From Gauging Accuracy of Quantity Estimates to Gauging Accuracy and Resolution of Measuring Physical Fields

For a numerical physical quantity v, because of the measurement imprecision, the measurement result ṽ is, in general, different from the actual value v of this quantity. Depending on what we know about the measurement uncertainty ∆v def = ṽ − v, we can use different techniques for dealing with this imprecision: probabilistic, interval, etc. When we measure the values v(x) of physical fields at different locations x (and/or different moments of time), then, in addition to the same measurement uncertainty, we also encounter another type of localization uncertainty: that the measured value may come not only from the desired location x, but also from the nearby locations. In this paper, we discuss how to handle this additional uncertainty. 1 Formulation of the Problem Need for data processing. In many real-life situations, we are interested in the value of a quantity which is difficult (or even impossible) to measure directly. For example, we may be interested in the distance to a star, or in the amount of water in an underground water layer. Since we cannot measure the corresponding quantity y directly, we measure it indirectly. Specifically, – we find easier-to-measure quantities x1, . . . , xn which are related to the desired quantity y by a known dependence y = f(x1, . . . , xn); – we measure the values of the auxiliary quantities x1, . . . , xn; and – we use the results x̃1, . . . , x̃n of measuring the auxiliary quantity to compute the estimate ỹ = f(x̃1, . . . , x̃n) for the desired quantity y. Example. To find the distance y to a star, we can use the following parallax method: – we measure the orientations x1 and x2 to this star at two different seasons, – we measure the the distance x3 between the spatial locations of the corresponding telescopes at these two seasons (i.e., in effect, the diameter of the earth orbit); – then, reasonably simply trigonometric computations enable us to describe the desired distance y as a function of the easier-to-measure quantities x1, x2, and x3. General case. In general, computations related to such indirect measurements form an important particular case of data processing. Need to take uncertainty into account. Measurements are never absolutely accurate. As a result, the measurement results x̃i are, in general, different from the actual (unknown) values xi of the measured quantities: ∆xi def = x̃i − xi 6= 0. Because of this, the result ỹ = f(x̃1, . . . , x̃n) of data processing is, in general, different from the actual (unknown) value y = f(x1, . . . , xn): ∆y def = ỹ − y 6= 0. Thus, in practical applications, we need to take this uncertainty into account. Interval uncertainty. In practice, we often only know the upper bound ∆i on the measurement errors ∆xi def = x̃i − xi: |∆xi| ≤ ∆i. In this case, the only information that we have about the actual values xi is that xi belongs to the interval xi def = [x̃i −∆i, x̃i + ∆i]. Under such interval uncertainty, we need to find the range of possible values of y: y = {f(x1, . . . , xn) : xi ∈ xi}. The problem of computing this range is known as interval computations; see, e.g., [4]. Need to measure physical fields. In practice, the situation is often more complex: the values that we measure can be: – values v(t) of a certain dynamic quantity v at a certain moment of time t – or, more generally, the values v(x, t) of a certain physical field v at a certain location x and at a certain moment of time t. For example, in geophysics, we are interested in the values of the density at different locations and at different depth. Need to take uncertainty into account when measuring physical fields. When we measure physical fields, not only we get the measured value ṽ ≈ v with some inaccuracy, but also the location x is not exactly known. Moreover, the sensor picks up the “averaged” value of v at locations close to the approximately known location x̃. In other words, in addition to inaccuracy ṽ 6= v, we also have a finite (spatial) resolution x̃ 6= x. Estimating uncertainty related to measuring physical fields: challenging problems. In general, the measured value ṽi differs from the averaged value vi by the measurement imprecision ∆vi = ṽi− vi. In the interval case, we know the upper bound ∆i on this measurement error |∆vi| ≤ ∆i. Thus, the averaged quantity vi can take any value from the interval [vi, vi], where vi def = ṽi−∆i and vi def = ṽi+∆i. Based on these bounds on vi, what can we learn about the original field v(x)? The answer to this questions depends on what we know about the averaging, i.e., on the dependence of vi on v(x). In principle, there are three possible situations: – sometimes, we know exactly how the averaged values vi are related to v(x); – sometimes, we only know the upper bound δ on the location error x̃−x (this is similar to the interval case); – sometimes, we do not even know δ. In the following sections, we describe how to process all these types of uncertainty. 2 Possibility of Linearization Sometimes, the signal v(x) that we are measuring is large, i.e., the values of the signal are much larger than the noise (and the measurement errors in general). In such situations, the measured values well represent the actual signal, and for many applications, the measurement errors can be safely ignored. The need to take into account measurement errors becomes important only when the signal v(x) is relatively weak. In this case, we can expand the dependence of vi on v(x) in Taylor series. To describe this expansion, let us first consider a simplified case in which there are only finitely many spatial points x, . . . , x, so the field v(x) is described by finitely many values v def = v ( x ) , j = 1, . . . , N . In this case, the dependence of the quantity vi on these values v, . . . , v can be expanded into Taylor series