title: An analysis of total correctness refinement models for partial relation semantics I

authors: Deutsch, Moshe

subjects: operation refinement

abstract: This is the first of a series of papers devoted to the thorough investigation of (total correctness) refinement based on an underlying partial relational model. In this paper we restrict attention to operation refinement. We explore four theories of refinement based on an underlying partial relation model for specifications, and we show that they are all equivalent. This, in particular, sheds some light on the relational completion operator (lifted-totalisation) due to Wookcock which underlines data refinement in, for example, the specification language Z. It further leads to two simple alternative models which are also equivalent to the others.

text: An analysis of total correctness refinement models for partial relation semantics I This is the first of a series of papers devoted to the thorough investigation of (total correctness) refinement based on an underlying partial relational model. In this paper we restrict attention to operation refinement. We explore four theories of refinement based on an underlying partial relation model for specifications, and we show that they are all equivalent. This, in particular, sheds some light on the relational completion operator (lifted-totalisation) due to Wookcock which underlines data refinement in, for example, the specification language Z. It further leads to two simple alternative models which are also equivalent to the others.

year: 2003

title: Uplift history and geomorphic development of the Southern Alps, New Zealand, assessed by fission track analysis

authors: Tippett, J. Mark

abstract: The uplift history and geomorphic development of the Southern Alps has been studied, using fission track analysis of basement samples in combination with measurement of the present geomorphology of the Southern Alps. 

During the late Cenozoic the Pacific plate has been converging obliquely with the Australia plate in South Island, New Zealand. A result of this convergence has been the growth of a major mountain range - the Southern Alps - at the leading edge of the Pacific plate. The results of fission track analysis of 140 samples from 13 transects across the 300 km-long central sector of the Southern Alps reported here, establish the late Cenozoic vertical kinematics (amount, age, and rate of rock uplift) of the Pacific crust underlying the Southern Alps. The pattern of late Cenozoic rock uplift is asymmetrical with respect to the Alpine Fault, being a maximum (19 km) immediately east of the central part of the Fault, with lesser values at the eastern (3 km), northern (10 km) and southern (8 km) extremities of the Southern Alps. The age of the start of rock uplift varies spatially across the Southern Alps, the earliest indications from fission track analysis being at 8 Ma ago at the southern end of the Southern Alps, decreasing to 5 Ma ago at the northern end and 3 Ma ago along the southeastern margin. This age variation reflects the longer time over which the southern parts of the Southern Alps have been in collision. The rate of propagation of rock uplift southeastwards into the Pacific plate has been 30 mm/a, nearly 4 times the late Cenozoic average rate of convergence normal to the plate boundary. Late Cenozoic mean rock uplift rates range from a maximum of ∼2.8 mm/a at the Alpine Fault to a minimum of ∼1.0 mm/a in the east. Uplift has accelerated over time, but only significantly since 1.3 ± 0.3 Ma ago. 

The fission track data in combination with topographic data have been analysed to quantify the response of the surface of the Pacific plate to the late Cenozoic continent-continent collision. The rates of mean surface uplift, the rates of work done against gravity during the mountain building, and the erosional, isostatic and tectonic components (amounts and rates) of rock uplift, have been derived and mapped over the central 300 km length of the Southern Alps. The data do not support the notion of erosional-mechanical coupling of the two wedges in the two-sided orogen model of Koons [1990], but are consistent with overriding of the Australia plate by the Pacific plate. The late Cenozoic rate of mean surface uplift in the Southern Alps ranges from <0.1 mm/a adjacent to the Alpine Fault to >0.3 mm/a over most of the area east of the Main Divide. The highest rates of mean surface uplift do not coincide spatially with the regions of highest mean elevation. The rate of work done in elevating the mean surface ranges from ∼2.5 mWm⁻² to ∼10 mWm⁻², although most areas have experienced similar rates of energy input of ∼7.5 mWm⁻². The amount of denudation ranges from ∼18 km adjacent to the Alpine Fault to ∼2 km along the southeast margin of the Southern Alps; the late Cenozoic mean rate of denudation ranges from ∼2.5 mm/a at the Fault to ∼0.5 mm/a in the southeast parts of the Southern Alps. The amount of tectonic uplift decreases from ∼4 km at the Alpine Fault to ∼1 km along the eastern margin of the Southern Alps. 

The fission track data have been used to assess the role of faulting in the late Cenozoic uplift of the Pacific plate crust. For the Moonlight, Ostler, Harper, Torlesse, Porter's Pass and Hope faults, the vertical offset in all cases lies within the uncertainty of the data, typically ± 2 km, and is less than 30% of the surrounding uplift. Over a scale of kilometres the uplift is continuous and regular, and deformation in the plate boundary zone can therefore be treated as if it were ductile. 

The regression of the mean surface, summit, and valley elevations on the amount, age, and rate of uplift for each of 82 sample sites is used to establish the nature of the relationship between uplift and geomorphology. The preferred regression models have uniform slope but varying elevation response between transects. Uplift explains 80-90 % of the variation in elevation. Substitution of space for time has allowed the evolution of landforms to be studied. To the east of the Main Divide, elevation and relief are proportional to and closely related to the age of initiation of rock uplift and the amount of rock uplift (r²>0.8). Mean surface uplift was delayed for ∼2 Ma after the start of rock uplift, a result of the stripping of a soft cover rock succession that prior to rock uplift overlaid the harder greywacke basement. Under a rock uplift rate of 0.8 km/Ma, uplift of the mean surface proceeded at 0.4 km/Ma, while the summits increased in elevation at a rate of 0.6 km/Ma and valleys increased in elevation at 0.2 km/Ma. The mountains east of the Main Divide have continued to increase in elevation and relief and evolve in form over time since the start of uplift. Mountain elevation has little relationship with late Cenozoic mean rock uplift rates of 0.8-1.0 km/Ma or inferred contemporary rock uplift rates (r²∼0.3). In contrast, to the west of the Main Divide, elevation is shown to be closely related to rock uplift rate (r²>0.8). Transects with higher rock uplift rates support higher topography. Landforms are therefore in a stable equilibrium with rock uplift rate, and the landscape contains no residual evidence of the total amount of rock uplift, or the age of uplift. Lithological variation appears to have no relationship with elevation. 

The new uplift and geomorphic data allow the construction of a new model of the geomorphic evolution of the Southern Alps. The model quantifies the development over time and space of rock uplift, mean surface elevation, exhumation of crustal section, and relief. The earliest indications of mean surface uplift are between 4 and 5 Ma ago at the Alpine Fault. Mean surface uplift propagated southeastward from the Alpine Fault at a rate of 30 km/Ma. At 3 Ma ago greenschist was exposed in the southern parts of the Southern Alps near Lake Wanaka, and since then has become exhumed along a narrow strip east of the Alpine Fault. Amphibolite grade schist has been exhumed adjacent to the Alpine Fault only in the last 0.3 Ma. At the southeast margin of the Southern Alps a 1.0-1.5 km step in the basement topography separates the Southern Alps from the non-uplifted plains to the east. The existing models of the geomorphic development of the Southern Alps - the dynamic cuesta model of J. Adams [1980 and 1985] and the numerical model of Koons [1989] - are compared with the new data. Particular constraints unrealised by these two earlier models include: the earlier age of rock uplift (8 Ma ago) and its spatial variation (8 Ma ago to 3 Ma ago) across the Southern Alps; the lower long-term rock uplift rate (0.8- 1.0 mm/a) of the Southern Alps for most of the late Cenozoic; the lag between the start of rock uplift and the start of mean surface uplift; and the patterns of rock uplift and erosion across the Southern Alps. The age of the start of rock uplift, and the amount and rate of rock uplift, all of which vary spatially, are considered to be the dominant influences on the development of the landscape in the Southern Alps.

text: Uplift history and geomorphic development of the Southern Alps, New Zealand, assessed by fission track analysis The uplift history and geomorphic development of the Southern Alps has been studied, using fission track analysis of basement samples in combination with measurement of the present geomorphology of the Southern Alps. 

During the late Cenozoic the Pacific plate has been converging obliquely with the Australia plate in South Island, New Zealand. A result of this convergence has been the growth of a major mountain range - the Southern Alps - at the leading edge of the Pacific plate. The results of fission track analysis of 140 samples from 13 transects across the 300 km-long central sector of the Southern Alps reported here, establish the late Cenozoic vertical kinematics (amount, age, and rate of rock uplift) of the Pacific crust underlying the Southern Alps. The pattern of late Cenozoic rock uplift is asymmetrical with respect to the Alpine Fault, being a maximum (19 km) immediately east of the central part of the Fault, with lesser values at the eastern (3 km), northern (10 km) and southern (8 km) extremities of the Southern Alps. The age of the start of rock uplift varies spatially across the Southern Alps, the earliest indications from fission track analysis being at 8 Ma ago at the southern end of the Southern Alps, decreasing to 5 Ma ago at the northern end and 3 Ma ago along the southeastern margin. This age variation reflects the longer time over which the southern parts of the Southern Alps have been in collision. The rate of propagation of rock uplift southeastwards into the Pacific plate has been 30 mm/a, nearly 4 times the late Cenozoic average rate of convergence normal to the plate boundary. Late Cenozoic mean rock uplift rates range from a maximum of ∼2.8 mm/a at the Alpine Fault to a minimum of ∼1.0 mm/a in the east. Uplift has accelerated over time, but only significantly since 1.3 ± 0.3 Ma ago. 

The fission track data in combination with topographic data have been analysed to quantify the response of the surface of the Pacific plate to the late Cenozoic continent-continent collision. The rates of mean surface uplift, the rates of work done against gravity during the mountain building, and the erosional, isostatic and tectonic components (amounts and rates) of rock uplift, have been derived and mapped over the central 300 km length of the Southern Alps. The data do not support the notion of erosional-mechanical coupling of the two wedges in the two-sided orogen model of Koons [1990], but are consistent with overriding of the Australia plate by the Pacific plate. The late Cenozoic rate of mean surface uplift in the Southern Alps ranges from <0.1 mm/a adjacent to the Alpine Fault to >0.3 mm/a over most of the area east of the Main Divide. The highest rates of mean surface uplift do not coincide spatially with the regions of highest mean elevation. The rate of work done in elevating the mean surface ranges from ∼2.5 mWm⁻² to ∼10 mWm⁻², although most areas have experienced similar rates of energy input of ∼7.5 mWm⁻². The amount of denudation ranges from ∼18 km adjacent to the Alpine Fault to ∼2 km along the southeast margin of the Southern Alps; the late Cenozoic mean rate of denudation ranges from ∼2.5 mm/a at the Fault to ∼0.5 mm/a in the southeast parts of the Southern Alps. The amount of tectonic uplift decreases from ∼4 km at the Alpine Fault to ∼1 km along the eastern margin of the Southern Alps. 

The fission track data have been used to assess the role of faulting in the late Cenozoic uplift of the Pacific plate crust. For the Moonlight, Ostler, Harper, Torlesse, Porter's Pass and Hope faults, the vertical offset in all cases lies within the uncertainty of the data, typically ± 2 km, and is less than 30% of the surrounding uplift. Over a scale of kilometres the uplift is continuous and regular, and deformation in the plate boundary zone can therefore be treated as if it were ductile. 

The regression of the mean surface, summit, and valley elevations on the amount, age, and rate of uplift for each of 82 sample sites is used to establish the nature of the relationship between uplift and geomorphology. The preferred regression models have uniform slope but varying elevation response between transects. Uplift explains 80-90 % of the variation in elevation. Substitution of space for time has allowed the evolution of landforms to be studied. To the east of the Main Divide, elevation and relief are proportional to and closely related to the age of initiation of rock uplift and the amount of rock uplift (r²>0.8). Mean surface uplift was delayed for ∼2 Ma after the start of rock uplift, a result of the stripping of a soft cover rock succession that prior to rock uplift overlaid the harder greywacke basement. Under a rock uplift rate of 0.8 km/Ma, uplift of the mean surface proceeded at 0.4 km/Ma, while the summits increased in elevation at a rate of 0.6 km/Ma and valleys increased in elevation at 0.2 km/Ma. The mountains east of the Main Divide have continued to increase in elevation and relief and evolve in form over time since the start of uplift. Mountain elevation has little relationship with late Cenozoic mean rock uplift rates of 0.8-1.0 km/Ma or inferred contemporary rock uplift rates (r²∼0.3). In contrast, to the west of the Main Divide, elevation is shown to be closely related to rock uplift rate (r²>0.8). Transects with higher rock uplift rates support higher topography. Landforms are therefore in a stable equilibrium with rock uplift rate, and the landscape contains no residual evidence of the total amount of rock uplift, or the age of uplift. Lithological variation appears to have no relationship with elevation. 

The new uplift and geomorphic data allow the construction of a new model of the geomorphic evolution of the Southern Alps. The model quantifies the development over time and space of rock uplift, mean surface elevation, exhumation of crustal section, and relief. The earliest indications of mean surface uplift are between 4 and 5 Ma ago at the Alpine Fault. Mean surface uplift propagated southeastward from the Alpine Fault at a rate of 30 km/Ma. At 3 Ma ago greenschist was exposed in the southern parts of the Southern Alps near Lake Wanaka, and since then has become exhumed along a narrow strip east of the Alpine Fault. Amphibolite grade schist has been exhumed adjacent to the Alpine Fault only in the last 0.3 Ma. At the southeast margin of the Southern Alps a 1.0-1.5 km step in the basement topography separates the Southern Alps from the non-uplifted plains to the east. The existing models of the geomorphic development of the Southern Alps - the dynamic cuesta model of J. Adams [1980 and 1985] and the numerical model of Koons [1989] - are compared with the new data. Particular constraints unrealised by these two earlier models include: the earlier age of rock uplift (8 Ma ago) and its spatial variation (8 Ma ago to 3 Ma ago) across the Southern Alps; the lower long-term rock uplift rate (0.8- 1.0 mm/a) of the Southern Alps for most of the late Cenozoic; the lag between the start of rock uplift and the start of mean surface uplift; and the patterns of rock uplift and erosion across the Southern Alps. The age of the start of rock uplift, and the amount and rate of rock uplift, all of which vary spatially, are considered to be the dominant influences on the development of the landscape in the Southern Alps.

year: 1992

title: Aihe Pepene

authors: Hart, Philip

abstract: Sharing the same illustrious whakapapa as his brother Reha Aperahama, Aihe Pepene was prominent when living in the Thames district before settling at Te Aroha in 1878. When speculators tried to acquire the Thames foreshore from Pepene and others in 1870, he became involved in the subsequent legal actions over unpaid promissory notes. Later he would acquire interests in many blocks of land, and received a steady income by leasing or selling these plus his share in the goldfields revenue. When the Aroha Block was considered by the land court for the last time, in 1878, he conducted the case for Ngati Rahiri.
He invested in four Hauraki goldfields, and was briefly an owner and skipper of two small river steamers, an unsuccessful endeavour that resulted in his being forced to sell more land to meet his debts. Once his 1880 and 1881 financial difficulties were resolved, no more such problems recurred. 
A leading rangatira in Hauraki generally as well as at Te Aroha, he would be elected to a Maori Committee that was soon revealed to have no significance. Closely involved with Pakeha, he assisted settlement, and his loyalty to the Crown was illustrated by his becoming an officer in the Thames Native Volunteer Corps. A member of the Church of England for many years, like many Maori he became a Mormon, a faith perhaps more appropriate to his private life, for he had more than one wife, notoriously eloping with the wife of a more senior rangatira despite already being married.

text: Aihe Pepene Sharing the same illustrious whakapapa as his brother Reha Aperahama, Aihe Pepene was prominent when living in the Thames district before settling at Te Aroha in 1878. When speculators tried to acquire the Thames foreshore from Pepene and others in 1870, he became involved in the subsequent legal actions over unpaid promissory notes. Later he would acquire interests in many blocks of land, and received a steady income by leasing or selling these plus his share in the goldfields revenue. When the Aroha Block was considered by the land court for the last time, in 1878, he conducted the case for Ngati Rahiri.
He invested in four Hauraki goldfields, and was briefly an owner and skipper of two small river steamers, an unsuccessful endeavour that resulted in his being forced to sell more land to meet his debts. Once his 1880 and 1881 financial difficulties were resolved, no more such problems recurred. 
A leading rangatira in Hauraki generally as well as at Te Aroha, he would be elected to a Maori Committee that was soon revealed to have no significance. Closely involved with Pakeha, he assisted settlement, and his loyalty to the Crown was illustrated by his becoming an officer in the Thames Native Volunteer Corps. A member of the Church of England for many years, like many Maori he became a Mormon, a faith perhaps more appropriate to his private life, for he had more than one wife, notoriously eloping with the wife of a more senior rangatira despite already being married.

year: 2016