Mapping complex biological processes across the landscape: the problem of non-stationarity

许多重要的过程都发生在地貌中。例如，某些土壤细菌、反硝化细菌在其呼吸作用中使用硝酸盐。其中一个产品是一氧化二氮，一种二氧化碳对全球变暖影响的270倍的气体。农田的管理会影响一氧化二氮的排放量。我们必须了解和测量这些过程，但受管理的土壤和植被是极不稳定的。它们在精细的空间尺度上(例如，在明显均匀的土壤的相邻土块之间)和在较粗的尺度上(例如，在坡底和山顶之间)变化。但我们必须从相对较少的观测中对这个复杂系统中的变量进行预测。科学家们用一种叫做地统计学的方法来做到这一点。我们假设一个区域上的一个变量是从一个随机过程中产生的。当然，我们知道这不是真的，但我们经常做出这样的假设：掷骰子上的数字取决于牛顿运动定律，但我们可以将其视为一个随机变量，在这里是一个从1到6的数字，无法预先预测，并且都以相同的概率发生。如果一个骰子不是一个完美的、统一的立方体，它就不会表现得这么好。如果我们多次掷骰子并记录出现的数字，我们可以估计出一个更好的真实骰子模型。在变异的地统计学图景中，我们认为我们在一组地点获得的数据是通过掷骰子获得的数字。该模型的唯一复杂之处在于，在距离较近的地点掷出的两个骰子上的数字比在相距较远的地点掷出的两个骰子上的数字更有可能相似。这就是所谓的空间相关性，地统计学要求我们能够描述它。这并不简单；对于任何两个地点a和b，我们在每个地点都只有一个观测，因此我们不能就它们的联合变化发表任何声明。解决办法是假设一个变量在地貌的一个部分相隔一定距离(例如10米)的两个观测之间的变化，以及相隔10米的另外两个观测之间的变化，可以说是关于空间变化的重复信息。通过这种方式，我们建立了一个空间相关性模型，称为变异函数，但它依赖于对潜在过程的假设，对于这个假设，两个地点之间的差异只取决于它们之间的距离，而不是它们所在的位置。这称为方差的平稳性。这种假设在现实中往往是不现实的。像反硝化这样的过程，在土壤中有泥炭斑块的潮湿低洼地区，比在排水良好的耕地中的可变性要大得多。在斜坡底部的混合沉积物中，土壤pH值在短距离内的变异性可能比在古老的侵蚀地表上的土壤pH的变异性大。这个项目的目的是开发地统计学方法来处理这种变化。我们的想法是将空间依赖视为空间位置的数学函数，在空间依赖模型中增加一些额外的参数。我们认为，虽然这不会从我们的分析中删除假设，但这些假设将比平稳性更可信。除了开发新的空间依赖模型外，我们还将开发探索性方法，以决定何时需要更复杂的空间模型，并帮助为特定问题设计它们。我们还将研究我们对过程的科学知识，通过数学描述，如何在变化复杂时帮助预测它们。我们将使用新的数据来测试和演示这些方法，这些数据是关于具有许多非常不同的土地用途的复杂农业景观中土壤的一氧化二氮排放量。这一重要变量将以一种复杂的方式变化，因此不能自信地假设平稳性。如果成功，该项目将提供工具来研究和预测许多不同的复杂变量，如土壤生物多样性和污染。

Many important processes happen in the landscape. For example, certain soil bacteria, denitrifiers, use nitrate in their respiration. One product of this is nitrous oxide, a gas with 270 times the impact of carbon dioxide on global warming. Management of farmland affects how much nitrous oxide is released. We must understand and measure these processes, but managed soils and vegetation are extremely variable. They vary at fine spatial scales (e.g. between adjacent clods of an apparently uniform soil) and at coarser scales (e.g. between the bottom of a slope and the top of a hill). But we must make predictions of variables in this complex system from relatively few observations. Scientists do this with methods called geostatistics. We assume that a variable over a region has arisen from a random process. We know that this is not true, of course, but we often make such assumptions: the number on a thrown die depends on Newton's laws of motion, but we can treat it as a random variable, here a number from 1 to 6 that cannot be predicted in advance, and all occur with equal probability. A die will not behave this well if it is not a perfect, uniform cube. We can estimate a better model for a real die if we throw it many times and record the numbers that turn up. In the geostatistical picture of variation, we think of our data, obtained at a set of sites, as numbers obtained by throwing a die. The only complication of the model is that the numbers on two dice thrown at sites near to each other are more likely to be similar than on two dice at sites that are further apart. This is called spatial dependence and geostatistics requires that we can describe it. This is not simple; for any two sites a and b we only have one observation at each, and from this we can make no statements about their joint variation. The solution is to assume that the variation between two observations of a variable separated by some distance (e.g. 10m) in one part of the landscape, and the variation between another two observations 10m apart are, as it were, duplicate information about the variability in space. In this way we build up a model of the spatial dependence, called the variogram, but it depends on this assumption of an underlying process for which the variation between two sites depends only on how far apart they are, not on where they are. This is called stationarity of the variance. This assumption is often unrealistic in the landscape. The variability of a process like denitrification in a wet low-lying area with peaty patches in the soil will be much greater than than in a well-drained, cultivated arable field. The variability of soil pH over short distances may be larger in mixed sediments at the bottom of a slope than on an old eroded land surface. The aim of this project is to develop geostatistical methods to deal with such variation. Our idea is to treat the spatial dependence as a mathematical function of spatial position, adding some extra parameters to the model of spatial dependence. We believe that, while this won't remove assumptions from our analysis, the assumptions will be more plausible than stationarity. As well as developing new models of spatial dependence we shall develop exploratory methods to decide when more complex spatial models are needed, and to help design them for particular problems. We shall also investigate how our scientific knowledge of processes, described mathematically, can be used to help predict them when the variation is complex. We shall test and demonstrate these methods using new data on the rates of nitrous oxide emissions from soils in complex farmed landscapes with many very different land uses. This important variable will vary in a complex way, so that stationarity cannot be assumed with confidence. If successful this project will provide tools to study and predict many different complex variables such as soil biodiversity and pollution.