Comparative assessment of linear and bilinear prism-based strategies for terrain correction computations

Abstract

We address the evaluation of the potential and of the gravitational attraction of mass distributions, assigned by means of a Digital Terrain Model (DTM), for terrain correction computations. In particular, we improve a recent analytical formulation based on the approximation of topographic masses by vertical prisms with linear top surfaces and compare the computational features of the enhanced formulation with alternative ones based on polyhedral modeling on bilinear approximation of the surface relief or on numerical approaches. The numerical tests carried out on the DTM of an area near Cassino (Italy) proved that the proposed enhanced analytical formulation, besides providing exact values of the potential and of the gravitational attraction, turns out to be faster than alternative approaches.

Appendices

Appendix 1

Aim of this appendix is to analytically compute the integral

$$\begin{aligned} I_{ln}= \int \limits _{x_1}^{x_2}\ln [wx+e+\sqrt{ax^2+bx+c}]dx \quad a>0 \end{aligned}$$

(81)

The idea of computing $I_{ln}$ using a software of symbolic mathematics cannot be pursued since the resulting expression, as already shown by Smith (2000), is rather cumbersome. For this reason we set

$$\begin{aligned} \sqrt{ax^2+bx+c}=\sqrt{a}(t-x) \end{aligned}$$

(82)

what yields

$$\begin{aligned} x=\frac{at^2-c}{b+2at} \end{aligned}$$

(83)

and

$$\begin{aligned} \sqrt{ax^2+bx+c}=\sqrt{a}\frac{at^2+bt+c}{b+2at} \end{aligned}$$

(84)

Thus, from (83), the differentials of $x$ and $t$ are related by

$$\begin{aligned} dx=\frac{2a(at^2+bt+c)}{(b+2at)^2}dt \end{aligned}$$

(85)

Setting

$$\begin{aligned}&\hat{f} = a (w+\sqrt{a}) \nonumber \\&\hat{g} = 2ae +\sqrt{a} b \\&\hat{h} = (\sqrt{a}-w)c + be\nonumber \end{aligned}$$

(86)

we have

$$\begin{aligned} I_{ln}= 2a \int \limits _{t_1}^{t_2}\ln \left[ \frac{\hat{f} t^2 + \hat{g} t + \hat{h}}{b+2at}\right] \frac{at^2+bt+c}{(b+2at)^2} dt \end{aligned}$$

(87)

where

$$\begin{aligned} t_i= \sqrt{ \frac{ax_i^2+bx_i+c}{a} } +x_i \qquad (i=1,2) \end{aligned}$$

(88)

on account of (82).

The expression (87) can be further simplified by setting

$$\begin{aligned} y= b+2at \end{aligned}$$

(89)

what yields, after some manipulation,

$$\begin{aligned} I_{ln}= \frac{1}{4a^2} \int \limits _{y_1}^{y_2}\ln \left[ \frac{\hat{f} y^2 + \hat{p} y + \hat{q}}{4a^2y}\right] \frac{ay^2+\hat{r}}{y^2} dy \end{aligned}$$

(90)

having set $y_i = b+2at_i$ and

$$\begin{aligned}&\hat{p} = 2a \hat{g} - 2b \hat{f} \nonumber \\&\hat{q} = \hat{f} b^2 -2ab \hat{g} + 4 a^2 \hat{h} \\&\hat{r} = 4 a^2 c - a b^2\nonumber \end{aligned}$$

(91)

The integral (90) has been computed using Mathematica 2013. In particular setting

$$\begin{aligned}&\varDelta = \sqrt{4\hat{f} \hat{q} - \hat{p}^2} \qquad \varGamma (y)=\hat{f} y^2 + \hat{p} y + \hat{q} \nonumber \\&LN(y) = \displaystyle { \ln \frac{\hat{f} y^2 + \hat{p} y + \hat{q}}{4a^2 y} } \nonumber \\&ATN(y) = \displaystyle {\arctan \frac{2\hat{f} y + \hat{p}}{\varDelta } } \end{aligned}$$

(92)

one has

$$\begin{aligned} I_{ln}&= \displaystyle {\frac{1}{4a^2} \Bigl \{ \Bigl (\frac{a}{\hat{f}}+\frac{\hat{r}}{\hat{q}}\Bigr )\varDelta [ATN(y_2)-ATN(y_1)] \Bigr . } \nonumber \\&\quad +\,\displaystyle { \hat{r}\frac{\hat{p}}{\hat{q}}\ln \frac{y_2}{y_1} + \Bigl (\frac{a\hat{p}}{2\hat{f}}-\frac{\hat{r}\hat{p}}{2\hat{q}}\Bigr ) \ln \frac{\varGamma (y_2)}{\varGamma (y_1)} } \nonumber \\&\quad +\, \displaystyle { \Bigl (\frac{\hat{r}}{y_2}-ay_2\Bigr ) [1-LN(y_2)] } \nonumber \\&\quad -\, \Bigl . \displaystyle { \Bigl (\frac{\hat{r}}{y_1}-ay_1\Bigr )[1-LN(y_1)] }\Bigr \} \end{aligned}$$

(93)

which has a considerably simpler expression than the one reported in Smith (2000).

Appendix 2

Aim of this appendix is to provide an explicit evaluation of the integral $K_{\varLambda _1 a}$ in (32) and the analogous one for $\varLambda _2$. To fix the ideas we shall make reference to a generic triangle $\varLambda$ by writing

$$\begin{aligned} K_{\varLambda a}=\displaystyle { \sum \limits _{j=1}^{3} ({\varvec{\rho }}_j\cdot {\varvec{\nu }}_j) \int \limits _{{l}_{j}}\ln [\psi ({\varvec{\rho }}(s_j),k) ] ds_j } \end{aligned}$$

(94)

where $\psi ({\varvec{\rho }}(s_j),k)$ denotes the value of the function (12) evaluated at the curvilinear abscissa $s_j$ spanning the $j$-th edge.

Setting $\lambda _j=s_j/l_j$ the previous expression can also be written as

$$\begin{aligned} K_{\varLambda a}=\displaystyle { \sum \limits _{j=1}^{3} ({\varvec{\rho }}_j\cdot {\varvec{\rho }}_{j+1}^\perp ) \int _{0}^{1}\ln [\psi ({\varvec{\rho }}(\lambda _j),k) ] d\lambda _j } \end{aligned}$$

(95)

where ${\varvec{\rho }}_j$ and ${\varvec{\rho }}_{j+1}^\perp$ are defined in (59).

Adopting in the previous integral the parameterization (58) for the $j$-th edge one has

$$\begin{aligned} \chi ({\varvec{\rho }}(\lambda _j),k) = a_j \lambda _j^2 + 2b_j\lambda _j + c_j \end{aligned}$$

(96)

and

$$\begin{aligned} {\mathbf g}\cdot {\varvec{\rho }}(\lambda _j)+k = d_j\lambda _j + e_j \end{aligned}$$

(97)

where it has been set

$$\begin{aligned}&a_j = ({\varvec{\rho }}_{j+1}-{\varvec{\rho }}_j)\cdot ({\varvec{\rho }}_{j+1}-{\varvec{\rho }}_j) + d_j^2 \nonumber \\&b_j = {\varvec{\rho }}_j\cdot ({\varvec{\rho }}_{j+1}-{\varvec{\rho }}_j) + d_je_j \nonumber \\&c_j = {\varvec{\rho }}_j\cdot {\varvec{\rho }}_j + e_j^2 \\&d_j = {\mathbf g}\cdot ({\varvec{\rho }}_{j+1}-{\varvec{\rho }}_j) \nonumber \\&e_j = {\mathbf g}\cdot {\varvec{\rho }}_j + k\nonumber \end{aligned}$$

(98)

Thus, the integral (95) becomes

$$\begin{aligned} K_{\varLambda a}=\displaystyle { \sum \limits _{j=1}^{3} ({\varvec{\rho }}_j\cdot {\varvec{\rho }}_{j+1}^\perp ) LN_{ja} } \end{aligned}$$

(99)

where it has been set

$$\begin{aligned} LN_{ja} =\displaystyle { \int _0^1 \ln [d_j\lambda _j+e_j + \sqrt{a_j\lambda _j^2 + 2b_j \lambda _j+c_j} ] d\lambda _j } \end{aligned}$$

(100)

In this way, we are finally led to compute an integral of the kind addressed in the Appendix 1.

To compute $K_{\varLambda b}$ in (33) and correctly take into account the singularities which can affect its evaluation we observe that, similarly to (99), we can write for a generic triangle $\varLambda$

$$\begin{aligned} K_{\varLambda b}=\displaystyle { \sum \limits _{j=1}^{3} ({\varvec{\rho }}_j\cdot {\varvec{\rho }}_{j+1}^\perp ) LN_{jb} } \end{aligned}$$

(101)

where

$$\begin{aligned} LN_{jb} =\displaystyle { \int _0^1 \ln [\sqrt{\bar{a}_j\lambda _j^2 + 2\bar{b}_j\lambda _j+\bar{c}_j} ] d\lambda _j } \end{aligned}$$

(102)

and $\bar{a}_j$, $\bar{b}_j$ and $\bar{c}_j$ are defined in (60). Actually $d_j=e_j=0$ in (98) due to the fact that now ${\mathbf g}={\mathbf o}$ and $k=0$.

It is simpler and computationally more effective to directly evaluate (102) rather than applying formula (81). To this end, setting

$$\begin{aligned} LN1_{j}&= \ln \,(\bar{a}_j+2\bar{b}_j+\bar{c}_j) \nonumber \\ LN2_{j}&= \ln \,\bar{c}_j\end{aligned}$$

(103)

and

$$\begin{aligned} ATN1_{j}&= \displaystyle { \arctan \frac{\bar{a}_j+\bar{b}_j}{\sqrt{\bar{a}_j\bar{c}_j-\bar{b}_j^2}} } \nonumber \\ ATN2_{j}&= \displaystyle { \arctan \frac{\bar{b}_j}{\sqrt{\bar{a}_j\bar{c}_j-\bar{b}_j^2}} } \end{aligned}$$

(104)

we finally have

$$\begin{aligned} LN_{jb}&= \displaystyle { -1 + \sqrt{\bar{a}_j\bar{c}_j-\bar{b}_j^2} (ATN1_{j}-ATN2_{j}) } \nonumber \\&\quad +\, \displaystyle { LN1_{j}+ \frac{\bar{b}_j}{\bar{a}_j} (LN1_{j}-LN2_{j}) } \end{aligned}$$

(105)

The previous expressions are certainly well defined on account of (64) and of the further property

$$\begin{aligned} \sqrt{\bar{a}_j\bar{c}_j-\bar{b}_j^2}&= ({\varvec{\rho }}_{j+1}\cdot {\varvec{\rho }}_{j+1})({\varvec{\rho }}_j\cdot {\varvec{\rho }}_j)-({\varvec{\rho }}_j\cdot {\varvec{\rho }}_{j+1})^2 \nonumber \\&= ({\varvec{\rho }}_j\cdot {\varvec{\rho }}_{j+1}^\perp )^2 \end{aligned}$$

(106)

Hence $LN1_{j}$ ($LN2_{j}$) in (103) becomes infinite if ${\varvec{\rho }}_{j+1}={\mathbf o}$ (${\varvec{\rho }}_j={\mathbf o}$) since $\bar{a}_j+2\bar{b}_j+\bar{c}_j=0$ ($\bar{c}_j=0$) in this case.

However, this produces no consequences from the practical point of view since $LN_{jb}$ in (105) is scaled by ${\varvec{\rho }}_j\cdot {\varvec{\rho }}_{j+1}^\perp$ in the expression (101) of $K_{\varLambda b}$. Actually, recalling (65), the computation of $LN1_i$ ($LN2_i$) can be skipped when ${\varvec{\rho }}_{i+1}={\mathbf o}$ (${\varvec{\rho }}_{i}={\mathbf o}$).

Analogously, should $\bar{a}_j\bar{c}_j-\bar{b}_j^2=0$, the quantities $ATN1_j$ and $ATN2_j$ would become numerically undefined but, in fact, their evaluation can be skipped since both of them are factored by the null quantity ${\varvec{\rho }}_j\cdot {\varvec{\rho }}_{j+1}^\perp$.