title: Multi-latin squares

authors: Cavenagh, Nicholas J.

subjects: latin square

abstract: A multi-latin square of order n and index k is an n×n array of multisets, each of cardinality k, such that each symbol from a fixed set of size n occurs k times in each row and k times in each column. A multi-latin square of index k is also referred to as a k-latin square. A 1-latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin square.
In this note we show that any partially filled-in k-latin square of order m embeds in a k-latin square of order n, for each n≥2m, thus generalizing Evans’ Theorem. Exploiting this result, we show that there exist non-separable k-latin squares of order n for each n≥k+2. We also show that for each n≥1, there exists some finite value g(n) such that for all k≥g(n), every k-latin square of order n is separable.
We discuss the connection between k-latin squares and related combinatorial objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares and k-latin trades. We also enumerate and classify k-latin squares of small orders.

text: Multi-latin squares A multi-latin square of order n and index k is an n×n array of multisets, each of cardinality k, such that each symbol from a fixed set of size n occurs k times in each row and k times in each column. A multi-latin square of index k is also referred to as a k-latin square. A 1-latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin square.
In this note we show that any partially filled-in k-latin square of order m embeds in a k-latin square of order n, for each n≥2m, thus generalizing Evans’ Theorem. Exploiting this result, we show that there exist non-separable k-latin squares of order n for each n≥k+2. We also show that for each n≥1, there exists some finite value g(n) such that for all k≥g(n), every k-latin square of order n is separable.
We discuss the connection between k-latin squares and related combinatorial objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares and k-latin trades. We also enumerate and classify k-latin squares of small orders.

year: 2010

title: Ecological effects of turbidity variations in and around dredging  areas in the Port of Tauranga

authors: Coppede Cussioli, Mariana

subjects: Dredging

abstract: Turbidity is the scattering of light in water bodies and is an important measurement for assessing water quality in coastal systems. Suspended particles in the water column can greatly impact on light penetration and measured turbidity levels. These suspended particles can originate from a range of natural and anthropogenic sources. This can include land use changes and soil erosion from surrounding catchment areas as well as resuspension from storms and dredging activities along coastlines. Although increased turbidity has the potential to affect coastal ecosystems, the interactions of different drivers have been poorly investigated. The objective of this study was to understand the sources, patterns and potential impacts of increased turbidity in a barrier-enclosed shallow lagoon. The study area, Tauranga Harbour (New Zealand) is a system affected by multiple stressors such as urban developments, industry, forestry, agricultural land use and a port facility. The Port of Tauranga is the largest export port in New Zealand and carries out regular maintenance dredging in the shipping channels. In this thesis, I focused on the effects of increased turbidity caused by the plumes generated during dredging activities and assessed the significance of these turbidity levels relative to background sediment inputs. The main body of this thesis covers three main areas: (1) the effects of turbidity on light attenuation (both light quantity and quality), (2) physiological response of sensitive species (Paphies australis) to increased turbidity, and (3) monitoring of dredging activity and plume footprints. 
Benthic plants such as seagrasses depend on light availability, which is an important controlling factor for primary production and ecological health. To determine the drivers modulating the light attenuation coefficient Kd(PAR) in the harbour, I carried out low-frequency (bi-monthly) measurements of light irradiance, suspended sediment concentration (TSS), chlorophyll-a and coloured dissolved organic matter (CDOM). I also measured light irradiance before and after dredging activities. Using these measurements in a multiple regression analysis allowed the main contributor to light attenuation in the harbour to be determined. Correlating the light attenuation coefficients from field measurements to turbidity levels recorded by turbidity sensors, I derived a regression model whereby turbidity data can be used as a proxy to estimate Kd(PAR). The turbidity records were collected by an array of six high-frequency sensors, deployed by the Port of Tauranga, which have been operating for approximately 3 years. The Kd(PAR) dataset derived from the turbidity measurements allowed the effect of storms and other relevant events such as dredging on light conditions to be assessed. The estimates of Kd(PAR) levels using this high-frequency dataset were considerably higher than those from the low-frequency dataset. Using these more representative Kd(PAR) values, I calculated thresholds of turbidity based on light requirements of New Zealand seagrass species, Zostera muelleri.
The influence of suspended material in the water column and its effect on light quality can largely depend on its origins (i.e. marine sources, such as dredging or terrestrial material from surrounding catchments). Terrestrial sediments usually differ in colour from marine sediments. Therefore, I investigated how different sediment colours (orange, grey and white), which were from different origins, affected underwater light quality. Results from a previous experiment using a modified water-holding tank and new spectrophotometer measurements showed that terrestrial based orange sediments changed the light quality more and filtered an exclusive range of wavelengths. The resultant wavelengths available were shown to be less effective for photosynthesis of some species, such as seagrasses. Among the sediments from marine sources, white sediments attenuated light more effectively compared to grey sediments; however, the spectral distribution of light was not modified by changes in suspended sediment concentration. 
Based on the range of turbidity experienced in estuarine waters in New Zealand, considering both background values and maintenance dredging events, I tested six treatments containing different TSS on the bivalve Paphies australis (pipi). The aim was to predict the short-term effects of increased TSS on the feeding behaviour of pipis and to model these responses to estimate threshold values. Pipis, like other species of bivalves, responded to increased sediment concentrations by using adaptive mechanisms, such as reduction in clearance rates and productions of pseudofaeces. These mechanisms showed efficiencies in increasing the quality of food ingested by pipis and thus regulating their energy acquisition in high turbidity treatments. However, above a threshold, responses in feeding rates indicated limitations of particle selection mechanisms. This suggests that further increases in sediment concentration could potentially constrain food acquisition and reduce pipi biomass. By including several feeding and digestion rates that have not been previously measured in pipis, this study contributes to modelling energetics of bivalves and in setting environmental limits for human activities in estuaries and harbours. 
With a clearer understanding of the effect of TSS on light conditions and pipi condition, I then determined the spatial and temporal footprint of the dredging plume. To do this, I monitored the 2014 maintenance dredging in Tauranga Harbour and used a process-based numerical modelling system (Delft3D) to simulate dredging plumes. To acquire observational data in the field, a boat-mounted acoustic Doppler current profiler (ADCP) recorded backscatter signals. These were converted to suspended sediment concentrations (TSS) using calibrations developed with water samples. The ADCP transects were carried out before, during, and after dredging within the direct dredging area and along the plume. These transects provided information about plume development with time and distance from the dredging area and were used in the model calibration and validation. Based on the length and width of plume footprints, I proposed the use of an index of plume symmetry to define vulnerability zones around dredging areas. The index showed the main deposition paths and how dredging location affected the plume footprint. The primary and secondary axis lengths were used to define areas of vulnerability, which were then related to sensitive groups of species in the harbour. From ADCP transects and model results, TSS in plumes and its quick dissipation time characterized the maintenance dredging plumes as having a low impact on the two species that were identified as vulnerable to dredging in Tauranga Harbour: seagrass Z. muelleri and bivalve P. Australis (pipi). The maximum sediment deposition from dredging was restricted directly within the dredging areas. The thickness of deposits under plumes that might have reached seagrass meadows were below thresholds that were likely to impact growth rates of Z. muelleri. However, plumes from terrestrial sources, due to its colour, can have a broader effect on seagrass photosynthesis compared with resuspended marine sediments.
This thesis attempts to provide a comprehensive understanding of turbidity variations and the associated ecological effects. It presents a number of important innovations in the field, including: (1) the development of a relationship between underwater light attenuation coefficients and turbidity; (2) the modelling of feeding and digestion rates of pipis, which have not been previously tested; (3) the development of a ‘plume symmetry’ index, and (4) the response of underwater light quality to sediment concentration and colour. It is recommended that future research adopts a greater and more regular sampling frequency in light and ecological measurements across coastal regions to better assess the interactions between natural and human induced changes.

text: Ecological effects of turbidity variations in and around dredging  areas in the Port of Tauranga Turbidity is the scattering of light in water bodies and is an important measurement for assessing water quality in coastal systems. Suspended particles in the water column can greatly impact on light penetration and measured turbidity levels. These suspended particles can originate from a range of natural and anthropogenic sources. This can include land use changes and soil erosion from surrounding catchment areas as well as resuspension from storms and dredging activities along coastlines. Although increased turbidity has the potential to affect coastal ecosystems, the interactions of different drivers have been poorly investigated. The objective of this study was to understand the sources, patterns and potential impacts of increased turbidity in a barrier-enclosed shallow lagoon. The study area, Tauranga Harbour (New Zealand) is a system affected by multiple stressors such as urban developments, industry, forestry, agricultural land use and a port facility. The Port of Tauranga is the largest export port in New Zealand and carries out regular maintenance dredging in the shipping channels. In this thesis, I focused on the effects of increased turbidity caused by the plumes generated during dredging activities and assessed the significance of these turbidity levels relative to background sediment inputs. The main body of this thesis covers three main areas: (1) the effects of turbidity on light attenuation (both light quantity and quality), (2) physiological response of sensitive species (Paphies australis) to increased turbidity, and (3) monitoring of dredging activity and plume footprints. 
Benthic plants such as seagrasses depend on light availability, which is an important controlling factor for primary production and ecological health. To determine the drivers modulating the light attenuation coefficient Kd(PAR) in the harbour, I carried out low-frequency (bi-monthly) measurements of light irradiance, suspended sediment concentration (TSS), chlorophyll-a and coloured dissolved organic matter (CDOM). I also measured light irradiance before and after dredging activities. Using these measurements in a multiple regression analysis allowed the main contributor to light attenuation in the harbour to be determined. Correlating the light attenuation coefficients from field measurements to turbidity levels recorded by turbidity sensors, I derived a regression model whereby turbidity data can be used as a proxy to estimate Kd(PAR). The turbidity records were collected by an array of six high-frequency sensors, deployed by the Port of Tauranga, which have been operating for approximately 3 years. The Kd(PAR) dataset derived from the turbidity measurements allowed the effect of storms and other relevant events such as dredging on light conditions to be assessed. The estimates of Kd(PAR) levels using this high-frequency dataset were considerably higher than those from the low-frequency dataset. Using these more representative Kd(PAR) values, I calculated thresholds of turbidity based on light requirements of New Zealand seagrass species, Zostera muelleri.
The influence of suspended material in the water column and its effect on light quality can largely depend on its origins (i.e. marine sources, such as dredging or terrestrial material from surrounding catchments). Terrestrial sediments usually differ in colour from marine sediments. Therefore, I investigated how different sediment colours (orange, grey and white), which were from different origins, affected underwater light quality. Results from a previous experiment using a modified water-holding tank and new spectrophotometer measurements showed that terrestrial based orange sediments changed the light quality more and filtered an exclusive range of wavelengths. The resultant wavelengths available were shown to be less effective for photosynthesis of some species, such as seagrasses. Among the sediments from marine sources, white sediments attenuated light more effectively compared to grey sediments; however, the spectral distribution of light was not modified by changes in suspended sediment concentration. 
Based on the range of turbidity experienced in estuarine waters in New Zealand, considering both background values and maintenance dredging events, I tested six treatments containing different TSS on the bivalve Paphies australis (pipi). The aim was to predict the short-term effects of increased TSS on the feeding behaviour of pipis and to model these responses to estimate threshold values. Pipis, like other species of bivalves, responded to increased sediment concentrations by using adaptive mechanisms, such as reduction in clearance rates and productions of pseudofaeces. These mechanisms showed efficiencies in increasing the quality of food ingested by pipis and thus regulating their energy acquisition in high turbidity treatments. However, above a threshold, responses in feeding rates indicated limitations of particle selection mechanisms. This suggests that further increases in sediment concentration could potentially constrain food acquisition and reduce pipi biomass. By including several feeding and digestion rates that have not been previously measured in pipis, this study contributes to modelling energetics of bivalves and in setting environmental limits for human activities in estuaries and harbours. 
With a clearer understanding of the effect of TSS on light conditions and pipi condition, I then determined the spatial and temporal footprint of the dredging plume. To do this, I monitored the 2014 maintenance dredging in Tauranga Harbour and used a process-based numerical modelling system (Delft3D) to simulate dredging plumes. To acquire observational data in the field, a boat-mounted acoustic Doppler current profiler (ADCP) recorded backscatter signals. These were converted to suspended sediment concentrations (TSS) using calibrations developed with water samples. The ADCP transects were carried out before, during, and after dredging within the direct dredging area and along the plume. These transects provided information about plume development with time and distance from the dredging area and were used in the model calibration and validation. Based on the length and width of plume footprints, I proposed the use of an index of plume symmetry to define vulnerability zones around dredging areas. The index showed the main deposition paths and how dredging location affected the plume footprint. The primary and secondary axis lengths were used to define areas of vulnerability, which were then related to sensitive groups of species in the harbour. From ADCP transects and model results, TSS in plumes and its quick dissipation time characterized the maintenance dredging plumes as having a low impact on the two species that were identified as vulnerable to dredging in Tauranga Harbour: seagrass Z. muelleri and bivalve P. Australis (pipi). The maximum sediment deposition from dredging was restricted directly within the dredging areas. The thickness of deposits under plumes that might have reached seagrass meadows were below thresholds that were likely to impact growth rates of Z. muelleri. However, plumes from terrestrial sources, due to its colour, can have a broader effect on seagrass photosynthesis compared with resuspended marine sediments.
This thesis attempts to provide a comprehensive understanding of turbidity variations and the associated ecological effects. It presents a number of important innovations in the field, including: (1) the development of a relationship between underwater light attenuation coefficients and turbidity; (2) the modelling of feeding and digestion rates of pipis, which have not been previously tested; (3) the development of a ‘plume symmetry’ index, and (4) the response of underwater light quality to sediment concentration and colour. It is recommended that future research adopts a greater and more regular sampling frequency in light and ecological measurements across coastal regions to better assess the interactions between natural and human induced changes.

year: 2018

title: Taking Taniwha seriously

authors: Kingsbury, Justine

subjects: intercultural understanding

abstract: Taniwha are powerful water creatures in te ao Māori (the Māori world/worldview). Taniwha sometimes afect public works in Aotearoa New Zealand: for example, consultation between government agencies and tangata whenua (the people of the land) about proposed roading developments sometimes results in the route being moved to avoid the dwelling place of a taniwha. Mainstream media responses have tended to be hostile or mocking, as you might expect, since on the face of it the dominant western scientifc worldview has no place for beings like taniwha. However, in the 2020s, there appears to be an increased willingness to engage with te ao Māori. In this spirit, this paper proposes a way for non-Māori to begin to take taniwha more seriously, taking as its starting point the work of Dan Hikuroa on the practical usefulness of taniwha pūrākau (traditional narratives) in encoding information about natural hazards. The focus of this paper is narrow, but aspects of the strategy it proposes may be generalisable both to other aspects of te ao Māori and to other bicultural and multicultural contexts.

text: Taking Taniwha seriously Taniwha are powerful water creatures in te ao Māori (the Māori world/worldview). Taniwha sometimes afect public works in Aotearoa New Zealand: for example, consultation between government agencies and tangata whenua (the people of the land) about proposed roading developments sometimes results in the route being moved to avoid the dwelling place of a taniwha. Mainstream media responses have tended to be hostile or mocking, as you might expect, since on the face of it the dominant western scientifc worldview has no place for beings like taniwha. However, in the 2020s, there appears to be an increased willingness to engage with te ao Māori. In this spirit, this paper proposes a way for non-Māori to begin to take taniwha more seriously, taking as its starting point the work of Dan Hikuroa on the practical usefulness of taniwha pūrākau (traditional narratives) in encoding information about natural hazards. The focus of this paper is narrow, but aspects of the strategy it proposes may be generalisable both to other aspects of te ao Māori and to other bicultural and multicultural contexts.

year: 2022-11-10