-Eastern Sections. Here, we used SQUID magnetic prospection maps for precisely reconstructing the canal course (Figure 5d).

Cross-Section Reference Geometries
We established five cross-section reference geometries based on two direct push sensing transects and three archaeological excavations. These geometries are the first interim results of our study.
The "WE cross-section" is situated in the western part of the West-East Section of the Fossa Carolina [38] (a in Figure 2; Figure 6). The second direct push sensing cross-section ("The Anomaly") is situated in the Northern Section (b in Figure 2). We detected the deepest level of the trench bottom at 5.5 m depth below modern surface (Figure 7).

Cross-Section Reference Geometries
We established five cross-section reference geometries based on two direct push sensing transects and three archaeological excavations. These geometries are the first interim results of our study.
The "WE cross-section" is situated in the western part of the West-East Section of the Fossa Carolina [38] (a in Figure 2; Figure 6). The second direct push sensing cross-section ("The Anomaly") is situated in the Northern Section (b in Figure 2). We detected the deepest level of the trench bottom at 5.5 m depth below modern surface (Figure 7).  About 300 m further north, we derived a cross-section reference geometry from an archaeological excavation that took place in 2013 (c in Figure 2). The excavation revealed a trench bottom approx. 3 m below surface (Figure 8a). Another excavation took place in 2016 that provided two cross-section reference geometries in the northernmost part of the canal. At excavation "2016 -S1" (d in Figure 2) the maximum depth of the trench bottom is approx. 2 m below surface (Figure 8b) and at excavation "2016 -S2" (e in Figure 2) the trench bottom was recovered at approx. 1.2 m below surface (Figure 8c). The general geometry over all cross-sections looks similar. Both cross-sections "2013" and "2016 -S1" show fairway width of approx. 5m and an almost finished canal construction. In contrast, the cross-section "2016 -S2" reveals a fairway width of just approx. 2.5 m.  About 300 m further north, we derived a cross-section reference geometry from an archaeological excavation that took place in 2013 (c in Figure 2). The excavation revealed a trench bottom approx. 3 m below surface (Figure 8a). Another excavation took place in 2016 that provided two cross-section reference geometries in the northernmost part of the canal. At excavation "2016 -S1" (d in Figure 2) the maximum depth of the trench bottom is approx. 2 m below surface (Figure 8b) and at excavation "2016 -S2" (e in Figure 2) the trench bottom was recovered at approx. 1.2 m below surface (Figure 8c). The general geometry over all cross-sections looks similar. Both cross-sections "2013" and "2016 -S1" show fairway width of approx. 5m and an almost finished canal construction. In contrast, the cross-section "2016 -S2" reveals a fairway width of just approx. 2.5 m. About 300 m further north, we derived a cross-section reference geometry from an archaeological excavation that took place in 2013 (c in Figure 2). The excavation revealed a trench bottom approx. 3 m below surface (Figure 8a). Another excavation took place in 2016 that provided two cross-section reference geometries in the northernmost part of the canal. At excavation "2016-S1" (d in Figure 2) the maximum depth of the trench bottom is approx. 2 m below surface (Figure 8b) and at excavation "2016-S2" (e in Figure 2) the trench bottom was recovered at approx. 1.2 m below surface (Figure 8c). The general geometry over all cross-sections looks similar. Both cross-sections "2013" and "2016-S1" show fairway width of approx. 5m and an almost finished canal construction. In contrast, the cross-section "2016-S2" reveals a fairway width of just approx. 2.5 m.

Application of Cross-Section Reference Geometries to Vibra-Coring and Additive Transects
The application of the cross-section reference geometries to the respective vibra-coring positions and additional trench bottom transects resulted in an interim data set of our study ( Figure  9). In total, we created 26 transects based on vibra-coring positions and 42 additional trench bottom transects.

Application of Cross-Section Reference Geometries to Vibra-Coring and Additive Transects
The application of the cross-section reference geometries to the respective vibra-coring positions and additional trench bottom transects resulted in an interim data set of our study (Figure 9). In total, we created 26 transects based on vibra-coring positions and 42 additional trench bottom transects.

3D-Model
The major step of our study is the development of a 3D-model of the entire Fossa Carolina trench bottom ( Figure 10). The interpolation of the all geometry transects resulted in the smooth integration of the canal geometry in the pre-modern landscape. This model is a raster layer and can be used like a DTM within a GIS environment. It has a spatial resolution of 0.5 x 0.5 m. Figure 9. Exemplary transfer of the cross-section reference geometry "2016/S1" to the vibra-coring transect "Märzkampagne".

3D-Model
The major step of our study is the development of a 3D-model of the entire Fossa Carolina trench bottom ( Figure 10). The interpolation of the all geometry transects resulted in the smooth integration of the canal geometry in the pre-modern landscape. This model is a raster layer and can be used like a DTM within a GIS environment. It has a spatial resolution of 0.5 × 0.5 m.

Volume Calculation
The result of the volume calculation can be separated into two data sets. First, the excavation volume calculation is based on the 3D-model of the Fossa Carolina and pre-modern DTM. The material that was moved during the construction has a volume of approx. 297,000 m³. Second, the present remnants of the dams have a volume of approx. 120,000 m³. This calculation is based on the pre-modern DTM and the present LiDAR DTM.

3D-Modelling Approach and Quality
Applications of GIS in geoarchaeological issues have become common recently [56,57]. The integration of geoarchaeological data with GIS tools offers various possibilities of data management and analysis [58,59]. However, modelling approaches that aim to reconstruct archaeological features are rare. Here, reconstructions are mainly on a small scale and include available spatial geometry information such as geophysics or excavation data. Smedt et al. [24] used a geophysical survey (electromagnetic induction) to reconstruct a medieval wetland reclamation. Further, Diamanti et al. [26] used geophysical survey data from electrical resistivity tomography to reconstruct buried city ruins. In contrast, Pickett et al. [17] reconstructed a medieval burial mound by means of data from archaeological excavations. Studies with a combination of geoarchaeological techniques are lacking.
For the first time, we developed an approach for the integration of different geoarchaeological data for a large-scale feature of 2.9 km length. We think that we produced reliable results but we have to discuss the quality and potential sources of uncertainty. The majority of our input data consists of published geoarchaeological cross-sections, archaeological excavations and a pre-modern

Volume Calculation
The result of the volume calculation can be separated into two data sets. First, the excavation volume calculation is based on the 3D-model of the Fossa Carolina and pre-modern DTM. The material that was moved during the construction has a volume of approx. 297,000 m 3 . Second, the present remnants of the dams have a volume of approx. 120,000 m 3 . This calculation is based on the pre-modern DTM and the present LiDAR DTM.

3D-Modelling Approach and Quality
Applications of GIS in geoarchaeological issues have become common recently [56,57]. The integration of geoarchaeological data with GIS tools offers various possibilities of data management and analysis [58,59]. However, modelling approaches that aim to reconstruct archaeological features are rare. Here, reconstructions are mainly on a small scale and include available spatial geometry information such as geophysics or excavation data. Smedt et al. [24] used a geophysical survey (electromagnetic induction) to reconstruct a medieval wetland reclamation. Further, Diamanti et al. [26] used geophysical survey data from electrical resistivity tomography to reconstruct buried city ruins. In contrast, Pickett et al. [17] reconstructed a medieval burial mound by means of data from archaeological excavations. Studies with a combination of geoarchaeological techniques are lacking.
For the first time, we developed an approach for the integration of different geoarchaeological data for a large-scale feature of 2.9 km length. We think that we produced reliable results but we have to discuss the quality and potential sources of uncertainty. The majority of our input data consists of published geoarchaeological cross-sections, archaeological excavations and a pre-modern DTM (Table 1). Direct push sensing data and geometric information derived from archaeological excavations provide excellethe trench bottom was recovered at approx. 1.2 m below nt depth accuracies. Vibra-coring data may have a coarser vertical resolution and uncertainties in depth accuracy due to the compaction of organic sediments [27]. The overall Root Mean Square Error (RMSE) of the pre-modern DTM [34] was calculated using vibra-coring, direct push sensing and archaeological excavation data that recovered a buried paleosol and, therefore, the former level of the pre-modern surface. The RMSE of 0.69 shows a general overestimation of the modelled pre-modern DTM levels (69 cm). Hence, the computed volume of the Fossa Carolina excavation material can be slightly lower in contrast to the calculated volume of the dams which would increase.

The Scientific History of Fossa Carolina Volume Calculations
First of all, Birzer [10] assumed an excavation volume only for the Central and West-East Section of approx. 80,000 m 3 . As the author had no reliable information about the total length of the canal, he estimated in the next step a 4.5 km long canal course with a constant trench bottom level. As a result, he estimated a total volume of approx. 450,000 m 3 for the entire canal course (Table 3). Hofmann [60] worked in detail with the building energetics. He hypothesised a single trench bottom level and no summit concept. With a length of 1.4 km, a 30 m width and an assumed depth of 6 m, he calculated a volume of approx. 130,000 m 3 . The most recent volume estimation is from Koch [8]. He drilled several cores and recovered that the detected trench bottom levels do not reach the level of the Altmühl River. Therefore, he concluded that the canal was presumably constructed as a summit canal and carefully assumed roughly several 100,000 m 3 of excavation material (Table 3). Our study presents for the first time a 3D-modelling approach that is based on precise excavation, vibra-coring and direct push sensing data sets. The resulting volume of approx. 297,000 m 3 represents an improved calculation in comparison with former estimations. With respect to the summit concept, no other authors assumed a volume as large as we calculated. We can summarise that the former lack of geoarchaeological data led to uncertain conclusions. Furthermore, this precise volume calculation is key for reliable future modelling of building energetics of this unique canal construction.
The spatial distribution of the excavated volume shows that the majority (54%) was excavated in the Central Section ( Figure 11). Further, the West-East Section has a proportion of roughly 32% of the volume and only 14% corresponds with the Northern and North-Eastern Sections (Table 4).

Where Has All the Material Gone?
It is obvious that more than half of the excavated volume is not stored in the remaining dams. Vibra-coring [40], direct push sensing [38] and archaeological excavation data [33] show that significant amounts of excavated material was already washed back into the open trench shortly after the construction site was abandoned. Upper trench fills reflect relocated material originating from the adjacent dams [32]. The canal and the corresponding dams were fundamentally modified even during modern times. Especially the northern canal sections, which were levelled for

Where Has All the Material Gone?
It is obvious that more than half of the excavated volume is not stored in the remaining dams. Vibra-coring [40], direct push sensing [38] and archaeological excavation data [33] show that significant amounts of excavated material was already washed back into the open trench shortly after the construction site was abandoned. Upper trench fills reflect relocated material originating from the adjacent dams [32]. The canal and the corresponding dams were fundamentally modified even during modern times. Especially the northern canal sections, which were levelled for agricultural purposes [61] and missing dam volumes in the central canal sections result from massive sand and loam mining activities for modern infrastructure and buildings [6,34].

Conclusions
Our 3D-modelling approach of the Early Medieval Fossa Carolina integrates archaeological excavations, direct push sensing and vibra-coring techniques, as well as the present LiDAR DTM, Fluxgate and SQUID magnetic surveys and a paleo-surface (pre-modern DTM) of the study area. We identified the buried canal trench by LiDAR DTM analysis of the present dam remnants and interpretation of magnetic survey maps. We transferred cross-section reference geometries (derived from archaeological excavations and direct push sensing transects) to vibra-coring positions. Finally, we interpolated the canal trench geometry to create the 3D-model of the Carolingian canal. The spatial and vertical accuracy of the model depends on the quality of its input data. The reference cross-sections are of high quality as well as the LiDAR DTM and pre-modern DTM. Because of the large spatial extent of the canal, we included several vibra-corings, which have at least cm depth accuracy. Our modelling routine minimises the uncertainties by creating cross-section reference geometries.
For the first time, the 3D-model provides a data-based calculation of the amount of material moved during the Early Medieval construction. The calculation of the earth volume was done by calculating the difference between the 3D-model and the pre-modern digital terrain model. The pre-modern DTM reflects a deconstructed landscape nearly free of human induced terrain changes. Altogether, approx. 297,000 m 3 of material was excavated during the Early Medieval construction time.
In comparison, we calculated the volume of the preserved dams. This was also done by subtracting the present shape from the pre-modern DTM. Approximately 120,000 m 3 material still remains in the dams. Nevertheless, more than the half of the excavation material was eroded, redistributed, or backfilled in the canal.